Mathematics of medical imaging with dynamic data

Diss. ETH No. 25178

Mathematics of medical imaging withdynamic data

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH(Dr. sc. ETH Zurich)

presented by

FRANCISCO ROMERO HINRICHSEN

BSc. Universidad de Chile

MSc. App. Math. Universite Paris Sud

born December 1st, 1989

citizen of Chile

accepted on the recommendation of

Prof. Dr. Habib Ammari, ETH Zurich, examiner

Prof. Dr. Josselin Garnier, Ecole Polytechnique, Paris, co-examiner

Prof. Dr. Knut Solna, University of Californa at Irvine, co-examiner

2018

Abstract

We consider three techniques in medical imaging: dynamic full-field optical coherence tomogra-phy, ultrafast ultrasound imaging, and nanoparticle imaging. For each of them, we study specificproblems pertaining modelization, signal processing and inverse problems. All studied problemsare supported by adequate numerical simulations.

For dynamic full-field optical coherence tomography, we model the forward operator underrandom medium assumptions for the particular case of cell imaging. We consider the signalas the contribution of two sources and study the resulting singular value decomposition obtainedwhen building a matrix composed of spatial and temporal values in each coordinate. We filterout the dominant component of the signal by removing the first singular values.

For ultrafast ultrasound imaging, we model the forward operator and obtain an explicit formulafor its point spread function. Furthermore, we study simplifications to such function and employthem to investigate the Doppler imaging problem. Similarly, as in optical coherence tomography,we use a filtering technique via a singular value decomposition.

Pertaining microbubble tracking with ultrafast ultrasound imaging, we propose a techniquebased on total variation minimization on the space of Radon measures. We prove existenceand unicity of solution of such a problem as well as numerical stability. Some examples areprovided pertaining the limits of this approach.

In the context of nanoparticle imaging, we use asymptotic analysis techniques to analyze the heatgeneration due to nanoparticles when illuminated at their plasmonic resonance. We considertwo-dimensional arbitrarily-shaped particles and we study the close-to-touching case.

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Resume

Dance cette these, nous considerons trois techniques d’imagerie medicale : la tomographie dy-namique par coherence optique plein champ, l’imagerie ultrasonore ultrarapide et l’imagerie denanoparticules. Pour chacune d’elles, nous etablissons une modelisation mathematique rigoureuseet nous introduisons des nouvelles techniques venues du traitement du signal pour la resolutiondu probleme d’imagerie correspondant.

Pour la tomographie par coherence optique plein champ, dans le cadre de l’imagerie sub-cellulaire,nous modelisons l’operateur direct en milieux aleatoires. Nous considerons le signal optiquecomme la somme des contributions de deux sources, incoherentes temporellement. Ensuite,nous etudions la decomposition par valeurs singulieres de la matrice de Cosarati dont les entreessont les mesures du champ optique a une position et a un temps donnes.

Pour l’imagerie ultrasonore rapide, nous modelisons l’operateur direct afin d’obtenir une for-mule explicite pour sa fonction d’etalement ponctuel. De plus, nous utilisons des versions sim-plifiees d’une telle fonction pour l’etude du probleme de l’imagerie Doppler. De meme, commedans le cas de la tomographie par coherence optique, nous appliquons aux donnees une techniquede filtrage par decomposition en valeurs singulieres.

Quant au suivi de microbulles avec l’imagerie ultrasonore ultrarapide, nous proposons une tech-nique basee sur la minimisation de la variation totale sur l’espace des mesures de Radon. Nousprouvons l’existence et l’unicite d’une solution au probleme de minimisation ainsi que sa sta-bilite numerique. Nous illustrons numeriquement les performances et les limites de l’approcheproposee.

Dans le cadre de l’imagerie des nanoparticules, nous utilisons des techniques d’analyse asympto-tique pour quantifier la generation de chaleur due aux nanoparticules plasmoniques lorsqu’ellessont eclairees a leurs frequences de resonances. Nous considerons des particules bidimension-nelles, de formes arbitraires et nous etudions en particulier le cas ou les particules sont prochesles unes des autres.

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To my parents, Leopoldo and Monica, and to you, distinguished reader.

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Contents

1 A signal separation technique for sub-cellular imaging using dynamic opticalcoherence tomography 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The dynamic forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Single particle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Multi-particle dynamical model . . . . . . . . . . . . . . . . . . . . . 4

1.3 Property analysis of the forward problem . . . . . . . . . . . . . . . . . . . . 61.3.1 Direct operator representation . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Eigenvalue analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 The inverse problem: Signal separation . . . . . . . . . . . . . . . . . . . . . 111.4.1 Analysis of the SVD algorithm . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Analysis of obtaining the intensity of metabolic activity . . . . . . . . 14

1.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.1 Forward problem measurements . . . . . . . . . . . . . . . . . . . . . 161.5.2 SVD of the measurements . . . . . . . . . . . . . . . . . . . . . . . . 171.5.3 Selection of cut-off singular value . . . . . . . . . . . . . . . . . . . . 181.5.4 Signal reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.5 Discussion and observations . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Appendix: Non-orthogonality of the eigenvectors of Fcc, Fcm, and Fmc . . . . . 23

2 Mathematical Analysis of Ultrafast Ultrasound Imaging 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The Static Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 The Static Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.1 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 The PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.3 Angle compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 The Dynamic Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.1 The quasi-static approximation and the construction of the data . . . . 362.4.2 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4.3 Multiple scatterer random model . . . . . . . . . . . . . . . . . . . . . 38

2.5 The Dynamic Inverse Problem: Source Separation . . . . . . . . . . . . . . . . 402.5.1 Formulation of the dynamic inverse problem . . . . . . . . . . . . . . 402.5.2 The SVD algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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2.5.3 Justification of the SVD in one-dimension . . . . . . . . . . . . . . . . 412.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.8 Appendix: The Justification of the Approximation of the PSF . . . . . . . . . . 50

3 Dynamic spike super-resolution and applications to Ultrafast ultrasound imag-ing 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 The space-velocity model . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2 The perfect low-pass case . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Exact recovery in absence of noise . . . . . . . . . . . . . . . . . . . . . . . . 583.3.1 Dual certificates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.3 Comments on the hypotheses of Theorem 3.3.2 . . . . . . . . . . . . . 62

3.4 Other constructions of dynamical dual certificates . . . . . . . . . . . . . . . . 663.4.1 Dual certificates with no static separation condition . . . . . . . . . . . 663.4.2 Dual certificates in presence of ghost particles . . . . . . . . . . . . . . 67

3.5 Stable reconstruction with noise . . . . . . . . . . . . . . . . . . . . . . . . . 693.6 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6.2 The measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7 Applications to ultrafast ultrasonography . . . . . . . . . . . . . . . . . . . . . 763.7.1 Fully automated imaging protocol . . . . . . . . . . . . . . . . . . . . 773.7.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Heat generation with plasmonic nanoparticles 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Setting of the problem and the main results . . . . . . . . . . . . . . . . . . . 824.3 Layer potentials for the Helmholtz equation in two dimensions . . . . . . . . . 854.4 Heat generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4.1 Small volume expansion of the inner field . . . . . . . . . . . . . . . . 874.4.2 Small volume expansion of the temperature . . . . . . . . . . . . . . . 914.4.3 Temperature elevation at the plasmonic resonance . . . . . . . . . . . . 954.4.4 Temperature elevation for two close-to-touching particles . . . . . . . . 97

4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.5.1 Single-particle simulation . . . . . . . . . . . . . . . . . . . . . . . . 984.5.2 Two particles simulation . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.7 Appendix: Asymptotic analysis of the single-layer potential in two dimensions 105

4.7.1 Layer potentials for the Laplacian in two dimensions . . . . . . . . . . 1064.7.2 Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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References 115

Curriculum Vitae 123

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Contents

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Introduction

Medical imaging is a rather old branch of science. We can pinpoint its birth at around 1895when the German physicist Wilhelm Rontgen, who studied what he called “x-rays”, publisheda famous picture of his wife’s hand through these rays (see Figure 0.1a); this is regarded as thefirst medical x-ray. From then, the pursuit of obtaining information from within a human bodyhas been of common interest among physicists, engineers, biologists, pharmacists, chemists,and most importantly for us, mathematicians; this leads us to study the mathematics involved inbiomedical imaging.

(a) First medical X-ray by Wilhelm Rontgen ofhis wife Anna Bertha, 1895.

(b) Image of rat brain blood vessels by Errico etal, 2015.

Figure 0.1: Contrast of different imaging techniques in history.

We could say that this area was first developed unknowingly by Johann Radon in 1917 [84], bystudying how to recover an unknown function with exclusively the information of its line inte-grals; such information is now denominated as the Radon Transform, and the inversion of suchoperator is the basis of what in 1967 would be the first commercial computerized tomographyscanner (CT-scan). Fast forward and nowadays there are diverse tomographic techniques withdifferent physical principles that are of interest in medical imaging. The mathematical study ofsuch methods allows us to grasp a better understanding of the underlying phenomena, and withsome skills and luck, it aids to enhance the techniques’ capabilities. Enhancement in biomedicalimaging can range from better image resolution, contrast, imaging speed, or simply computerfeasibility by reducing the computational cost of processing the data.

As the title of this dissertation suggests, we are motivated by the mathematical study of somemedical imaging techniques where the time-dependent information is relevant. We are interestedin exploiting this time dependency to both, enhance the methods themselves, or in uncovering

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Introduction

hidden features of interest. In particular, the studied imaging techniques in this work are dy-namic full-field optical coherence tomography, ultrafast ultrasound imaging, and nanoparticleimaging:

• Optical coherence tomography (OCT) [60] is an interferometry-based imaging technique;it consists in shining light on a target and afterwards measuring the backscattered lightwith the purpose of getting detailed low-depth information of a target tissue. Currently, itis widely used in clinical settings, where its resolution is still being enhanced [75]. Initiallyin OCT, to obtain a cross-sectional image of a plane parallel to the surface it was requiredto transversely scan, lately, there have been developed OCT alternatives, like dynamicfull-field OCT, that significantly reduce the sampling speed by avoiding the scanning step[23].

• Ultrasound imaging is a technique that relies on emitting acoustic pulses into a targetmedium and afterward measuring the backscattered waves to reconstruct an image [89].Conventionally, it relies on focusing ultrasonic waves along lines and reconstructing themsequentially; by composing this process along many lines, it reconstructs a plane image.Ultrafast ultrasound imaging relies in emitting a single plane acoustic wave, and with it,reconstructs the whole planar zone of interest [90]; it improves significantly the samplingrate, as its ultrafast name suggests.

• Nanoparticle imaging is a technique relying on resonant nanoparticles. These particlesare typically made of gold or silver, and, given their small shape, they behave as meta-materials; they exhibit what is called a ”plasmonic resonance“ when exposed to certainelectromagnetic frequencies. While undergoing resonance they increment their scatteringcross section [62], acts as acoustic sources [93], and produce localized heat [54]. Further-more, they can be engineered to attach to specific tissues and from outside they can beexited employing light that penetrates the tissue [94]. These particles promise uses in thelocalization of a target tissue or to thermally ablate it.

Contributions

As a basis for all the mentioned imaging techniques, mathematical modeling is a must for furtheranalysis. Especially relevant for the case of UUI where, as an emergent technique, there was noexisting modeling that properly described the associated point spread function. From anotherperspective, modeling of measured signals is also a requirement to study the respective inverseproblems, especially relevant when we are considering random fluctuations on it because ofmicroscopic effects.

One classical interest in signal processing is to filter out undesirable signals. In the case oftime-dependent images, one proposed way to separate them is by means of the singular valuedecomposition (SVD), also called spatiotemporal filtering [42]. Our contribution was to under-stand why such a technique would work, in terms of spatial and temporal correlations of thesignals, and study it numerically in a setting where we know the ground truth. Furthermore, weextended the technique for OCT for the particular case of metabolic imaging of cells; the interest

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Introduction

in such information relates to [23], where it is expected that it will be possible to classify the ma-lignancy related to tumor studies. The signal separation technique is treated in both Chapters 1and 2, in the case of OCT and UUI, respectively.

Thanks to the high sample rate of UUI, a new super-resolution technique was achieved in [49](see Figure 0.1b); it consists in tracking individual micro-bubbles in the bloodstream with thepurpose of super localizing small vessels deep in tissue. The proposed technique suffers fromtwo main drawbacks: there is a plethora of discarded data and the employed tracking methodsare computationally expensive. Our contribution, presented in Chapter 3, is a method to ame-liorate the before mentioned drawbacks that consist in posing the imaging technique as an L1minimization problem; it follows a similar approach to the one given at [36] but extended to adynamical setting. We studied the emergent problem in a rigorous way and, show the superiorityof our approach, via numerical simulations.

As a last contribution presented in Chapter 4, we are motivated by the study of nanoparticleimaging in a modeling way: we want to quantify the generated heat with respect to its shape,illuminating frequency and intensity. Such results are mainly relevant for thermal ablation oftumors, although we are also motivated by employing the heat information for the localizationof target tissues [78]. In relation to the dynamic imaging motivation of the work for the case ofnanoparticles, instead of a relevant time information, we have a relevant frequency informationthat can be helpful to help localize and tune medical processes. The results obtained in this thesisare published in [4, 5, 16, 17]

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Introduction

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1 A signal separation technique forsub-cellular imaging using dynamicoptical coherence tomography

1.1 Introduction

Since dynamic properties are essential for a disease prognosis and a selection of treatment op-tions, a number of methods to explore these dynamics has been developed. When optical imag-ing methods are used to observe cell-scale details of a tissue, the highly-scattering collagenusually dominates the signal, obscuring the intra-cellular details. A challenging problem is toremove the influence of the collagen in order to have a better imaging inside the cells.

There have been many studies on optical imaging to extract useful information. In [68] the au-thors use stochastic method, which follows from a probabilistic model for particle movements,and then they express the autocorrelation function of the signal in terms of some parametersincluding different components of the velocity and the fraction of moving particles. Those pa-rameters are then estimated using a fitting algorithm. In [63, 71], the autocorrelation function ofthe signal can be written as a complex-valued exponential function of the particle displacements.Through the relation between the real and imaginary parts of this autocorrelation function, theauthors analyze the temporal autocorrelation on the complex-valued signals to obtain the mean-squared displacement (MSD) and also time-averaged displacement (TAD) (which is the veloc-ity) of scattering structures. Very recently, Apelian et. al. [23] use difference imaging method,which consists in directly removing the stationary parts from the images by taking differencesor standard deviations. The motivation of this chapter comes from [23].

Some researchers use Doppler optical coherence tomography to obtain high resolution tomo-graphic images of static and moving constituents simultaneously in highly scattering biologicaltissues, for example, [38] and in [47, Chapter 21].

In this chapter, using dynamic optical coherence tomography we introduce a signal separationtechnique for sub-cellular imaging and give a detailed mathematical analysis of extracting use-ful information. This includes giving a new multi-particle dynamical model to simulate themovement of the collagen and metabolic activity, and also providing some results relating theeigenvalues and the feasibility of using singular value decomposition (SVD) in optical imaging,which as far as we know is original.

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1 A signal separation technique for sub-cellular imaging using OCT

There are three main contributions. First, we give a new model as an extension of the singleparticle optical Doppler tomography, which allows us to justify the SVD approach for the sep-aration between the collagen signal and metabolic activity signal. Then we perform eigenvalueanalysis for the operator with the intensity as an integral kernel, and prove that the largest eigen-value corresponds to the collagen. This means that using a SVD of the images and removingthe part corresponding to the largest eigenvalue is a viable method for removing the influence ofcollagen signals. Finally, based on SVD, we give a new method for isolating the intensity of themetabolic activity.

This chapter is structured as follows. In Section 2 we introduce our multi-particle dynamicalmodel based on a classical model in [47]. In Section 3, we discuss the forward operator with totalsignal as its integral kernel, and give its eigenvalue analysis, showing that the part correspondingto the collagen signal have rank one, which provides the theoretical foundation for using SVD.In Section 4, we discuss the mathematical rationality for using a SVD method and the methodof isolating the metabolic signal. In Section 5 we give some numerical experiments to validateour approach. Some concluding remarks are presented in the final section.

1.2 The dynamic forward problem

Optical Coherence Tomography (OCT) is a medical imaging technique that uses light to capturehigh resolution images of biological tissues by measuring the time delay and the intensity ofbackscattered or back-reflected light coming from the sample. The research on OCT has beengrowing very fast for the last two decades. We refer the reader, for instance, to [48, 50, 51, 60,80, 87, 92]. This imaging method has been continuously improved in terms of speed, resolutionand sensitivity. It has also seen a variety of extensions aiming to assess functional aspects of thetissue in addition to morphology. One of these approaches is Doppler OCT (called ODT), whichaims at visualizing movements in the tissues (for example, blood flows). ODT is based on theidentical optical design as OCT, but additional signal processing is used to extract informationencoded in the carrier frequency of the interferogram.

The purpose of this chapter is to analyze the mathematics of ODT in the context of its applicationfor imaging sub-cellular dynamics. We prove that a signal separation technique performs welland allows imaging of sub-cellular dynamics. We refer the reader to [2, 3, 4] for recently devel-oped signal separation approaches in different biomedical imaging frameworks. These includeultrasound imaging, photoacoustic imaging, and electrical impedance tomography.

1.2.1 Single particle model

We first consider a single moving particle. In [47, Chapter 21], the optical Doppler tomographyis modeled as follows. Assume that there is one moving particle at a point x in the sample Ω and

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denote by ν the z-component of its velocity. Then the ODT signal generated by this particle isgiven by

ΓODT (x, t) = 2∫ ∞

0S 0(ω)K(x, ω)KR(x, ω) cos

(2πω

(τ +

∆

c

)+ 2πω

2nvtc

)dω, (1.1)

where ω is the frequency, S 0(ω) is the spectral density of the light source, K(x, ω) and KR(x, ω)are the reflectivities of the sample and the reference mirror respectively, n is the index of re-fraction, c is the speed of the light, τ is the time delay on the reference arm, and ∆ is the pathdifference between the reference arm and sample arm.

Since cos is an even function, the above integral can be rewritten as

ΓODT (x, t) =

∫ ∞

−∞

S 0(ω)K(x, ω)KR(x, ω)e2πωi(τ+ ∆c )+2πωi 2nvt

c dω. (1.2)

a b

Figure 1.1: a) Illustration of the imaging setup. b) A particle moves from A to B covering a distance of vt.When the particle is at B, the light travels an additional distance of 2vt inside a medium withrefrative index n, so the effective path-length of the sample arm increases by 2nvt.

To give an explanation for the exponential term of the above formula, we choose a suitablecoordinate system such that the beam propagates along the z-direction, and suppose that theparticle moves in this direction from point A to point B with velocity v, which also meanscovering a distance of vt (see Figure 1.1). Physically, the received signal ΓODT is determined bythe effective path-length difference between the sample and reference arms. In addition, for thismoving particle the effective path-length difference is represented by the quantity cτ+ ∆ + 2nvt,which could also be seen as the z-coordinate of the particle (see Figure 1.1).

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Note that (1.2) is only applicable to a single particle at x moving with a constant velocity v. For aparticle with a more general movement, the path-length difference is no longer a linear functionwith respect to t. Nevertheless, we define ϕ(t) as the z-coordinate of the particle at time t, whichis a generalization of cτ + ∆ + 2nvt. Also, in our case the reference arm is a mirror, so withoutloss of generality, we make the assumption that KR(x, ω) = 1. Then the following expression forsignal ΓODT (x, t) holds

ΓODT (x, t) =

∫ ∞

−∞

S 0(ω)K(x, ω)e2πωi( 2nc ϕ(t))dω.

This is not just a simplification of the model (1.1), but also a small modification, since theparticles with regular and random movements produce difference signals. Here we look intomore details of particle movements. For the sake of simplicity, we assume that the collagenparticles move with a constant speed v, so ϕ(t) = ϕ(0) + vt. On the other hand, for the particlesbelonging to the metabolic activity part, ϕ(t) behaves as a random function, since we do not havemuch information with regard to them.

Remark. Formula (1.2.1) is derived in [47] by considering what is essentially our ϕ(t) (writtenas ∆d there, see formula (21.11) and (21.15) of [47].) This justifies our treatment for generalparticles above. We emphasize that we generalized the model in [47] to accommodate particleswith variable velocities.

1.2.2 Multi-particle dynamical model

We have seen the effect of the image ΓODT (x, t) for one moving particle. We now consider themore realistic case of a medium (could be cell or tissue) with a large number of particles inmotion. In actual imaging, for each pixel which we denote also by x, there would be manyparticles, all with different movement patterns.

We choose an appropriate coordinate system, such that for any particle on the plane z = 0, itseffective path-length difference is zero. Let L be the coherence length. Physically, only theparticles with path-length difference smaller than L, or equivalently z ∈ [−L, L], will be presentin the image. In fact, if the differences between the two arms are larger than the coherencelength, then the lights from two arms do not interfere anymore, and thus do not contribute tothe received signal. This means that the imaging region is a ”thin slice” within the sample withthickness 2L (see Figure 1.2). Then we divide the slice into small regions, such that each regioncorresponds to a pixel of the final image. See Figure 1.2 for the imaged small region, which isgiven by x × [−L, L], and for the correspondence between them and pixels of the final image.

Since there are many particles in this region, we describe their distribution using a density func-tion p. Moreover, for any function f (z), we have that the integral

∫ z2

z1f (z)p(x, z, t)dz is equal

to the sum of f (z) over all particles in x × [z1, z2]. We know that the received light intensity inthe small region x × [−L, L] could be seen as the sum of light intensity over all particles in this

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Figure 1.2: One ”slice” in the sample, and its division into small regions corresponding to the pixels ofthe image.

region. Therefore for uniform medium, we can write it as an integral in terms of the densityfunction p(x, z, t),

ΓODT (x, t) =

∫ ∞

−∞

∫ L

−LS 0(ω)K(x, ω)e2πωi( 2n

c z)p(x, z, t)dω dz,

noting that the reflectivity coefficient K must be the same for all involved particles. According tothe definition of p(x, z, t), we consider it as the sum of the density function of collagen particlesand the density function of metabolic activity particles, namely,

p(x, z, t) = pc(x, z, t) + pm(x, z, t). (1.3)

Consequently, their respective reflectivities will be denoted Kc and Km, giving us the ODT mea-surements formula

ΓODT (x, t) = ΓcODT (x, t) + Γm

ODT (x, t), (1.4)

where ΓcODT (x, t) corresponds to the collagen signal and Γm

ODT (x, t) corresponds to the metabolicactivity signal, with formulas

ΓjODT (x, t) =

∫ ∞

−∞

∫ L

−LS 0(ω)K j(x, ω)p j(x, z, t)e2πωi( 2n

c z)dω dz, for j ∈ c,m . (1.5)

Physically, since the collagen moves as a whole, we could assume that the collagen particlesmove with one uniform (and very small) velocity v0, which means any such particles will be atposition z + v0t at time t. Let qc(x, z) denote the density function of all the collagen particles

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inside area x with initial vertical position z. Then we have

pc(x, z + v0t, t) = qc(x, z). (1.6)

Furthermore, from this expression we could see when t = 0, qc(x, z) = pc(x, z, 0).

In the case of metabolic activity we do not assume any condition on the density function pm(x, v, z),because there is no physical law of motions for us to use. In the numerical experiments, becauseof the large number of particles, a random medium generator is used to simulate the particledistribution while keeping the computational cost low.

Since x is a small area inside the sample, when we choose x, it could include both collagenparticles and metabolic activity particles. The aim is to separate the two classes of particles. Inpractice, the contributions of collagen particles to the intensity is much larger than the contri-butions of the metabolic activity. This allows us to understand that the reflectivity of collagenparticles Kc is much larger (realistic quantities are about 102 to 104 times) than the reflectivityof metabolic activity particles Km, and

|ΓcODT (x, t)| |Γm

ODT (x, t)|. (1.7)

In this section, we have given a multi-particle dynamical model, to separate the collagen signaland the metabolic activity signal. The next step is to analyze the properties of this model.

1.3 Property analysis of the forward problem

1.3.1 Direct operator representation

Based on the multi-particle dynamical model, in order to analyze the properties of collagen andmetabolic activity, we first represent their corresponding operators.

Let S be the integral operator with the kernel ΓODT (x, t), which is a real-valued function given by(1.5). Similarly, Let S c and S m be integral operators with kernels Γc

ODT (x, t) and ΓmODT (x, t), re-

spectively. The collagen signal has high correlation between different points, while the metabolicsignals have relatively lower correlation, so it would be useful to look at the correlation ofthe whole signal. The formula

∫ΓODT (x, t)ΓODT (y, t) dt represents the correlation between two

points x and y, which is exactly the integral kernel of the operator S S ∗, where S ∗ is the adjointoperator of S . We denote the kernel of S S ∗ by

F(x, y) =

∫ T

0ΓODT (x, t)ΓODT (y, t)dt, (1.8)

for some fixed T > 0. Substituting the representation of ΓODT (x, t) in (1.4) into (1.8), we arriveat

F(x, y) = Fcc(x, y) + Fcm(x, y) + Fmc(x, y) + Fmm(x, y),

6


where for j, k ∈ c,m, F jk(x, y) is given by

F jk(x, y) =

∫R2×[−L,L]2×[0,T ]

S 0(ω1)S 0(ω2)K j(x, ω1)Kk(y, ω2)p j(x, z1, t)

× pk(y, z2, t)e4πin

c (ω1z1−ω2z2)dω1dω2dz1dz2dt,(1.9)

with z1, z2 ∈ [−L, L] and ω1, ω2 ∈ R, t ∈ [0,T ]. Likewise, the integral operators with kernelF jk(x, y) are exactly the operators S jS ∗k for j, k ∈ c,m. In the case of the collagen signal, notethat the operator S cS ∗c contains the solely collagen information.

First we consider its kernel Fcc. Applying the uniform movements of collagen particles (1.6)along the z-direction yields

Fcc(x, y) =

∫R2×[−L,L]2×[0,T ]

S 0(ω1)S 0(ω2)Kc(x, ω1)Kc(y, ω2)qc(x, z1 − v0t)

× qc(y, z2 − v0t)e4πin

c (ω1z1−ω2z2)dω1dω2dz1dz2dt.(1.10)

In order to simplify this expression even further, let us introduce a couple of assumptions.

Physically, since the scale of collagen and inter-cellular structures (such as collagen) are muchlarger than the coherence length L, the particle distribution inside a small slice |z| < L should bemore or less uniform. Therefore, it is reasonable to assume that qc(x, z) does not actually dependon z inside such a small slice, namely, qc(x, z) = qc(x).

Furthermore, in practice the tissue being imaged is nearly homogeneous, and therefore the re-flectivity spectrum, (or more intuitively, the ”color” of the tissue) should stay the same every-where. The only difference in reflectivity between two points should be a difference of totalreflectivity (using our ”color” analogy, the two points would look like, e.g. ”different shadesof red”, and not ”red and yellow”). Therefore, for any two pixels x1 and x2, by looking at thereflectivities Kc(x1, ω) and Kc(x2, ω) as functions of frequency ω, they are directly proportional.Thus it is reasonable to assume that Kc(x, ω) could be written in the variable separation formKc1(x)Kc2(ω).

Under these two assumptions, the expression of Fcc(x, y) can be simplified considerably:

Fcc(x, y) =T Kc1(x)Kc1(y)qc(x)qc(y)

×

∫R2×[−L,L]2

S 0(ω1)S 0(ω2)Kc2(ω1)Kc2(ω2)e4πin

c (ω1z1−ω2z2)dω1dω2dz1dz2

=T Kc1(x)Kc1(y)qc(x)qc(y)

×

∫[−L,L]2

F (S 0Kc2)(−

4πnz1

c

)F (S 0Kc2)

(4πnz2

c

)dz1dz2

(1.11)

where the Fourier transform of a function f (ω) is defined as F f (τ) =∫R

f (ω)e−iωτdω.

7


This is the fundamental formula for analyzing collagen signal, since from this formula, we couldsee that Fcc(x, y) is variable separable with respect to x and y. This property gives us a hint tocompute the eigenvalues of the collagen signal.

For the correlation terms Fcm(x, y) and Fmc(x, y), which contains both the collagen and metabolicactivity signals, we use again the uniform movement assumption for pc while keeping themetabolic part pm. Inserting (1.6) into (1.9), we have

Fmc(x, y) =Kc1(y)qc(y)∫

[−L,L]2×[0,T ]F (S 0Kc2)

(4πnz2

c

)× F (S 0Km)

(x,−

4πnz1

c

)pm(x, z1, t)dz1dz2dt,

(1.12)

and

Fcm(x, y) = Kc1(x)qc(x)∫

[−L,L]2×[0,T ]F (S 0Kc2)

(−

4πnz1

c

)× F (S 0Km)

(y,

4πnz2

c

)pm(y, z2, t)dz1dz2dt.

(1.13)

From representations (1.12) and (1.13), we could see that Fmc(x, y) and Fcm(x, y) have alsovariable separated forms with respect to x and y.

In the case of the metabolic activity kernel Fmm(x, y), by keeping the representation pm, it isclear that

Fmm(x, y) =

∫[−L,L]2×[0,T ]

F (S 0Km)(x,−

4πnz1

c

)F (S 0Km)

(y,

4πnz2

c

)× pm(x, z1, t)pm(y, z2, t)dz1dz2dt.

(1.14)

To sum up, the main feature of our multi-particle dynamical model is that, except the solemetabolic activity signal, all the other parts have kernels of variable separable form. There-fore, it is important to relate this property to the separation of the signals. This will be the aimof the next subsection.

1.3.2 Eigenvalue analysis

Since the light source has a limited frequency range, the function S 0(ω) has compact sup-port. Therefore, using (1.5), the functions Γc

ODT (x, t) and ΓmODT (x, t) are given by integrals of

a bounded integrand over a bounded region, so they are bounded functions on Ω × [0,T ].

Then the operators S c and S m are integral operators with L2 kernels. Therefore, they are Hilbert-Schmidt integral operators by definition. Since the composition of two Hilbert-Schmidt opera-

8


tors is trace-class (see [53]), the operators S cS ∗c, S cS ∗m, S mS ∗c, and S mS ∗m are trace-class opera-tors, and their trace is given by

Tr(S cS ∗c) = ‖S c‖2HS , Tr(S cS ∗m) = 〈S c, S m〉HS ,

Tr(S mS ∗m) = ‖S m‖2HS , Tr(S mS ∗c) = 〈S m, S c〉HS ,

where the Hilbert-Schmidt inner product is written by

〈S j, S k〉HS =

∫Ω×[0,T ]

ΓjODT (x, t)Γk

ODT (x, t)dx dt,

and the Hilbert-Schmidt norm is written by

‖S c‖2HS =

∫Ω×[0,T ]

|ΓjODT (x, t)|2dx dt,

for any j, k ∈ m, c.

In order to argue for the feasibility of using a SVD, we will calculate the corresponding eigen-values, showing that the collagen signal has one very large eigenvalue relative to the metabolicactivity. We assume that the eigenvalues are ordered decreasingly, so λ1 is the largest one.

We first recall that for an operator A with rank one, the unique non-zero eigenvalue λ is equalto the trace of A. From the expression of Fcc(x, y), we could see that Fcc(x, y) has rank onebecause of the separable form with respect to x and y, so the operator S cS ∗c has only one nonzeroeigenvalue, which we denote by λ(S cS ∗c). Now we compare λ(S cS ∗c) and the eigenvalues of theoperator S mS ∗m.

Lemma 1.3.1. Let S cS ∗c and S mS ∗m be the integral operators with kernels Fcc and Fmm definedin (1.11) and (1.14), respectively. If the intensities of collagen and metabolic activity satisfy(1.7), then we have

λ(S cS ∗c) λi(S mS ∗m), ∀i ≥ 1.

Proof. On one hand, Fcc(x, y) has rank one, so it is clear that

λ(S cS ∗c) = Tr(S cS ∗c). (1.15)

On the other hand, since the eigenvalues of operator S mS ∗m are all positive, any eigenvalueλi(S mS ∗m) satisfies

λi(S mS ∗m) < Σ∞i=1λi(S mS ∗m) = Tr(S mS ∗m). (1.16)

Then it suffices to prove that Tr(S cS ∗c) Tr(S mS ∗m). From the definition of trace of an operator,we readily get Tr(S cS ∗c) =

∫x∈Ω Fcc(x, x)dx. Substituting the expression (1.8) into the above

9


formula yields

Tr(S cS ∗c) =

∫x∈Ω

12

∫ ∞

−∞

ΓcODT (x, t)Γc

ODT (x, t)dtdx

=12

∫x∈Ω

∫ ∞

−∞

|ΓcODT (x, t)|2dtdx.

The same analysis can be carried out by looking at Tr(S mS ∗m),

Tr(S mS ∗m) =12

∫x∈Ω

∫ ∞

−∞

|ΓmODT (x, t)|2dtdx.

Recall that the intensity of collagen signal is much larger than metabolic activity signal, whichis the assumption in (1.7). Hence, we obtain the trace comparison Tr(S cS ∗c) Tr(S mS ∗m).

Now we compare the eigenvalue λ(S cS ∗m) with λ(S cS ∗c) and λ1(S mS ∗m).

Lemma 1.3.2. Let S cS ∗c, S cS ∗m and S mS ∗m be the integral operators with kernels defined in(1.11), (1.12) and (1.14). Then their eigenvalues satisfy

λi(S cS ∗m) ≤√λ(S cS ∗c)λ1(S mS ∗m)

for all i.

Proof. Recall the definition of the operator norm of an operator A, namely, ‖A‖OP = sup ‖Av‖‖v‖ , v ∈

V with v , 0, which yields λ(S cS ∗m) 6 ‖S cS ∗m‖OP. Since the operator norm is equal to the largestsingular value, direct calculation shows that

‖S cS ∗m‖OP ≤ ‖S c‖OP‖S ∗m‖OP

= σ1(S c)σ1(S ∗m)

=√λ(S cS ∗c)

√λ1(S mS ∗m),

where σ1 denotes the largest singular value.

In this section, we discussed eigenvalue analysis for the forward operator of multi-particle dy-namical model. More explicitly, we showed that the largest eigenvalue corresponds to the colla-gen signal, the middle eigenvalues mix the collagen signal and metabolic activity signal, and theremaining eigenvalue corresponds to the metabolic activity signal. Also in our model the solelycollagen signal has rank one, which provides a good reason to use SVD in solving the inverseproblem.

10

1.4 The inverse problem: Signal separation


Our main purpose in this chapter is to image the dynamics of metabolic activity of cells. Highlybackscattering structures like collagen dominate the dynamic OCT signal, masking structureswith low-backscattering such as metabolic activity. As shown in the modeling part, we dividethe scattering particles in the tissue into the high-backscattering collagen part, and the low-backscattering metabolic activity part. Based on this division, the resulting image ΓODT (x, t)could also be written as the sum of the collagen Γc

ODT (x, t) and the metabolic activity partΓm

ODT (x, t). The inverse problem is to recover the intensity of metabolic activity of cells fromthe image ΓODT (x, t). In this chapter, we use the singular value decomposition (SVD) methodto approximate the metabolic activity part, then using a particular formula (see (1.23)) to get itscorresponding intensity.

1.4.1 Analysis of the SVD algorithm

Since we have proved the high-backscattering signal corresponds to a rank one kernel and thispart is far larger than the rest, it is natural to associate it to the first singular value of the SVDexpansion. We claim that in order to remove the high backscattering signal, it is reasonableto remove the first term in the SVD expansion of the image. Given the orthogonal nature ofthe SVD decomposition, it is not possible to ensure a clean signal separation. Nonetheless atthe end of this section, we provide a result that illustrates how the error committed with thisdecomposition is actually small.

Let x1, x2, . . . , x j, . . . denote the pixels of the image. Define the matrices A, Ac ∈ Cnx×nt by

A j,k = ΓODT (x j, tk)

(Ac) j,k = ΓcODT (x j, tk),

where j ∈ 1, ..., nx, k ∈ 1, ..., nt.

Recall that a non-negative real number σ is a singular value for a matrix A if and only if thereexists unit vectors u ∈ Rnx and v ∈ Rnt such that

Av = σu and A∗u = σv,

where the vectors u and v are called left-singular and right-singular vectors of A for the singularvalue σ.

Assume that the singular values of A are ordered decreasingly, that is, σ1 ≥ σ2 ≥ . . . , and letui and vi be the singular vectors for σi. We emphasize that the vectors ui and vi are orthonormalsets in Cnx and Cnt , respectively. Thus, the SVD of the matrix A is given by

A = Σnti=1σiuivi

T . (1.17)

11


Total signal ΓODT First term in theSVD of ΓODT

collagen signalΓc

ODT

Matrix A A1 Ac

First singular value σ1 σ1 σc

First singular vector u1, v1 u1, v1 uc, vc

Other singular values σ2 > σ3 >

· · · > σi

0 0

Table 1.1: Singular values and singular vectors.

Since the matrix A is composed of a large rank one part Ac and a small part coming frommetabolic signal Γm

ODT , we can say that A − Ac is ”relatively small” with respect to A. It iswell known that the first term in the SVD expansion of A is the rank one matrix A1 such that‖A − A1‖op is minimal. Therefore, it is natural to think that Ac is ”close” to A1 in some way.But A1 = σ1u1v1

T is generally not the same as Ac, because as we will see in Appendix 1.7, theeigenvectors of the kernels Fcc, Fcm and Fmc are generally not orthogonal. Since SVD alwaysgives an orthogonal set of eigenvectors, we conclude that the SVD approach itself does notgive the eigenvectors exactly. Nevertheless, we can show that the SVD result is still a goodapproximation to the true eigenvectors.

To bridge the gap between the collagen signal ΓcODT and the first term of SVD expansion of

ΓODT , we investigate the relationship between their singular values and singular vectors. ΓcODT

has only one nonzero singular value σc, with the corresponding singular vectors uc and vc.

We claim in the following theorem that the singular value σ1 and the corresponding singularvector u1 are good approximations of the singular value σc and singular vector uc. See Table 1.1for the notations of their singular values and singular vectors.

Theorem 1.4.1. Let σi, ui, vi, Ac, uc and vc be described in Table 1.1. Assume that the collagensignal dominates, that is,

‖A − Ac‖op

‖Ac‖op= 1/N (1.18)

for a large N. Then there exist constants C > 0 and ε ∈ −1, 1 such that

|σc − σ1|

σc≤ C/N,

and

‖uc − εu1‖l2 ≤ C/N.

Proof. We define a matrix-valued function

F : s 7→ (Ac + sN(A − Ac))∗(Ac + sN(A − Ac)). (1.19)

12


Through this construction of F, we obtain

F(0) = A∗cAc and F(

1N

)= A∗A.

Applying Rellich’s perturbation theorem on hermitian matrices F (see, for example, [85]) to getthe following two properties. There exists a set of n analytic functions λ1(s), λ2(s), . . . such thatthey are all the eigenvalues of F(s). Also, there exists a set of vector-valued analytic functionsu1(s), u2(s), . . . such that F(s)ui(s) = λi(s)ui(s), and 〈ui(s), u j(s)〉 = δi j.

In view of the definition of ui(s) and λi(s), we show four useful properties, for some ε ∈ −1, 1,

u1(0) = uc, u1(1/N) = εu1,

λ1(0) = σ2c = ‖Ac‖

2op, λ1(1/N) = σ2

1,(1.20)

where the last property comes from the fact λ1(1/N) is the largest eigenvalue of F(1/N) = A∗Awhen N 1.

The objective is to get upper bounds for ‖uc − εu1‖l2 and |σc − σ1|. Using (1.20), we haveuc − εu1 = u1(0) − u1(1/N) and σc − σ1 =

√λ1(0) −

√λ1(1/N). Since u1(s) and λ1(s) are

analytic, a Taylor expansion at 0 yields

‖uc − εu1‖l2 = ‖u′1(0)

N‖l2 + O(1/N3/2),

|σc − σ1| =λ′1(0)

2√λ1(0)N

+ O(1/N2).(1.21)

The next step is to seek for proper upper bounds for λ′1(0) and u′1(0).

For the upper bound of λ′1(0), we differentiate F(s)ui(s) = λi(s)ui(s) with respect to s and thentake s = 0 to obtain

F′(0)ui(0) + F(0)u′i(0) = λi(0)u′i(0) + λ′i(0)ui(0). (1.22)

Since we always have ‖ui(s)‖`2 = 1, a direct calculation shows that

〈ui(s), u′i(s)〉 =12

dds‖ui(s)‖2 = 0.

By taking an inner product of both sides of (1.22) with ui(0), we get

λ′i(0) = λ′i(0)‖ui(0)‖2l2= 〈ui(0), F′(0)ui(0)〉 + 〈ui(0), F(0)u′i(0)〉

= 〈ui(0), F′(0)ui(0)〉 + 〈F(0)ui(0), u′i(0)〉

= 〈ui(0), F′(0)ui(0)〉 + λi(0)〈ui(0), u′i(0)〉

= 〈ui(0), F′(0)ui(0)〉.

13


Hence, λ′i(0) satisfies |λ′i(0)| ≤ ‖F′(0)‖op. By the definition of F(s), we have ‖F′(0)‖op =

N‖A∗c(A − Ac) + (A − Ac)∗Ac‖ ≤ 2N‖Ac‖op‖A − Ac‖op. Replacing N with (1.18) yields |λ′i(0)| ≤2‖Ac‖

2op. Therefore, by inserting the expression λ1(0) in (1.20) into (1.21), we get |σc − σ1| ≤

σcN + O(1/N2).

For the upper bound of u′1(0), we look again at (1.22). By taking an inner product with u′1(0), weimmediately obtain

〈u′1(0), F′(0)u1(0)〉 + 〈u′1(0), F(0)u′1(0)〉 = λ1(0)‖u′1(0)‖2l2 .

Recall that the matrix Ac is of rank one. So, there exists a positive constant c, such that A∗cAc =

cu1(0)uT1 (0), which reads

F(0)u′1(0) = cu1(0)(uT1 (0)u′1(0)) = cu1(0) ˙u1(0), u′1(0) = 0.

Therefore, direct calculation shows that ‖u′1(0)‖l2 ≤‖F′(0)u1(0)‖l2

λ1(0) ≤‖F′(0)‖op

‖Ac‖2op≤ 2.

The rest of the proof follows by substituting the above bound into (1.21), then we have ‖uc −

εu1‖l2 ≤2N + O(1/N3/2).

Remark 1. Theorem 1.4.1 shows that the eigenvector difference of two classes is of the order of1N , where N could be seen as the ratio between collagen signal and metabolic signal, so when N islarge enough, the difference could be ignored. Therefore, it is reasonable to use the eigenvectorsof the SVD to approximate the true eigenvectors.

Remark 2. In the proof of Theorem 1.4.1, we did not use any representation of A and Ac. So, ina more general case, for any matrix A = Ac + o(Ac) where rank of Ac is 1, the first singular valueand first singular vector of A could be used to approximate the singular value and the singularvector of Ac.

1.4.2 Analysis of obtaining the intensity of metabolic activity

Recall that our objective is to get the intensity of the metabolic activity after removing theinfluence of the collagen signal. We have proved that the largest singular value corresponds to thecollagen signal, and the following few singular values correspond to the correlation part betweencollagen signal and metabolic activity; the rest of the singular values contains information relatedto the metabolic activity.

Let T be the set of these remaining singular values. In practice, we only know the total signalΓODT (x, t) (or the matrix A). By performing a SVD for ΓODT (x, t), we take the terms only corre-sponding to the singular values in T in the SVD expansion. The next problem is to reconstructthe intensity of the particle movements of metabolic activity. In the numerical experiments, itis observed that the sum

∑i∈T σ

2i |ui(x j)|2 gives a very good approximation to the intensity of

metabolic activity at the pixel x j. A theorem is given to explain why it works.

14


Physically, we expect the metabolic activity signal to be centered around 0, so in each pixel x j,the norm ‖Am(x j, t)‖2`2 could be seen as the standard deviation of the metabolic signal, whichcould represent the intensity of metabolic activities in pixel x j. However, the eigenvectors of theoperators with the kernels Fcc, Fcm and Fmc are not orthogonal (this statement may be justifiedby arguing as in Appendix 1.7). Thus when using a SVD, we do not get the exact ”pure”metabolic activity signal Am, but only an approximation, which we denote by Am1 . We first givean interpretation that

∑i∈T σ

2i |ui(x j)|2 could be written as a `2 norm of the matrix Am1 .

Theorem 1.4.2. Let A be the matrix after the discretization of ΓODT (x, t) with respect to x andt, such that the j-th row of A corresponds to the pixel x j, and the k-th column of A correspondsto the time tk. Let T be a subset of singular values of A, and Am1 be the result of taking only thesingular values in T from A. Then for any pixel x j, we have∑

i∈T

σ2i |ui(x j)|2 =

∑k

|Am1(x j, tk)|2. (1.23)

Proof. We apply the SVD algorithm to the matrix A to get A = US V∗, where U = (u1, u2, . . . ),V = (v1, v2, . . . ) are unitary matrices, and S is a diagonal matrix containing the singular valuesof A.

We construct a new diagonal matrix S T , which is obtained from S by keeping all the singularvalues in T , but changing everything else to zero. By the definition of Am1 , we readily deriveAm1 = US T V∗.

Note that σiui(x j) is the element at row j, column i of the matrix US . By construction of S T , weknow σiui(x j) is the element at row j, column i of the matrix US T for every i ∈ T . Therefore,the sum

∑i∈T σ

2i |ui(x j)|2 is equal to the square-sum of the j-th row of the matrix US T , which

gives‖US T (x j, ·)‖2`2 =

∑i∈T

σ2i |ui(x j)|2. (1.24)

Moreover, the relation Am1 = (US T )V∗ means that for each x j, Am1(x j, ·) = (US T )(x j, ·)V∗.

A direct calculation from the definition of `2 norm of vectors shows that∑k

|Am1(x j, tk)|2 = ‖Am1(x j, ·)‖2`2 = Am1(x j, ·)Am1(x j, ·)∗.

Using V∗V = I and substituting (US T )(x j, ·)V∗ for Am1(x j, ·) yields

∑k

|Am1(x j, tk)|2 = US T (x j, ·)(US T (x j, ·))∗ = ‖US T (x j, ·)‖2`2 . (1.25)

Combining (1.24) and (1.25) completes the proof.

15


Then let us look at the `2 norm of the difference between the two matrices Am and Am1 . Pro-ceeding as in the proof of Theorem 1.4.1, we can estimate ‖Am − Am1‖. When N in (1.18)is large enough, it is reasonable to approximate Am by Am1 . This fact enables us to say that‖Am1(x j, t)‖2`2 ≈ ‖Am(x j, t)‖2`2 for each pixel x j.

Therefore, we conclude that∑

i∈T σ2i |(ui) j|

2 over the set T of remaining singular values is indeeda good approximation for the metabolic activity intensity.

1.5 Numerical experiments

In this section we model the forward measurements of our problem. Using the SVD decompo-sition we filter out the signal, finally obtaining images of the hidden weak sources.

1.5.1 Forward problem measurements

To simulate the signal measurements using formula (1.5), we only need to simulate the densityfunction p(x, z, t) of the media to be illuminated. For each pixel x, there are two types of su-perimposed media. One is the collagen media characterized for having a strong signal and slowmovement. The second medium is the metabolic activity, that has a fast movement relative tothe time samples. According to [23], the collagen signal intensity is around 100 times strongerthan the metabolic one.

Given these properties, both media are modeled differently. The collagen particles are simulatedas an extended random medium on z that displaces slowly on time; see [66]. For each pixel x, anindependent one-dimensional random medium rx(·) is generated, and then p(x, z, t) = rx(z + tv)with v being the constant movement velocity. The metabolic activity is simulated as an uniformwhite noise, whose intensity represents its magnitude. Background or instrumental noise isadded everywhere in a similar fashion, but with smaller intensity.

After the medium is simulated, formula (1.5) is applied to reproduce the measured signal, wherefor integration purposes, the broadband light is approximated by Dirac deltas in certain frequen-cies. All the model parameters are set such that we obtain similar measurements to the onesobtained in [23]. In Figure 1.3 we can see, for a single pixel, the simulated signal as a functionof time.

In the following, we consider a two-dimensional 21x21 grid of pixels. The collagen signal,albeit being generated by an independent random media, has the same parameters everywhere,thus sharing a similar behavior. In Figure 1.4, we present the considered metabolic activityintensity map and two snapshots at different times of the total signal.

16


Total Signal

0 100 200 300 400 500 600 700 800

Sample

3884.5

3885

3885.5

3886

3886.5

3887

Inte

nsity

Collagen Signal Metabolic Signal

0 100 200 300 400 500 600 700 800

Sample

3885

3885.2

3885.4

3885.6

3885.8

3886

3886.2

3886.4

3886.6

Inte

nsity

0 100 200 300 400 500 600 700 800

Sample

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Inte

nsity

Figure 1.3: At the image on top, we can see the total signal measured at a fixed pixel. If decomposed intothe one corresponding to the collagen structures and metabolic signal, we obtain the bottomimages, left- and right-hand side, respectively.

Metabolic activity map Snapshot of measurements Snapshot of measurements

5 10 15 20

2

4

6

8

10

12

14

16

18

20

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

5 10 15 20

2

4

6

8

10

12

14

16

18

20

3883.5

3884

3884.5

3885

3885.5

3886

3886.5

3887

3887.5

5 10 15 20

2

4

6

8

10

12

14

16

18

20

3883.5

3884

3884.5

3885

3885.5

3886

3886.5

3887

3887.5

Figure 1.4: On the left we can see the considered metabolic map, it describes the intensity of the metabolicsignal presented in Figure 1.3. The other two images correspond to raw sampling of the mediaat different times.

1.5.2 SVD of the measurements

To use the singular value decomposition on the signal, we reshape the raw data ΓODT (x, t) undera Casorati matrix form, where the two-dimensional pixels on the x variable are rearranged as aone-dimensional variable, and hence the total signal is written as a matrix A where each dimen-sion corresponds, respectively, to the space and time variables. The total signal consists on the

17


addition of the metabolic and collagen signals, namely A = Am + Ac. Our objective is to recoverthe spatial information of the metabolic signal Am.

We apply the SVD decomposition (1.17) over the total signal A, where the dimension of eachspace corresponds to the amount of pixels of the image and the time samples, respectively. Eachspace vector ui point out which pixels are participating in the ith singular value. To obtain animage of the pixels participating in a particular subset of singular values T ⊂ N, we use thefollowing formula (see Section 4.2 for why it works):

I( j) =

√∑i∈T

σ2i ui( j)2, (1.26)

where the indices j are for indexing the image’s pixels. When the signal has mean 0, formula(1.26) corresponds to the standard deviation that was already considered as an imaging formulain [23].

In Figure 1.5, we can see an image of each space vector ui ordered by their associated singularvalues, these vectors correspond to the decomposition of the total signal A. The other twopictures below correspond to the singular space vectors but for each unmixed signal Ac and Am,separately. As it can be seen, the spatial vectors of both signals get mixed in the total signal, butthe metabolic activity ones get embedded in a clustered fashion, although there is a distortion ofthese vectors, this is unavoidable given the nature of the SVD.

The location of the spatial vectors is related to their respective singular values, that are presentedin Figure 1.6. It is observed that the moment in which the spatial vectors of the total signal startto look like the ones from the metabolic activity, is close to the moment in which the singularvalues from the metabolic activity get close to those in the total signal. In a mathematical way,we say that the index j ∈ N in which the spatial vectors ui start to resemble those of the metabolicactivity, corresponds to

j = argminσ j(A) < σ1(Am) − k, with k small. (1.27)

In practice, for the tested examples (up to 24x24 grid of pixels, and 500 to 1000 time samples)k ≈ 10 achieve the best results.

The clustered behavior of the singular vectors arise from the model itself, as it generates fastdecaying singular values for the collagen signal, whereas the metabolic singular values decay ina more slow fashion. Hence, it is possible to assign an interval of the total signal space vectorsas an approximation to the metabolic activity Am.

1.5.3 Selection of cut-off singular value

The before mentioned criteria to choose an adequate interval of singular-space vectors to applythe imaging formula (1.26) is not possible in practice, as we have no a priori information onwhere the metabolic singular values σi(Am) lie. Since the idea is to consider an interval of

18


A singular space-vectors

20 40 60 80 100 120 140

Index i

50

100

150

200

250

300

350

400

Ima

ge

's p

ixe

ls

Ac singular space-vectors Am singular space-vectors

20 40 60 80 100 120 140

Index i

50

100

150

200

250

300

350

400

Image's

pix

els

20 40 60 80 100 120 140

Index i

50

100

150

200

250

300

350

400

Ima

ge

's p

ixe

ls

Figure 1.5: These images are the space matrices obtained with the SVD, where each column of thesematrices corresponds to a singular space-vector, naturally ordered from left to right by theirrespective singular value index. Notice that each row of these matrices represents each pixelof the image presented in Figure 1.4 and the color intensity indicates how much weight eachsingular vector placed in that particular pixel has. From the bottom-right image we can seethat the first singular space-vectors from the metabolic signal are clearly clustered on somepixels (that correspond to the zones in where there is metabolic activity), whereas from thebottom-left image we notice that the singular space-vectors do not concentrate in any specificlocation, as the collagen is present everywhere. Observing the image on the top, that is thetotal signal, we notice that after dropping the firsts singular space-vectors, we also observe aclustered behavior. This information is the one we seek to recover from the total signal.

singular space vectors, the first and last elements must be defined. The length of the intervalcorresponds to the range of the matrix Am, with some added terms coming from the matrix Ac.This can be left as a free parameter to be decided by the controller. As a general guideline, itcorresponds to the quantity of pixels in which it is expected to find the metabolic activity.

For the considered first singular space vector, also called cut-off one, there is a criteria that arisesfrom the model. Given the differences between the metabolic signal and the collagen signal,the latter in the time variable has some regularity and self correlation. This characteristic istransferred to the first singular time-vectors. In Figure 1.7 we can see plots of these time-vectorsfor each signal.

19


0 50 100 150 200 250 300 350 400 450

Index i

0

5

10

15

20

25

30

35

40

45

50

Ideal cut-off point

Total singular values

Collagen s.v.

Metabolic s.v.

Figure 1.6: Singular values for the signals. The circle represents the optimal starting index j at which weshould consider the singular space-vectors of the total signal to contain mostly information onthe singular space-vectors of the metabolic activity. The first singular value of the total signaland the collagen signal is outside the plot, with an approximate value of 2.3 × 106.

Total signal singular time-vector Collagen signal singular time-vector Metabolic signal singular time-vector

0 100 200 300 400 500 600 700 800

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 100 200 300 400 500 600 700 800

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 100 200 300 400 500 600 700 800

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Figure 1.7: Plots of the singular time-vectors for each signal, at the second singular value. The collagentime vectors are more regular and correlated compared to the metabolic signal, albeit thisproperty is gradually loosened as we augment the index of the time vectors. Since the SVD ofthe total signal is dominated by the collagen signal, its time-vectors inherit the same property.

Our proposed technique consists in measuring the regularity of the time vectors using the totalvariation semi-norm, the smaller the value the more regular. With this information we seek toestimate the argminσ j(A) < σ1(Am) and thus, using formula (1.27), select the cut-off point. Inthe case of a discrete signal, the total variation can be stated as

| v |TV =

N−1∑i=1

|v(i + 1) − v(i)|.

Applying the total variation norm to the total signal singular time-vectors vi, we can see that theregularity drops until arriving to, in mean, a slowly increasing plateau. To find it, in an operatorfree way, it is possible to fit a two piece continuous quadratic spline in the total variation plot,and estimate the first metabolic singular value index as the index l in which the spline changes.In our simulations this l is a good approximation for the first singular value of the metabolic

20


activity, meaning that σl ≈ σ1(Am); see Figure 1.8. Thus by subtracting the value k in formula(1.27), we obtain a cut-off point. Keep in mind that this considered method does not make useof a priori information.

0 50 100 150 200 250 300 350 400 450 500

Index of singular values

0

5

10

15

20

25

30

35

40

45

50

Selected cut-off point

Total singular values

Collagen s.v.

Metabolic s.v.

Singular time-vector TVnorm

Figure 1.8: Same plot as in Figure 1.6, but including the total variation of the singular time-vectors of thetotal signal. The total variation is scaled to fit the plot with the singular values.

1.5.4 Signal reconstruction

Employing the cut-off criteria in subsection 1.5.3 and formula (1.26) to our simulation, we canreconstruct the metabolic activity. In Figure 1.9 we have on the left-hand side the best possi-ble reconstruction using the SVD technique, that is, the one we could do if we could isolatecompletely the signal Am from the total signal A. On the right-hand side, we have the actualreconstruction. It is worth mentioning that we are not able to reconstruct the exact metabolicmap, as formula (1.26) is used on the simulated media, and thus the image obtained out of theisolated signal Am is the one we are aiming to reconstruct.

1.5.5 Discussion and observations

Since the SVD uses information of all pixels simultaneously to filter out the collagen signal, thistechnique works better the larger the considered image size is, as the main point is to use thejoint information of all the pixels in the image, in contrast to frequency filtering that considersonly point-wise information. Numerically, this effect is notorious, as the larger the image size,the more clustered are the singular space-vectors associated to the metabolic activity and thus itis easier to filter out the collagen signal.

With respect to the time samples, keep in mind that all simulations considered a fixed samplerate, and it is observed that the filtering process degrades if too many time samples are considered(i.e., lengthier measurement experiences). When this happens (for our 21x21 grid size, this is

21


Best possible reconstruction Achieved reconstruction

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

Figure 1.9: Reconstruction of the metabolic map presented in Figure 1.4. The left image correspond tousing directly formula (1.26) on the isolated Am signal. The right-hand side image correspondto using our reconstruction method on the total signal. Once the images are normalized, thecommitted error with respect to the original metabolic map is 0.011 and 0.017, respectively

above 1000 time samples), the singular values of the collagen signal start decaying in a slowerrate, accomplishing a less clustered behavior of the metabolic singular space-vectors, and thusachieving a worse signal separation. Therefore, if there are too many available time samples, onepossible recommendation is to do several reconstructions using subsets of these time samplesand then averaging the results.

The reasons of this behavior when too many time samples are accessible, are specific to thesingular value decomposition and not the sample rate or the noise level. It is possible that thesample rate gets small enough to the point that we could track specific metabolic movementsinside the cell, if this is the case, the model becomes invalid and new assumptions must beplaced.

1.6 Conclusion

In this chapter, we performed a mathematical analysis of extracting useful information for sub-cellular imaging based on dynamic optical coherence tomography. By using a novel multi-particle dynamical model, we analyzed the spectrum of the operator with the intensity as anintegral kernel, and showed that the dominant collagen signal has rank one. Therefore, a SVDapproach can theoretically separate the metabolic activity signal from the collagen signal. Weproved that the SVD eigenvectors are good approximation to the collagen signal, showing thatthe SVD approach is feasible and reliable as a method to remove the influence of collagensignals. And we also discovered a new formula that gives the intensity of metabolic activityfrom the SVD analysis. This is further confirmed by our numerical results on simulated datasets.

22

1.7 Appendix: Non-orthogonality of the eigenvectors of Fcc, Fcm, and Fmc

1.7 Appendix: Non-orthogonality of the eigenvectors of Fcc, Fcm,and Fmc

In this appendix we will illustrate the fact that the eigenvectors of the kernels Fcc(x, y), Fcm(x, y)and Fmc(x, y) are in general not orthogonal. Since all of them have variable separable forms withrespect to x and y, which is the basis of our analysis, so here we only prove the nonorthogonalitybetween eigenvectors of the kernels Fcc(x, y) in (1.11) and Fcm(x, y) in (1.12).

Let A be the matrix obtained from discretizing the signal ΓODT . The singular values of A arethe square roots of the eigenvalues of the matrix A∗A, and the singular vectors of A are thecorresponding eigenvectors of A∗A. We notice that A∗A is a discretization of the integral kernelF(x, y). We first demonstrate the relation between kernels with variable separable forms andeigenvectors.

Lemma 1.7.1. For any function f (x, y) ∈ L2(Ω × Ω), if there exist functions f1(x) and f2(y),such that f (x, y) = f1(x) f2(y), then the operator T : L2(Ω)→ L2(Ω) with kernel f (x, y) can haveonly one nonzero eigenvalue. Furthermore, the eigenvector of T corresponding to the uniquenonzero eigenvalue is given by f1(x) or f2(y).

Proof. Using the variable separation form f (x, y) = f1(x) f2(y) yields

(Th)(x) =

∫Ω

f (x, y)h(y)dy

=

∫Ω

f1(x) f2(y)h(y)dy

= f1(x)∫

Ω

f2(y)h(y)dy.

(1.28)

Suppose that λ is a nonzero eigenvalue of T , and g(x) is the corresponding eigenvector, so thatTg(x) = λg(x). Comparing this with (1.28) implies

λg(x) = f1(x)∫

Ω

f2(y)g(y)dy.

Therefore, the eigenvector g(x) is a multiple of f1(x), and the corresponding eigenvalue λ =∫Ω

f2(y) f1(y)dy.

23


Denote the functions ϕc(x) and ϕm(x) by

ϕc(x) =Kc1(x)qc(x),

ϕm(x) =

∫[−L,L]2×[0,T ]

F (S 0Kc2)(4πnz2

c

)× F (S 0Km)

(x,−

4πnz1

c

)pm(x, z1, t)dz1dz2dt.

Then the kernels Fcc and Fcm can be written as

Fcc(x, y) = C1ϕc(x)ϕc(y),

Fcm(x, y) = C2ϕc(x)ϕm(y),

where C1 and C2 are constants.

Applying Lemma 1.7.1 to the kernels Fcc and Fcm, we know that the corresponding eigenvectorsare ϕc and ϕm respectively.

Since this integral∫Ωϕc(x)ϕm(x)dx depends much on the random term pm(x, z, t), it will not be

zero almost all of the time. Hence, in our construction, the vectors ϕc and ϕm are in general notorthogonal.

24

2 Mathematical Analysis of UltrafastUltrasound Imaging

2.1 Introduction

Conventional ultrasound imaging is performed with focused ultrasonic waves [88, 89]. Thisyields relatively good spatial resolution, but clearly limits the acquisition time, since the entirespecimen has to be scanned. Over the last decade, ultrafast imaging in biomedical ultrasound hasbeen developed [42, 76, 90]. Plane waves are used instead of focused waves, thereby limiting theresolution but increasing the frame rate considerably, up to 20,000 frames per second. Ultrafastimaging has been made possible by the recent technological advances in ultrasonic transducers,but the idea of ultrafast ultrasonography dates back to 1977 [34]. The advantages given by thevery high frame rate are many, and the applications of this new modality range from blood flowimaging [28, 42], deep superresolution vascular imaging [49] and functional imaging of the brain[72, 73] to ultrasound elastography [52]. In this chapter we focus on blood flow imaging.

A single ultrafast ultrasonic image is obtained as follows [76]. A pulsed plane wave (focused onthe imaging plane – see Figure 2.1b) insonifies the medium, and the back-scattered echoes aremeasured at the receptor array, a linear array of piezoelectric transducers. These spatiotemporalmeasurements are then beamformed to obtain a two-dimensional spatial signal. This is what wecall static inverse problem, as it involves only a single wave, and the dynamics of the medium isnot captured. The above procedure yields very low lateral resolution, i.e. in the direction parallelto the wavefront, because of the absence of focusing. In order to solve this issue, it was proposedto use multiple waves with different angles: these improve the lateral resolution, but have thedrawback of reducing the frame rate.

For dynamic imaging, the above process is repeated many times, which gives several thousandimages per second. In blood flow imaging, we are interested in locating blood vessels. One of themain issues lies in the removal of the clutter signal, typically the signal scattered from tissues,as it introduces major artifacts [30]. Ultrafast ultrasonography allows to overcome this issue,thanks to the very high frame rate. Temporal filters [28, 72, 73], based on high-pass filtering thedata to remove clutter signals, have shown limited success in cases when the clutter and bloodvelocities are close (typically of the order of 10−2 m·s−1), or even if the blood velocity is smallerthan the clutter velocity. A spatiotemporal method based on the singular value decomposition(SVD) of the data was proposed in [42] to overcome this drawback, by exploiting the differentspatial coherence of clutter and blood scatterers. Spatial coherence is understood as similarmovement, in direction and speed, in large parts of the imaged zone. Tissue behaves with higher

25

2 Mathematical Analysis of Ultrafast Ultrasound Imaging

spatial coherence when compared to the blood flow, since large parts of the medium typicallymove in the same way, while blood flow is concentrated only in small vessels, which do notshare necessarily the same movement direction and speed. This explains why spatial propertiesare crucial to perform the separation.

In this chapter, we provide a detailed mathematical analysis of ultrasound ultrafast imaging.To our knowledge, this is the first mathematical work addressing the important challenges ofthis emerging and very promising modality. Even though in this work we limit ourselves toformalize the existing methods, the mathematical analysis provided gives important insights,which we expect will lead to improved reconstruction schemes.

The contributions of this chapter are twofold. First, we carefully study the forward and inversestatic problems. In particular, we derive the point spread function (PSF) of the system, in theBorn approximation for ultrasonic wave propagation. We investigate the behavior of the PSF,and analyze the advantages of angle compounding. In particular, we study the lateral and ver-tical resolutions. In addition, this analysis allows us to fully understand the roles of the keyparameters of the system, such as the directivity of the array and the settings related to anglecompounding.

Second, we consider the dynamic problem. The analysis of the PSF provided allows to study theDoppler effect, describing the dependence on the direction of the flow. Moreover, we considera random model for the movement of blood cells, which allows us to study and justify the SVDmethod for the separation of the blood signal from the clutter signal, leading to the reconstructionof the blood vessels’ geometry. The analysis is based on the empirical study of the distribution ofthe singular values, which follows from the statistical properties of the relative data. We provideextensive numerical simulations, which illustrate and validate this approach.

This chapter is structured as follows. In Section 2.2 we describe the imaging system and themodel for wave propagation. In Section 2.3 we discuss the static inverse problem. In particular,we describe the beamforming process, the PSF, and the angle compounding technique. In Sec-tion 2.4 the dynamic forward problem is considered: we briefly discuss how the dynamic dataare obtained and analyze the Doppler effect. In Section 2.5 we focus on the source separation tosolve the dynamic inverse problem. We discuss the random model for the refractive index andthe method based on the SVD decomposition of the data. In Section 2.6 numerical experimentsare provided. Some concluding remarks and outlooks are presented in the final section.

2.2 The Static Forward Problem

The imaging system is composed of a medium contained in R3+ := (x, y, z) ∈ R3 : z > 0

and of a fixed linear array of transducers located on the line z = 0, y = 0. This linear array ofpiezoelectric transducers (see [89, Chapter 7]) produces an acoustic illumination that is focusedin elevation—in the y coordinates, near the plane y = 0—and has the form of a plane wave inthe direction k ∈ S 1 in the x, z coordinates (see Figure 2.1b). Typical sizes for the array lengthand for the penetration depth are about 10−1 m.

26

2.2 The Static Forward Problem

Time (s)×10 -6

-1.5 -1 -0.5 0 0.5 1 1.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(a) The real part of the input pulse f .

x

z

y

Receptor array

Imaging plane

(b) The imaging system. The incident wave issupported near the imaging plane y = 0,within the focusing region bounded by thetwo curved surfaces.

Figure 2.1: The pulse f of the incident wave ui and the focusing region.

We make the assumption that the acoustic incident field ui can be approximated as

ui (x, y, z, t) = Az (y) f(t − c−1

0 k · (x, z)),

where c0 is the background speed of sound in the medium. The function Az describes the beamwaist in the elevation direction at depth z (between 4 · 10−3 m and 10−2 m). This is a simplifiedexpression of the true incoming wave, which is focused by a cylindrical acoustic lens locatednear the receptor array (see [89, Chapters 6 and 7]). The function f is the waveform describingthe shape of the input pulse:

f (t) = e2πiν0tχ (ν0t) , χ (u) = e−u2

τ2 , (2.1)

where ν0 is the principal frequency and τ the width parameter of the pulse (see Figure 2.1a).Typically, ν0 will be of the order of 106 s−1. More precisely, realistic quantities are

c0 = 1.5 · 103 m·s−1, ν0 = 6 · 106 s−1, τ = 1. (2.2)

Let c : R3 → R+ be the speed of sound and consider the perturbation n given by

n (x) =1

c2 (x)−

1c2

0

.

We assume that supp n ⊆ R3+. The acoustic pressure in the medium satisfies the wave equation

∆u (x, t) −1

c2 (x)∂2

∂t2 u (x, t) = 0, x ∈ R3,

27


with a suitable radiation condition on u − ui. Let G denote the Green’s function for the acousticwave equation in R3 [9, 95]:

G(x, t, x′, t′) = −(4π)−1

|x − x′|δ((

t − t′)− c−1

0

∣∣∣x − x′∣∣∣) .

In the following, we will assume that the Born approximation holds, i.e. we consider only firstreflections on scatterers, and neglect subsequent reflections [9, 39] (in cases when the Bornapproximation is not valid, nonlinear methods have to be used). This is a very common approx-imation in medical imaging, and is justified by the fact that soft biological tissues are almostacoustic homogeneous, due to the high water concentration. In mathematical terms, it con-sists in the linearization around the constant sound speed c0. In this case, the scattered waveus := u − ui is given by

us (x, t) =

∫R

∫R3

n(x′

) ∂2ui

∂t2

(x′, t′

)G

(x, t, x′, t′

)dx′dt′, x ∈ R3, t ∈ R+,

since contributions from n∂2t us are negligible. Therefore, inserting the expressions for the

Green’s function and for the incident wave yields

us (x, t) = −

∫R3

(4π)−1

|x − x′|n(x′

)Az′

(y′)

f ′′(t − c−1

0

((x′, z′) · k +

∣∣∣x − x′∣∣∣)) dx′,

where we set x = (x, y, z) and x′ = (x′, y′, z′). Since the waist of the beam in the y direction issmall compared to the distance at which we image the medium, we can make the assumption∣∣∣x − (

x′, y′, z′)∣∣∣ ' ∣∣∣x − (

x′, 0, z′)∣∣∣ , x = (x, 0, 0) ∈ R3,

so that the following expression for us holds for x = (x, 0, 0) ∈ R3 and t > 0:

us (x, t)=

∫R2

−(4π)−1

|x − (x′, 0, z′)|f ′′

(t − c−1

0

((x′, z′) · k +

∣∣∣x − (x′, 0, z′

)∣∣∣)) n(x′, z′)dx′dz′,

where n is given by

n(x′, z′) :=∫R

n(x′

)Az′

(y′)

dy′, x′ = (x′, y′, z′) ∈ R3. (2.3)

Since our measurements are only two-dimensional (one spatial dimension given by the lineararray and one temporal dimension), we cannot aim to reconstruct the full three-dimensionalrefractive index n. However, the above identity provides a natural expression for what can bereconstructed: the vertical averages n of n. Since Az is supported near y = 0, n reflects thecontribution of n only near the imaging plane. In physical terms, n contains all the scatterers inthe support of Az; these scatterers are in some sense projected onto y = 0, the imaging plane.For simplicity, with an abuse of notation from now on we shall simply denote n by n, sincethe original three-dimensional n will not play any role, due to the dimensionality restriction

28

2.3 The Static Inverse Problem

discussed above. Moreover, for the same reasons, all vectors x and x′ will be two-dimensional,namely, x = (x, z) and similarly for x′. In view of these considerations, for x = (x, 0) ∈ R2 andt > 0 the scattering wave takes the form

us (x, t) = −

∫R2

(4π)−1

|x − x′|f ′′

(t − c−1

0

(x′ · k +

∣∣∣x − x′∣∣∣)) n

(x′

)dx′. (2.4)

It is useful to parametrize the direction k ∈ S 1 of the incident wave by k = kθ = (sin θ, cos θ) forsome θ ∈ R; in practice, |θ| ≤ 0.25 [76].


The static inverse problem consists in the reconstruction of n (up to a convolution kernel) fromthe measurements us at the receptors, assuming that n does not depend on time. This processprovides a single image, and will be repeated many times in order to obtain dynamic imaging,as it is discussed in the next sections.

2.3.1 Beamforming

The receptor array is a segment Γ = (−A, A) × 0 for some A > 0. The travel time from thereceptor array to a point x = (x, z) and back to a receptor located in u0 = (u, 0) is given by

τθx (u) = c−10 (x · kθ + |x − u0|) .

The beamforming process [76, 89] consists in averaging the measured signals on Γ at t = τθx (u),which results in the image

sθ(x, z) :=∫ x+Fz

x−Fzus

(u0, τ

θx (u)

)du, x = (x, z) ∈ R2

+ := (x, z) ∈ R2 : z > 0.

The dimensionless aperture parameter F indicates which receptors are chosen to image the loca-tion x = (x, z), and depends on the directivity of the ultrasonic array (in practice, 0.25 ≤ F ≤ 0.5[76]). In general, F depends on the medium roughness and on θ, but this will not be consideredthis work. The above identity is the key of the static inverse problem: from the measurementsus((u, 0), t) we reconstruct sθ(x, z).

We now wish to understand how sθ is related to n. In order to do so, observe that by (2.4) wemay write for x ∈ R2

+

sθ(x, z) = −

∫x′∈R2

n(x′

) ∫ x+Fz

x−Fz

(4π)−1

|x′ − u0|f ′′

(τθx (u) − τθx′ (u)

)du dx′

=

∫x′∈R2

gθ(x, x′

)n(x′

)dx′,

(2.5)

29


×1013

-8

-6

-4

-2

0

2

4

6

8

(a) The exact PSF given in(2.6).

×1013

-8

-6

-4

-2

0

2

4

6

8

(b) The approximation of thePSF given in (2.9).

×1013

-8

-6

-4

-2

0

2

4

6

8

(c) The approximation of thePSF given in (2.10).

Figure 2.2: The real part of the point spread function g0 and its approximations are shown in these figures(with parameters as in (2.1) and (2.2), and F = 0.4). The size of the square shown is 2 mm ×2 mm, and the horizontal and vertical axes are the x and z axes, respectively. The relative errorin the L∞ norm is about 7% for the approximation shown in panel (b) and about 9% for theapproximation shown in panel (c).

where gθ is defined as

gθ(x, x′

)= −

∫ x+Fz

x−Fz

(4π)−1

|x′ − u0|f ′′

(τθx (u) − τθx′ (u)

)du, (2.6)

(see Figure 2.2a for an illustration in the case when θ = 0). In other words, the reconstructionsθ is the result of an integral operator given by the kernel gθ applied to the refractive index n.Thus, the next step is the study of the PSF gθ (x, x′), which should be thought of as the imagecorresponding to a delta scatterer in x′.

2.3.2 The PSF

In its exact form, it does not seem possible to simplify the expression for g further: we willhave to perform some approximations. First, observe that setting hθx,x′(u) = τθx (u) − τθx′ (u) forx, x′ ∈ R2

+ we readily derive

(hθx,x′)′(u) = c−1

0

(u − x|x − u0|

−u − x′

|x′ − u0|

)≈ c−1

0

(u − x|x′ − u0|

−u − x′

|x′ − u0|

)= c−1

0x′ − x|x′ − u0|

,

for x close to x′ (note that, otherwise, the magnitude of the PSF would be substantially lower).As a consequence, by (2.6) we have

gθ(x, x′

)≈

c0(4π)−1

x − x′

∫ x+Fz

x−Fz(hθx,x′)

′(u) f ′′(hθx,x′(u)

)du

=c0(4π)−1

x − x′[f ′(hθx,x′(x + Fz)) − f ′(hθx,x′(x − Fz))

].

(2.7)

30


In order to simplify this expression even further, let us do a Taylor expansion of wθ±(x, z) :=

hθx,x′(x ± Fz) with respect to (x, z) around (x′, z′). Direct calculations show that

wθ±(x′, z′) = 0, ∇wθ

±(x′, z′) =c−1

0√

1 + F2

( √1 + F2 sin θ ∓ F, 1 +

√1 + F2 cos θ

),

whence

hθx,x′(x ± Fz)≈c−1

0√

1 + F2

((1 +

√1 + F2 cos θ

)(z − z′)+

( √1 + F2 sin θ ∓ F

)(x − x′)

).

Substituting this expression into (2.7) yields

gθ(x, x′) ≈ gθ(x − x′), (2.8)

where

gθ(x) =c0

4πx

f ′ c−1

0√

1 + F2

((1 +

√1 + F2 cos θ

)z +

( √1 + F2 sin θ − F

)x)

− f ′ c−1

0√

1 + F2

((1 +

√1 + F2 cos θ

)z +

( √1 + F2 sin θ + F

)x) , (2.9)

(see Figure 2.2b for an illustration in the case θ = 0), thereby allowing to write the image sθgiven in (2.5) as a convolution of gθ and the refractive index n, namely

sθ(x) =

∫x′∈R2

gθ(x − x′)n(x′

)dx′ = (gθ ∗ n)(x), x ∈ R2

+.

The validity of this approximation, obtained by truncating the Taylor expansion of wθ± at the

first order, is by no means obvious. Indeed, by construction, the pulse f (t) is highly oscillating(ν0 ≈ 6 · 106 s−1), and therefore even small variations in t may result in substantial changes inf (t). However, this does not happen, since if (x, z) is not very close to (x′, z′) then the magnitudeof the PSF is very small, if compared to the maximum value. The verification of this fact is quitetechnical, and thus is omitted; the details may be found in Appendix 2.8.

Remark 2.3.1. From this expression, it is easy to understand the role of the aperture parameterF, which depends on the directivity of the array. Ignoring the second order effect in F andtaking, for simplicity θ = 0, we can further simplify the above expression as

g0(x) ≈c0

4πx

[f ′

(c−1

0 (2z − Fx))− f ′

(c−1

0 (2z + Fx))].

It is clear that F affects the resolution in the variable x: the higher F is, the higher the resolutionis. Moreover, the aperture parameter affects also the orientation of the diagonal tails in the PSF.These two phenomena can be clearly seen in Figure 2.3. In general, the higher the apertureis the better for the reconstruction; as expected, the intrinsic properties of the array affects the

31


×1013

-4

-3

-2

-1

0

1

2

3

4

(a) The PSF with F =

0.2.

×1013

-6

-4

-2

0

2

4

6

(b) The PSF with F =

0.3.

×1013

-8

-6

-4

-2

0

2

4

6

8

(c) The PSF with F =

0.4.

×1014

-1

-0.5

0

0.5

1

(d) The PSF with F =

0.5.

Figure 2.3: The exact PSF with different values of the aperture parameter F (with parameters as in (2.1)and (2.2), and θ = 0). The size of the square shown is 2 mm × 2 mm, and the horizontal andvertical axes are the x and z axes, respectively.

×1013

-8

-6

-4

-2

0

2

4

6

8

(a) The PSF with θ = 0.

×1013

-8

-6

-4

-2

0

2

4

6

8

(b) The PSF with θ =

0.1.

×1013

-8

-6

-4

-2

0

2

4

6

8

(c) The PSF with θ =

0.2.

×1013

-8

-6

-4

-2

0

2

4

6

8

(d) The PSF with θ =

0.3.

Figure 2.4: The exact PSF with different values of the angle θ (with parameters as in (2.1) and (2.2), andF = 0.4). The size of the square shown is 2 mm × 2 mm, and the horizontal and vertical axesare the x and z axes, respectively.

reconstruction.

Remark 2.3.2. It is also easy to understand the role of the angle θ. In view of

gθ(x) ≈c0

4πx

[f ′

(c−1

0 ((1 + cos θ)z + (sin θ − F)x))− f ′

(c−1

0 ((1 + cos θ)z + (sin θ + F)x))],

an angle θ , 0 substantially gives a rotation of the PSF; see Figure 2.4.

We have now expressed gθ as a convolution kernel. In order to better understand the differentroles of the variables x and z, it is instructive to use the actual expression for f given in (2.1).

32


Since f ′(t) = ν0e2πiν0tχ(ν0t) with χ(t) = 2πiχ(t) + χ′(t), we can write

f ′ c−1

0√

1 + F2

((1 +

√1 + F2 cos θ

)z +

( √1 + F2 sin θ ± F

)x)

= ν0e2πiν0c−1

0√1+F2

((1+√

1+F2 cos θ)z+

(√1+F2 sin θ±F

)x)

· χ

ν0c−10

√1 + F2

((1 +

√1 + F2 cos θ

)z +

( √1 + F2 sin θ ± F

)x)

≈ ν0e2πiν0c−10 (2z+(θ±F)x)χ

(2ν0c−1

0 z),

where we have approximated the dependence on F and θ at first order around F = 0 and θ = 0in the complex exponential (recall that F and θ are small) and at zero-th order (F = 0 andθ = 0) inside χ: the difference in the orders is motivated by the fact that the variations of thecomplex exponentials have much higher frequencies than those of χ, since several oscillationsare contained in the envelope defined by χ, as it can be easily seen in Figure 2.1a (and similarlyfor χ′). This approximation may be justified by arguing as in Appendix 2.8. Inserting thisexpression into (2.9) yields

gθ(x) ≈c0

4πx

[ν0e2πiν0c−1

0 (2z+(θ−F)x)χ(2ν0c−1

0 z)−ν0e2πiν0c−1

0 (2z+(θ+F)x)χ(2ν0c−1

0 z)]

= −iν0c0

2πxχ(2ν0c−1

0 z)

e4πiν0c−10 ze2πiν0c−1

0 θx sin(2πν0c−10 Fx),

whence for every x = (x, z) ∈ R2

gθ(x) ≈ −iν20Fχ

(2ν0c−1

0 z)


0 θx sinc(2πν0c−10 Fx), (2.10)

where sinc(x) := sin(x)/x (see Figure 2.2c). This final expression allows us to analyze the PSFgθ, and, in particular, its different behaviors with respect to the variables x and z. Consider forsimplicity the case θ = 0 (with τ = 1). In view of the term χ

(2ν0c−1

0 z), the vertical resolution is

approximately 0.8 · ν−10 c0; similarly, in view of the term sinc(2πν0c−1

0 Fx), the horizontal resolu-tion is approximately 1

2F ν−10 c0. Even though horizontal and vertical resolutions are comparable,

in terms of focusing and frequencies of oscillations the PSF has very different behaviours in thetwo directions. Indeed, we can observe that the focusing in the variable z is sharper than that inthe variable x: the decay of χ is much stronger than the decay of sinc. Moreover, in the variablez we have only high oscillations, while in the variable x the highest oscillations are at least fourtimes slower (2 = 4 1

2 ≥ 4F), and very low frequencies are present as well, due to the presenceof the sinc. As it is clear from Figure 2.2, this approximation introduces evident distortions ofthe tails, as it is expected from the approximation F = 0 inside χ; however, the center of thePSF is well approximated. Similar considerations are valid for the case when θ , 0: as observedbefore, this simply gives a rotation.

The same analysis may be carried out by looking at the expression of the PSF in the frequencydomain. For simplicity, consider the case θ = 0: the general case simply involves a translation in

33


0 0.2 0.4 0.6 0.80

1

2

3

4

5

(a) The Fourier trans-form of the exactPSF g0 given in(2.6).

0 0.2 0.4 0.6 0.80

1

2

3

4

5

(b) The Fourier trans-form of the approx-imation of the PSFg0 given in (2.9).

0 0.2 0.4 0.6 0.80

1

2

3

4

5

(c) The Fourier trans-form (2.11) of theapproximation ofthe PSF g0 given in(2.10).

0 0.2 0.4 0.6 0.80

1

2

3

4

5

(d) The Fourier trans-form of the PSF gac

Θ

given in (2.13), forΘ = 0.25.

Figure 2.5: The absolute values of the Fourier transforms of the point spread functions and its approx-imations (with parameters as in (2.1) and (2.2), and F = θ = 0). The frequency axes arenormalized by ν0c−1

0 : the PSF is a low pass filter with cut-off frequency Fν0c−10 with respect

to the variable x and a band pass filter around 2ν0c−10 with respect to z.

the frequency domain with respect to x. Thanks to the separable form of gθ given in (2.10), theFourier transform may be directly calculated, and results in the product of the Fourier transformof χ and the Fourier transform of the sinc. More precisely, we readily derive

F gθ(ξx, ξz)

=

∫R2

gθ(x, z)e−2πi(xξx+zξz) dxdz

≈ −iν20F

∫R

sinc(2πν0c−10 Fx)e−2πixξx dx

∫R

χ(ν0c−1

0 z)

e−2πi(−2ν0c−10 +ξz)z dz.

Thus, since the Fourier transform of the sinc may be easily computed and is a suitable scaledversion of the rectangle function, we have

F gθ(ξx, ξz) ≈ −iν20F

12ν0c−1

0 F1[−F,F]

(c0ν−10 ξx

) ∫R

χ(ν0c−1

0 z)

e−2πi(−2ν0c−10 +ξz)z dz

= −ic0ν0

21[−F,F]

(c0ν−10 ξx

) 1ν0c−1

0

F χ

−2ν0c−10 + ξz

ν0c−10

,whence

F gθ(ξx, ξz) ≈ −ic20 1[−F,F]

(c0ν−10 ξx

)F χ

(−2 + ν−1

0 c0ξz)/2. (2.11)

Therefore, up to a constant, the Fourier transform of the PSF is a low-pass filter in the variablex with cut-off frequency Fν0c−1

0 and a band pass filter in z around 2ν0c−10 (since χ is a low-pass

filter). This explains, from another point of view, the different behaviors of gθ with respect tox and z. This difference is evident from Figure 2.5, where the absolute values of the Fouriertransforms of the different approximations of the PSF are shown.

34


2.3.3 Angle compounding

We saw in the previous subsection that, while very focused in the direction z, the PSF is notvery focused in the direction x due to the presence of the sinc function; see (2.10). In orderto have better focusing, it was proposed in [76] to use multiple measurements corresponding tomany angles in an interval θ ∈ [−Θ,Θ] for some 0 ≤ Θ ≤ 0.25. The reason why this techniqueis promising is evident from Figure 2.4: adding up several angles together will result in anenhancement of the center of the PSF, and in a substantial reduction of the artifacts caused bythe tails in the direction x. Let us now analyze this phenomenon analytically.

In a continuous setting, angle compounding corresponds to setting

sacΘ (x) =

12Θ

∫ Θ

−Θ

sθ(x) dθ, x ∈ R2+. (2.12)

Thus, by linearity, the corresponding PSF is given by

gacΘ (x, x′) =

12Θ

∫ Θ

−Θ

gθ(x, x′) dθ, x, x′ ∈ R2+. (2.13)

Let us find a simple expression for gacΘ

. By using (2.8), we may write gacΘ

(x, x′) ≈ gacΘ

(x − x′),where gac

Θis given by gac

Θ(x) = 1

2Θ

∫ Θ

−Θgθ(x) dθ, so that the image may be expressed as

sacΘ (x) = (gac

Θ ∗ n)(x), x ∈ R2+. (2.14)

Thus, in view of the approximation (2.10), we can write

gacΘ (x) = −

iν20F

2Θ

∫ Θ

−Θ

χ(2ν0c−1

0 z)


0 θx sinc(2πν0c−10 Fx) dθ

= −iν20Fχ

(2ν0c−1

0 z)

e2iν0c−10 z sinc(2πν0c−1

0 Fx) sinc(2πν0c−10 Θx).

Therefore, we immediately obtain

gacΘ (x) = g0(x) sinc(2πν0c−1

0 Θx), x ∈ R2. (2.15)

This expression shows that the PSF related to angle compounding is nothing else than the PSFrelated to the single angle imaging with θ = 0 multiplied by sinc(2πν0c−1

0 Θx). Thus, for Θ = 0we recover gθ for θ = 0, as expected. However, for Θ > 0, this PSF enjoys faster decay inthe variable x. See Figure 2.6 for an illustration of gac

Θand gac

Θand a comparison with gθ and

Figure 2.5d for an illustration of the Fourier transform of gacΘ

.

To sum up the main features of the static problem, we have shown that the recovered image maybe written as sac

Θ= gac

Θ∗ n, where gac

Θis the PSF of the imaging system with measurements taken

at multiple angles. The ultrafast imaging technique is based on obtaining many of these imagesover time, as we discuss in the next section.

35


×1013

-8

-6

-4

-2

0

2

4

6

8

(a) The PSF gθ with θ = 0.

×1013

-8

-6

-4

-2

0

2

4

6

8

(b) The PSF gacΘ

with Θ =

0.25.

×1013

-8

-6

-4

-2

0

2

4

6

8

(c) The PSF gacΘ

with Θ = 0.25.

Figure 2.6: A comparison of the PSF related to the single illumination with the PSF associated to multipleangles (with parameters as in (2.1) and (2.2), and F = 0.4). The better focusing in the variablex for gac

Θis evident, as well as the good approximation given by gac

Θ. The size of the square

shown is 2 mm× 2 mm, and the horizontal and vertical axes are the x and z axes, respectively.

2.4 The Dynamic Forward Problem

2.4.1 The quasi-static approximation and the construction of the data

The dynamic imaging setup consists in the repetition of the static imaging method over timeto acquire a collection of images of a medium in motion. We consider a quasi-static model:the whole process of obtaining one image, using the image compounding technique discussedin Subsection 2.3.3, is fast enough to consider the medium static, but collecting several imagesover time gives us a movie of the movement over time. In other words, there are two time scales:the fast one related to the propagation of the wave is considered instantaneous with respect tothe slow one, related to the sequence of the images.

In view of this quasi-static approximation, from now on we neglect the time of the propagationof a single wave to obtain static imaging. The time t considered here is related to the slow timescale. In other words, by (2.14) at fixed time t we obtain a static image s(x, t) of the mediumn = n(x, t), namely

s(x, t) =(gac

Θ ∗ n( · , t))

(x). (2.16)

Repeating the process for t ∈ [0,T ] we obtain the movie s(x, t), which represents the main datawe now need to process. As mentioned in the introduction, our aim is locating the blood vesselswithin the imaged area, by using the fact that s(x, t) will be strongly influenced by movementsin n.

2.4.2 The Doppler effect

Measuring the medium speed is an available criterion to separate different sources; thus, we wantto see the influence on the image of a single particle in movement, as by linearity the obtained

36


conclusions naturally extend to a group of particles. For a single particle, we are interested inobserving the generated Doppler effect in the reconstructed image, namely, peaks in the Fouriertransform away from zero.

Intuitively, Figure 2.5d shows that there is a clear difference in the movements depending on theirorientation. We want to explore this difference in a more precise way. Let us consider n(x, z, t) =

δ(0,vt)(x, z), i.e. a single particle moving in the z direction with velocity v. The resulting image,as a function of time, is obtained via equations (2.15) and (2.16):

s(x, z, t) ≈∫R2

gacΘ (x − x′, z − z′)δ(0,vt)(x′, z′)dx′dz′

= gacΘ (x, z − vt)

= g0(x, z − vt) sinc(2πν0c−10 Θx).

Therefore, arguing as in (2.11), we obtain that the Fourier transform with respect to the timevariable t of the image is given by

Ft(s)(x, z, ξ) ≈∫R

g0(x, z − vt)e−2πiξtdt sinc(2πν0c−10 Θx)

=1v

e−2πi ξzv F2 (g0) (x,−ξ

v) sinc(2πν0c−1

0 Θx),

where F2 is the Fourier transform with respect to the variable z. Adopting approximation (2.10),we obtain

Ft(s)(x, z, ξ)≈−1v

iν20Fe−2πi ξzv sinc(2πν0c−1

0 Θx) sinc(2πν0c−10 Fx)F (χ)

−ξ

2ν0c−10 v− 1

.Given the shape of χ, its Fourier transform has a maximum around 0, thus we can see a peak of|Ft(s)(x, z, ξ)| when ξ is around −2ν0c−1

0 v, and so we have the Doppler effect.

In the case when the particle is moving parallel to the detector array, namely n(x, z, t) = δ(vt,0)(x, z),following an analogous procedure as before, we obtain

s(x, z, t) ≈ g0(x − vt, z) sinc(2πν0c−10 Θ(x − vt)),

and applying the Fourier transform in time yields

Ft(s)(x, z, ξ) ≈1v

e−2πi ξxv F (g0(·, z) sinc(2πν0c−1

0 Θ·))(−ξ

v

).

Using approximation (2.10), the convolution formula for the Fourier transform and the knowntransform of the sinc function, gives

Ft(s)(x, z, ξ) ≈ −ie−2πi ξx

v

4Θvν0c0χ(2ν0c−1

0 z)e4πiν0c−10 z(1[−F,F] ∗ 1[−Θ,Θ])

− ξ

vν0c−10

.

37


The convolution of these characteristic functions evaluated at η is equal to the length of interval[−F + η, F + η] ∩ [−Θ,Θ], because

(1[−F,F] ∗ 1[−Θ,Θ])(η)=

∫R

1[−F,F](η − s)1[−Θ,Θ](s)ds=

∫R

1[−F+η,F+η](s)1[−Θ,Θ](s)ds.

Since both intervals are centered at 0, this value is maximized for η (and thus ξ) around 0, likein the static case, and so the observed Doppler effect is very small.

These differences are fundamental to understand the capabilities of the method for blood flowimaging. This phenomenon will be experimentally verified in Section 2.6.

2.4.3 Multiple scatterer random model

We have seen the effect on the image s(x, z, t) of a single moving particle. We now consider themore realistic case of a medium (either blood vessels or tissue) with a large number of particlesin motion. This will allow to study the statistical properties of the resulting measurements.

We consider a rectangular domain Ω = (−Lx/2, Lx/2) × (0, Lz), which consists in N point par-ticles. Let us denote the location of particle k at time t by ak(t). In the most general case, eachparticle is subject to a dynamics

ak(t) = ϕk (uk, t) , ak(0) = uk, (2.17)

where (uk)k=1,...,N are independent uniform random variables on Ω and (ϕk)k=1,...,N are indepen-dent and identically distributed stochastic flows: for instance, they can be the flows of a stochas-tic differential equation or the deterministic flows of a partial differential equation. Thus, the ak’sare independent and identically distributed stochastic processes. In view of these considerations,we consider the medium given by

n (x, t) =C√

N

N∑k=1

δak(t) (x) , (2.18)

where C > 0 denotes the scattering intensity and 1√N

is the natural normalization factor in viewof the central limit theorem.

To avoid minor issues from boundary effects, which are of no interest to us in the analysisof this problem, we assume the periodicity of the medium. In other words, we consider theperiodization

np(x, t) =∑l∈Z2

n(x + l · L, t), (2.19)

where L = (Lx, Lz). Let g (x) :=∑

l∈Z2 gacΘ

(x + l · L) be the periodic PSF, which is more conve-nient than gac

Θ(given by (2.15)) for a Ω-periodic medium. The dynamic image s is then given

38


by

s(x, t) = (gacΘ ∗ np (·, t)) (x) = (g ∗ n( · , t))(x) =

C√

N

N∑k=1

g (x − ak (t)) .

Let us also assume for the sake of simplicity that, at every time t, ak (t) modulo Ω is a uniformrandom variable on Ω, namely

E∑l∈Z2

w(ak(t) + l · L) = |Ω|−1∫R2

w(y) dy, w ∈ L1(R2). (2.20)

As a simple but quite general example, it is worth noting that in the case when ak(t) = uk + F(t),where F(t) is any random process independent of uk, the above equality is satisfied, since

E∑l∈Z2

w(uk + F(t) + l · L) = |Ω|−1E∑l∈Z2

∫Ω

w(y + F(t) + l · L)dy = |Ω|−1∫R2

w(y) dy,

where the expectation in the first term is taken with respect to uk and F(t), while in the secondterm only with respect to F(t).

We now wish to compute the expectation of the random variables present in the expression fors(x, t). By (2.10) and (2.15), since gac

Θis a derivative of a Schwartz function in the variable z, we

have∫R2 gac

Θ(y)dy = 0. Thus, by (2.20) the expected value may be easily computed as

E (g (x − ak (t))) = E∑l∈Z2

gacΘ (x − ak(t) + l · L) = |Ω|−1

∫R2

gacΘ (y)dy = 0. (2.21)

Let (xi)i=1,...,mx and(t j)

j=1,...,mtbe the sampling locations and times, respectively. The data may

be collected in the Casorati matrix S N ∈ Cmx×mt defined by

S N(i, j) = s(xi, t j).

By (2.21), according to the multivariate central limit theorem, the matrix S N converges in dis-tribution to a Gaussian complex matrix S ∈ Cmx×mt , the distribution of which is entirely deter-mined by the following correlations, for i, i′ = 1, . . . ,mx and j, j′ = 1, . . . ,mt

E(S (i, j)) = 0,

Cov(S (i, j), S (i′, j′)

)= C2E

(g(xi − a1

(t j))

g(xi′ − a1

(t j′

))), (2.22)

Cov(S (i, j), S (i′, j′)

)= C2E

(g(xi − a1

(t j))

g(xi′ − a1

(t j′

))). (2.23)

More precisely, let w ∈ Cmxmt be a column vector containing all the entries of S . Let v ∈ C2mxmt

and V ∈ C2mxmt×2mxmt be defined by

v =(w1,w1,w2,w2, ...,wmxmt ,wmxmt

)T and V = E(vvT

).

39


The covariance matrix V can be easily computed from (2.22) and (2.23). Then the probabilitydensity function f of v can be expressed as [32]:

f (v) =1

πmxmt det (V)12

exp(−

12

v∗V−1v).

Moreover, it is possible to generate samples from this distribution: if X is a complex unit varianceindependent normal random vector, and if

√V is a square root of V , then

√VX is distributed

like v. This allows for simulations of sample image sequences for a large number of particleswith a complexity independent of the number of particles.

The analysis carried out here will allow us to study the distribution of the singular value of thematrix S , depending on the properties of the flows ϕk. This will be the key ingredient to justifythe correct separation of blood and clutter signals by means of the SVD of the measurements.

2.5 The Dynamic Inverse Problem: Source Separation

2.5.1 Formulation of the dynamic inverse problem

As explained in the introduction, the aim of the dynamic inverse problem is blood flow imaging.In other words, we are interested in locating blood vessels, possibly very small, within themedium. The main issue is that the signal s(x, t) is highly corrupted by clutter signal, namely, thesignal scattered from tissues. In the linearized regime we consider, we may write the refractiveindex n as the sum of a clutter component nc and a blood component nb, namely n = nc +

nb. Blood is located only in small vessels in the medium, whereas clutter signal comes fromeverywhere: by (2.3), since blood vessels are smaller than the focusing height, even pixelslocated in blood vessels contain reflections coming from the tissue. Let us denote the locationof blood vessels by Ωb ⊂ Ω. The inverse problem is the following: can we recover Ωb from thedata s(x, t) = sc(x, t) + sb(x, t)? Here, sc and sb are given by (2.16), with n replaced by nc and nb,respectively. In this section, we provide a quantitative analysis of the method described in [42]based on the singular value decomposition (SVD) of s.

2.5.2 The SVD algorithm

We now review the SVD algorithm presented in [42]. The Casorati matrix S ∈ Cmx×mt is definedas in previous section by

S (i, j) = s(xi, t j

), i ∈ 1, ...,mx , j ∈ 1, ...,mt .

40

2.5 The Dynamic Inverse Problem: Source Separation

Without loss of generality, we further assume that mt ≤ mx. We remind the reader that the SVDof S is given by

S =

mt∑k=1

σkukvkT ,

where(u1, ..., umx

)and

(v1, ..., vmt

)are orthonormal bases of Cmx and Cmt , and σ1 ≥ σ2 ≥ ... ≥

σmt ≥ 0. For any K ≥ 1, S K =∑K

k=1 σkukvkT is the best rank K approximation of S in the

Frobenius norm. The SVD is a well-known tool for denoising sequences of images; see forexample [61]. The idea is that since singular values for the clutter signal are quickly decayingafter a certain threshold, the best rank K approximation of S will contain most of the signalcoming from the clutter, provided that K is large enough. This could be used to recover clutterdata, by applying a “denoising” algorithm, and keeping only S K . But it can also be used torecover the blood location, by considering the “power Doppler”

S b,K (i) :=mt∑

k=K+1

σ2k |uk|

2(i) =

mt∑j=1

|(S − S K) (i, j)|2 , i ∈ 1, ...,mx .

As we will show in the following subsection, clutter signal can be well approximated by a low-rank matrix. Therefore, S K will contain most of the clutter signal for K large enough. In thiscase, even if the intensity of total blood reflection is small, S − S K will contain more signalcoming from the blood than from the clutter and therefore high values of S b,K (i) should belocated in blood vessels.

Before presenting the justification of this method, let us briefly provide a heuristic motivationby considering the SVD of the continuous data given by

s(x, t) =

∞∑k=1

σkuk(x)vk(t).

In other words, the dynamic data s is expressed as a sum of spatial components uk movingwith time profiles vk, with weights σk. Therefore, since the tissue movement has higher spatialcoherence than the blood flow, we expect the first factors to contain the clutter signal, and theremainder to provide information about the blood location via the quantity S b,K .

2.5.3 Justification of the SVD in one-dimension

We will assume that the particles of the blood and of the clutter have independent dynamicsdescribed by (2.17)-(2.19). We add the subscripts b and c to indicate the dynamics of blood andclutter, respectively.

In this subsection, using the limit Gaussian model presented in Section 2.4.3, we present thestatistics of the singular values in a simple one-dimensional model. These are useful to under-stand the behavior of SVD filtering. The results of Section 2.4.3 allow to simulate large numberof sample signals s, given that we can compute the covariance matrices (2.22) and (2.23). Since

41


these matrices are very large, we restrict ourselves to the one-dimensional case, so that all sam-pling locations xi are located at x = 0, and are thus characterized by their depth zi. We willtherefore drop all references to x in the following. We also consider very simplified dynamics,which can be thought of as local descriptions of the global dynamics at work in the medium. Letab = a1,b and ac = a1,c be the random variables for the dynamics of blood and clutter particles,respectively, as introduced in (2.17). The dynamics is modelled by a Brownian motion withdrift, namely,

aα (t) = uα + vαt + σαBt, α ∈ b, c .

Here, uα represents the position of the particle at time t = 0, and is uniformly distributed in(0, Lα), where Lb Lc. The deterministic quantity vα is the mean velocity of the particles.In order to take into account the random fluctuations of the particles in movement, we added adiffusion term σαBt, where Bt is a Brownian motion and σ2

α is a diffusion coefficient quantifyingthe variance of the fluctuations of the particle position relative to the mean trajectory. We alsomake the simplifying assumption that the diffusion terms are independent over different particles.More precisely, we have the following conditional expectation and variance:

E (aα (t)| uα) = uα + vαt, Var (aα (t)| uα) = tσ2α.

The difference between clutter and blood dynamics is in the diffusion coefficient: in the case ofclutter, since it is an elastic displacement, σ2

c ≈ 0. For simplicity, from now on we set σc = 0.In the case of blood, which is modelled as a suspension of cells in a fluid, we have σ2

b = σ2 > 0.This coefficient is expressed in m2s−1, and models the random diffusion in a fluid transportingred blood cells due to turbulence in the fluid dynamics and collisions between cells. In practice,σ2 is much larger than the diffusion coefficient of microscopic particles in a static fluid, anddepends on the velocity vb [37]. As for the mean velocities, in the most extreme cases, vb and vc

can be of the same order, even though most of the time vb > vc.

Let S b and S c denote the data matrix constructed in Section 2.4.3, related to blood and cluttersignal, respectively. We now compute the covariance matrix V of S α:

Cov(S α(i, j), S α(i′, j′))

=C2αE

(g(zi − aα

(t j))

g(zi′ − aα

(t j′

)))=

C2α

LE

∫ L

0g(zi − y − vαt j − σαvαBt j

)g(zi′ − y − vαt j′ − σαvαBt j′

)dy

=C2αECgg

(zi − zi′ + vα

(t j′ − t j + σα(Bt j′ − Bt j)

)),

where Cgg (z) = 1L

∫ L0 g(y)g(z + y)dy and Cb and Cc denote the intensity of the blood and clutter

signals, respectively. The expectation operator is taken over all possible positions uα and allpossible drifts Bt j and Bt j′ in the first line, and only over all drifts in the second and third lines.By standard properties of the Brownian motion, Bt j′ − Bt j is Gaussian distributed, of expectedvalue 0 and variance

∣∣∣t j − t j′∣∣∣ and so it has the same distribution as Bt j′−t j . Thus, in the case of

42

2.6 Numerical Experiments

the blood, we can write

Cov(S b(i, j), S b(i′, j′)

)= C2

bECgg(zi − zi′ + vb(t j′ − t j + σbBt j′−t j)

).

Likewise,

Cov(S b(i, j), S b(i′, j′)

)= C2

bECgg(zi − zi′ + vb(t j′ − t j + σbBt j′−t j)

),

where Cgg (z) = 1L

∫ L0 g(y)g(z + y)dy. The tissue model is then given by σc = 0, and is therefore

deterministic given the initial position. Thus

Cov(S c(i, j), S c(i′, j′)

)= C2

cCgg(zi − zi′ + vc

(t j′ − t j

)),

Cov(S c(i, j), S c(i′, j′)

)= C2

cCgg(zi − zi′ + vc

(t j′ − t j

)).

On one hand, in the case of blood, since Cgg and Cgg are oscillating and with very small support(see Figures 2.7a and 2.7b), the integration done when taking the expectation in the blood caseshould yield small correlations as long as |t j′ − t j| is large enough. On the other hand, in the caseof clutter, correlations will be high between the two signals as long as zi−zi′ and vc

(t j − t j′

)are of

the same order and almost cancel out. This heuristic is confirmed by numerical experiments. InFigure 2.7c, we compare the clutter model and the blood model in one dimension: velocities arein the z direction, and we only consider points aligned on the z axis. As we can see, correlationsare quickly decaying as we move away from (0, 0) in the case of blood. In the case of clutter,there are correlations at any times at the corresponding displaced locations.

Once the correlation matrix is computed, we can generate a large number of samples to study thedistribution of the singular values in different cases. In Figure 2.8a, we compare the distributionin the two models (blood and clutter), using the Gaussian limit approximation for the simula-tions, with the same intensity for both models. A comparison with a white noise model with thesame variance shows that blood and noise have approximately the same singular value distribu-tion. On the contrary, the distribution of the singular values of clutter presents a much largertail. A comparison of the distribution of the singular values for the clutter model at differentvelocities shows no real difference in the tail of the distribution (Figure 2.8b).

As a consequence, the clutter signal sc is well approximated by a low-rank matrix, and theblood signal can be thought of as if it were only noise. Therefore, the SVD method act as adenoising algorithm and extracts the clutter signal, according to the discussion in the previoussubsection.


In this section, we consider again a more realistic two-dimensional model, given by (2.17). Thisframework will allow us to simulate generic blood flow imaging sequences from particles. Thedynamics of blood and clutter are modelled as follows. Let us assume that clutter is subject to

43


-1 -0.5 0 0.5 1

#10 -3

-4

-3

-2

-1

0

1

2

3#10 25

(a) The real part of Cgg.

-1 -0.5 0 0.5 1

#10 -3

-3

-2

-1

0

1

2

3

4#10 26

(b) The real part of Cgg.

Clutter: jCov(S(0; 0); S(z; t))j

z (m) #10 -3

-5 0 5

t(s

)

-0.05

0

0.05

#10 26

0

0.5

1

1.5

2

2.5

3

3.5Clutter: jCov(S(0; 0); S(z; t))j

z (m) #10 -3

-5 0 5

t(s

)

-0.05

0

0.05

#10 25

0

1

2

3

4

Blood: jCov(S(0; 0); S(z; t))j

z (m) #10 -3

-5 0 5

t(s

)

-0.05

0

0.05

#10 26

0

0.5

1

1.5

2

2.5

3Blood: jCov(S(0; 0); S(z; t))j

z (m) #10 -3

-5 0 5

t(s

)

-0.05

0

0.05

#10 25

0

1

2

3

4

(c) Absolute values of the correlations in the clutter model (σ = 0, vc = 10−2 m·s−1) and in theblood model (σ2 = 10−6 m2s−1, vb = 10−2 m·s−1).

Figure 2.7: Correlations of the Casorati matrix.

44


#10 14

0 2 4 6 8 10

#10 -14

0

0.5

1

1.5

Clutter

Blood

Noise

(a) The clutter model (σ = 0, vc = 10−2 m·s−1),the blood model (σ2 = 10−6 m2s−1, vb =

10−2 m·s−1) and a white noise model withsame variance as the blood.

#10 14

0 2 4 6 8 10

#10 -14

0

0.5

1

1.5

2

2.5

Clutter (0:5 cm.s!1)

Clutter (1 cm.s!1)

Clutter (2 cm.s!1)

(b) The clutter model with different velocities.

Figure 2.8: The distribution of the singular values of the Casorati matrix S in different cases.

a deterministic and computable flow ϕc. The randomness of the motion of red blood cells invessels is modelled by a stochastic differential equation, given by

dy = vb (t, y) dt + σ(y) dBt, (2.24)

where Bt is a two dimensional Brownian motion and σ is determined by the effective diffusioncoefficient K = 1

2σ2. In blood vessels, this diffusion coefficient is proportional to the product γr2

where γ is the shear stress in the vessel, and r is the radius of red blood cells. As in the previoussection, let ac = a1,c and ab = a1,b. Let ϕb be the flow associated to (2.24). We assume that ϕb

represents the dynamics of blood particles, relative to overall clutter movement, so that

ac (t) = ϕc (uc, t) , ϕc(uc, 0) = uc, (2.25)

andab (t) = ϕc (ϕb (ub, t) , t) , ϕb(ub, 0) = ub. (2.26)

The dynamics of all the other particles are then taken to be independent realizations of thesame dynamics. The velocity field vb and the clutter dynamics ϕc are computed beforehandand correspond to the general blood flow velocity and to an elastic displacement, respectively.In our experiments, we let ϕc be an affine displacement of the medium, changing over time: aglobal affine transformation, with slowly varying translation and shearing applied to the mediumat each frame, namely

ϕc(u, t) =[

1 w1(t)0 1

]u +

[w2(t)w3(t)

],

45


x #10 -3

-2 -1 0 1 2

z

#10 -3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

#10 16

-6

-4

-2

0

2

4

Figure 2.9: Single frame of ultrafast ultrasound (real part).

where wi are smooth and slowly varying (compared to ϕb) functions such that wi(0) = 0. As forthe blood velocity flow vb, it is parallel to the blood vessels, with its intensity decreasing awayfrom the center of the blood vessel [89, Section 11.3]. More precisely, vb is a Poiseuille laminarflow, namely the mean blood flow velocity is half of the maximum velocity, which is the fluidvelocity in the center of the vessel.

The relative blood displacements bk, j = ϕk,b(ub,k, t j

)are computed according to the following

discretization of the stochastic differential equation (2.24):

bk, j+1 = bk, j + δtvb(t j, bk, j

)+√δtσ

(bk, j

)Xk, j + o (δt) ,

where(Xk, j

)are centered independent Gaussian random variables and δt = t j+1 − t j is taken

to be constant. The blood particle positions ak,b(t j)

are then computed simply by applying theprecomputed flow ϕc.

In order to validate the SVD approach, we explore the effects of the blood velocity and of thedirection of the blood vessels on the behavior of the singular values and on the quality of thereconstruction. In each case, the clutter displacement is the same composition of time-varyingshearing and translation, and the mean clutter velocity is 1 cm·s−1. We choose Cc = 5 andCb = 1, for the same density of scatterers from clutter and blood: per unit of area, the clutterintensity is therefore five times higher than the blood intensity. A single frame of ultrafastultrasound imaging is presented in Figure 2.9: it is clear that without further processing, it isimpossible to locate the blood vessels.

In Figure 2.10, the results for various velocities and orientations are presented. The reconstruc-tion intensities are expressed in decibels, relatively to the smallest value in the image. The SVDmethod allows for reconstruction of blood vessels, even if the maximum blood velocity is closeto, or oven lower than, the mean velocity of clutter. We always use the threshold K = 20. Aswe can see, due to the better resolution in the z direction discussed in Section 2.3, vessels ori-ented parallel to the receptor array have a reconstruction with a better resolution. But due to theoscillating behavior of the PSF in the z direction, and the low-pass filter behavior of the PSF

46


-2 0 2

1

2

3

4

0

10

20

30

40

-2 0 2

0

2

4

0

0.5

1

1.5

2

0 20 4010 16

10 18

10 20

ClutterBloodBoth

-2 0 2

1

2

3

4

0

10

20

30

40

-2 0 2

0

2

4

0

0.5

1

1.5

2

0 20 4010 15

10 20

ClutterBloodBoth

(a) Maximum blood velocity: 2 cm·s−1; mean clutter velocity: 1 cm·s−1.

-2 0 2

1

2

3

4

0

10

20

30

-2 0 2

0

2

4

0

0.2

0.4

0.6

0.8

1

0 20 4010 15

10 20

ClutterBloodBoth

-2 0 2

1

2

3

4

0

10

20

30

-2 0 2

0

2

4

0

0.2

0.4

0.6

0.8

1

0 20 4010 15

10 20

ClutterBloodBoth

(b) Maximum blood velocity: 1 cm·s−1; mean clutter velocity: 1 cm·s−1.

-2 0 2

1

2

3

4

0

10

20

30

-2 0 2

0

2

4

0

0.1

0.2

0.3

0.4

0.5

0 20 4010 15

10 20

ClutterBloodBoth

-2 0 2

1

2

3

4

0

10

20

30

-2 0 2

0

2

4

0

0.1

0.2

0.3

0.4

0.5

0 20 4010 15

10 20

ClutterBloodBoth

(c) Maximum blood velocity: 0.5 cm·s−1; mean clutter velocity: 1 cm·s−1.

Figure 2.10: The SVD method for different velocities and orientations. In each case, we have from leftto right: the blood velocity and location, the reconstructed blood location, the decay of thesingular values. The squares are 5 mm × 5 mm, and the horizontal and vertical axes are thex and z axes, respectively. The parameters used are those given in (2.1) and (2.2), F = 0.4and Θ = 7. The density of particles for both blood and clutter is 2,000 per mm2, andσ = 2.5 · 10−5.

47


0 0.02 0.04 0.06 0.08 0.1

-5

0

5×10

15

(a) Flow parallel to the receptor array.0 0.02 0.04 0.06 0.08 0.1

-1

-0.5

0

0.5

1×10

16

(b) Flow perpendicular to the receptor array.

Figure 2.11: Time behavior of a single pixel (real part), located in a constant velocity flow.

-2 0 2

1

2

3

40

10

20

30

-2 0 2

1

2

3

40

10

20

-2 0 2

1

2

3

40

10

20

-2 0 2

1

2

3

40

5

10

15

(a) Flow parallel to the receptor array.

-2 0 2

1

2

3

40

10

20

30

-2 0 2

1

2

3

40

10

20

30

-2 0 2

1

2

3

40

10

20

-2 0 2

1

2

3

40

10

20

(b) Flow perpendicular to the receptor array.

Figure 2.12: Effect of the threshold K on the reconstruction. From left to right: K = 10, 20, 30, 40.

in the x direction, the sensitivity is better for vessels oriented perpendicularly to the receptorarray, and the SVD method is able to reconstruct smaller vessels with lower velocities. Thisfollows from the discussion in Subsection 2.4.2. In order to visualize this phenomenon evenbetter, Figure 2.11 presents the time behavior of a single pixel from the data of Figure 2.10c.We can clearly see the Doppler effect in the case when the flow is perpendicular to the receptorarray, and the low frequency behavior of the signal in the case when it is parallel to the receptorarray.

In Figure 2.12, results of an investigation on the effect of the threshold K on the reconstructionare presented. Except for K, the parameters of Figure 2.10b are used. If the threshold is toolow, the reconstruction is not satisfactory and artefacts appear everywhere in the reconstructedimage. If the threshold is too high, the reconstruction still works but the contrast becomes lower.With our parameters, K = 20 seems to produce the best results.

In order to further validate the method, we consider the impact of measurement noise on therecovery. To this end, we add independent white Gaussian noise to the data, and consider thequality of the reconstruction as a function of the noise intensity. Let us define the contrast ofthe reconstruction as the ratio between the mean intensity of the reconstructed image inside and

48


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1noise level

1

1.5

2

2.5

3

3.5

4

4.5

contr

ast

oriented along xoriented along z

(a) Contrast as a function of noise level

-2 -1 0 1 2

1

2

3

4

0

5

10

15

20

25

30

35

(b) 0% noise-2 -1 0 1 2

1

2

3

4

0

5

10

15

20

(c) 2.5% noise-2 -1 0 1 2

1

2

3

4

0

5

10

15

(d) 5% noise-2 -1 0 1 2

1

2

3

4

0

2

4

6

8

10

(e) 7.5% noise

Figure 2.13: Effect of noise on the reconstruction. The parameters are the same used in Figure 2.10b.

49


outside the blood domain. The parameters of Figure 2.10b are used. Blood intensity is fivetimes lower than clutter intensity, and therefore a noise intensity of 10% corresponds to half theintensity of blood. In Figure 2.13, sample reconstructions at different noise levels are provided.We can conclude that contrast is robust to moderate levels of noise, since blood vessels can stillbe identified up to 7.5% of noise if they are oriented along the z axis, and up to 2.5% of noiseif they are oriented along the x axis. Figure 2.13 also clearly quantifies the better contrast forvessels oriented along the z axis.

2.7 Concluding Remarks

In this chapter, we have provided for the first time a detailed mathematical analysis of ultra-fast ultrasound imaging. By using a random model for the movement of the blood cells, wehave shown that an SVD approach can separate the blood signal from the clutter signal. Ourmodel and results open a door for a mathematical and numerical framework for realizing super-resolution in ultrafast ultrasound imaging by tracking microbubbles [49], as will be seen in thenext chapter. It would be interesting to generalize our approach in this chapter to acousto-opticimaging based on the use of ultrasound plane waves instead of focused ones, which allows toincrease the imaging rate drastically [67].

2.8 Appendix: The Justification of the Approximation of thePSF

This appendix is devoted to the formal justification of the PSF approximation (2.8) which wasobtained by truncating the Taylor expansion of wθ

± at the first order: we shall show here that theerror caused by this truncation is small. For simplicity, we shall consider only the case whenz = z′ and θ = 0: the general case may be tackled in a similar way. Without loss of generality,we may set x′ = 0 and suppose x ≥ 0. We also suppose that we are not too close to the detectors,namely z ≥ 10−2 m. Moreover, in order to be able to be quantitative, we consider the particularcase when F = 0.4 and τ = 1.

The expression of the PSF that we want to approximate is (see (2.7))

g(x) := g0((x, z), (0, z)) =c0

4πx[f ′(w+(x)) − f ′(w−(x))

],

where w±(x) is given by

w±(x) := h0x,x′(x ± Fz) = c−1

0

( √1 + F2z −

√z2 + (x ± Fz)2

).

50

2.8 Appendix: The Justification of the Approximation of the PSF

(Note that, for simplicity of notation, we have removed the dependence of w on θ and z.) Animmediate calculation shows that

w±(0) = 0, w′±(0) =∓c−1

0 F√

1 + F2, w′′±(x) =

−c−10 z2(

(x ± Fz)2 + z2)3/2 .

Hence, there exists ξx ∈ [0, x] such that

w±(x) =∓c−1

0 F√

1 + F2x + cx

x2

2, |cx| = |w′′±(ξx)| ≤ c−1

0 z−1.

Therefore, the absolute error E(x) due to the truncation of the Taylor series of w± at first orderis given by

E(x) = c0(4π)−1 [E+(x) − E−(x)] ,

where

E±(x) =1x

f ′ ∓c−1

0 F√

1 + F2x + cx

x2

2

− f ′ ∓c−1

0 F√

1 + F2x

.We now consider two cases, depending on x. First, consider the case when x > 5 · 10−3 m. Fromthe above calculations we immediately have

|E(x)| ≤ c0(4π)−1 4x

∥∥∥ f ′∥∥∥∞≤

25

c0103ν0 ≤ 3.7 · 1012.

Next, consider the case when x ≤ 5·10−3 m. By using again the mean value theorem we obtain

E±(x) = cxx2

f ′′(θx), θx =∓c−1

0 F√

1 + F2x + δxcx

x2

2

for some δx ∈ [0, 1]. Since | f ′′(t)| is even and decreasing for t > 0, we have that

|E±(x)| ≤ c−10

x2z

∣∣∣∣∣∣∣ f ′′ c−1

0 F√

1 + F2x − c−1

0x2

2z

∣∣∣∣∣∣∣ ,

since the inequality x ≤ 5 · 10−3 m guarantees thatc−1

0 F√

1+F2x − c−1

0x2

2z > 0. Therefore we have

|E(x)| ≤ (4π)−1xz−1

∣∣∣∣∣∣∣ f ′′ c−1

0 F√

1 + F2x − c−1

0x2

2z

∣∣∣∣∣∣∣ .

Let us look at the right hand side of this inequality. As x → 0 the error tends to 0: this isexpected, because of the Taylor expansion around 0. On the other hand, for big x, the value of

| f ′′(c−1

0 F√

1+F2x − c−1

0x2

2z )| is very small, since | f ′′(t)| decays very rapidly for large t. Therefore, themaximum of the right hand side is attained in a point x∗ ∈ (0, 0.005). The value in this point

51


may be explicitly calculated, and we have

|E(x)| ≤ 4 · 1012, 0 ≤ x ≤ 5 · 10−3 m.

To summarize the above derivation, we have shown that the absolute error E(x) is bounded by

|E(x)| ≤ 4 · 1012, x ≥ 0. (2.27)

We now wish to estimate the relative error ‖E‖∞ / ‖g‖∞. In order to do this, let us compute g(0).Since the Taylor expansion becomes exact as x → 0, we may very well compute g(0) by usingthe approximated version. Thus, setting G = F/

√1 + F2 we have

g(0) = limx→0−

c0

4πx

[f ′(c−1

0 Gx) − f ′(−c−10 Gx)

]= lim

x→0−G(4π)−1

f ′(c−10 Gx) − f ′(0)

c−10 Gx

+f ′(−c−1

0 Gx) − f ′(0)

−c−10 Gx

= −2G(4π)−1 f ′′(0),

whence |g(0)| ≥ 8.8 · 1013 by a direct calculation of | f ′′(0)|. Finally, combining this inequalitywith (2.27) allows to bound the relative error by

‖E‖∞‖g‖∞

≤ 5%.

We have proven that the relative error of the approximation obtained by truncating the Taylorexpansions of w± at the first order is less than 5%. This has been proven only in the particularcase when z = z′: the general case may be done by extending the above argument to twodimensions.

52

3 Dynamic spike super-resolution andapplications to Ultrafast ultrasoundimaging

3.1 Introduction

It is well-known that the resolution of any wave imaging method is limited by the diffractionlimit [6]. Super-resolution is understood as any technique whose resolution surpasses this fun-damental limit, which is of order of half the operating wavelength. More precisely, the super-resolution problem can be stated as follows: given low frequency measurements of a medium,typically obtained with a convolution by a low pass filter, reconstruct the original medium witha resolution exceeding the diffraction limit. Since high frequencies are completely lost in themeasuring process, it is impossible to solve this problem in the general case. It is then natural tofocus on the particular case when the medium is made of a finite number of point sources, withunknown locations and intensities. This framework finds applications in many imaging modal-ities, from the pioneering work on super-resolved fluorescence microscopy [29, 45, 57, 58, 91](Nobel Prize in Chemistry 2014 [1]) to the works on super-focusing in locally resonant media[8, 19, 20, 69, 70] and the more recent findings in ultrafast ultrasound localization microscopy[44, 49].

In mathematical terms, each point source is represented as a Dirac delta wiδxi , with unknownlocations xi ∈ X ⊆ Rd and intensities wi ∈ C. We have the sparse spike reconstruction problem:recover

µ =

N∑i=1

wiδxi

from the measurements y = F µ, where

F :M(X)→ Rn (3.1)

is the measurement operator from the set of Radon measuresM(X) defined on X. SinceM(X) isinfinite dimensional, F is not injective, and therefore one has to use regularization to recover µ.The common choice in this context is an infinite dimensional variant of the `1 minimization:

minν∈M(X)

‖ν‖TV subject to F ν = y. (3.2)

53

3 Dynamic spike super-resolution and applications to Ultrafast ultrasound imaging

Mathematical theory on this problem has greatly advanced over the past years. It includes stablereconstruction of spikes with separation in one and multiple dimensions [35, 36], robust recoveryof positive spikes in the case of a Gaussian point spread function with no condition of separation[27], exact reconstruction for positive spikes in a general setting [41] and the correspondingstability properties [43, 81].

Whenever the dynamics of the medium is relevant, as in the case of blood vessel imaging, thesuper-resolution problem for dynamic point reflectors µt arises. In this case, at each time stepone measures F µt and needs to reconstruct both the locations of the spikes and their dynam-ics: we call this problem the dynamic spike super-resolution problem (see Figure 3.1). Thecurrent approach for this problem is to perform a static reconstruction at each time step (usingthe method discussed above), and then to track the spikes to obtain their velocities [49]. Thisapproach suffers from three main drawbacks: first, a lot of data are discarded whenever staticreconstruction cannot be performed because of particles being too close, second, the informa-tion from neighboring frames is ignored in the first step of the reconstruction and, third, trackingalgorithms are computationally expensive.

In this chapter we propose a new method for this dynamical super-resolution problem based ona fully dynamical inversion scheme, in which the spikes’ locations and velocities are simultane-ously reconstructed. After assuming a (local) linear movement of the spikes, we lift the problemto the phase space, in which each spike becomes a particle with location and velocity. Thesuper-resolution problem is then set in this augmented domain, and the minimization performeddirectly with the full dynamical data. We provide a theoretical investigation of this technique(the analysis shares some common aspects with the one presented in [46] for a similar prob-lem), including exact and stable recovery properties, as well as extensive numerical simulations.These simulations show the great potential of this approach for practical applications, far beyondthe predictions of the theory, which we believe can be further developed.

As mentioned above, one of the main motivations and applications of this work is ultrafastultrasound localization microscopy [44, 49]. As shown in Chapter 2, the resolution of ultrafastultrasonography, which is based on the use of planes waves instead of the usual focused waves,is determined by the wavelength of the incident wave, and by other factors such as the lengthof the receptor array and the range of angles used in angle compounding [4]. Due to diffractiontheory, the highest resolution is half a wavelength, which is of the order of 300 µm. Thus, inblood vessel imaging, blood vessels separated by less than 300 µm cannot be distinguished.As in fluorescence microscopy, randomly activated micro-bubbles in the blood may be used toproduce very localized spikes in the observations, giving rise to a dynamical super-resolutionproblem [49], in which both the locations and the velocities of the bubbles are of interest (thevelocities are also used to estimate the thickness of the blood vessels). This is a framework whenour approach can be immediately applied.

This chapter is structured as follows. In Section 3.2 we describe the dynamical super-resolutionproblem and discuss the method introduced in this work: the phase-space lifting. In Sections 3.3and 3.4 we study the exact recovery issue in absence of noise, while in Section 3.5 we prove astability result for noisy measurements. In Section 3.6 we provide several numerical simulationswhich validate the method and the theoretical results. In Section 3.7 we discuss the applications

54

3.2 Setting the stage

of this technique to ultrafast ultrasound localization microscopy. Finally, Section 3.8 containssome concluding remarks and future perspectives.

(a) Position and velocity of the particles: the dots correspond to the positions of particles in the middle frame (t = t0),and the arrows to the displacement of the particles from the first frame to the last one.

t−2 t−1 t0 t1 t2

(b) Corresponding measured sequence ykk.

Figure 3.1: Illustration of the dynamic spike super-resolution problem, with parameters d = 2, N = 2 andK = 2 and F is a convolution operator by a Gaussian point spread function.


3.2.1 The space-velocity model

Let us now introduce our model for super-resolution of dynamic spikes. Instead of consideringa single measure µ, we consider a time-varying measure µt, where t ∈ [−δ, δ], and δ > 0 definesour observation window. Since δ is expected to be small, we can approximate the dynamicsof each point linearly. Therefore we model each point source as a particle displacing with aconstant velocity:

µt =

N∑i=1

wiδxi+vit, t ∈ [−δ, δ],

where vi ∈ Rd. The measurement vector is then composed of uniform samples in the observation

window tk = kτ for k ∈ −K,−K + 1, . . . ,K − 1,K, with K = δ/τ ∈ N (whereN denotes the setof all positive integers):

yk = F µtk , k ∈ [−K,K].

Figure 3.1 illustrates the space-velocity model in two dimensions.

55


In this work, we show that under certain conditions, we are able to recover the positions xi,the velocities vi and the weights wi simultaneously with infinite precision, using a sparse spikerecovery method.

From now on we will assume that the phase space particles, understood as positions xi andvelocities vi, lie inside the domain

Ω =(x, v) ∈ R2d : x + kτv ∈ [0, 1]d, ∀k ∈ [−K,K]

, (3.3)

that is the space-velocity domain in which, for all the considered time samples, the locations ofthe particles stay inside [0, 1]d. Let T = (xi, vi)Ni=1 denote the set of particles. Furthermore, theset of associated weights wi will be considered inK, where K can be either C or R.

The measurement operator F : M([0, 1]d)→ Cn is assumed to be of the form

F ν = (〈ν, ϕl〉)nl=1 , ν ∈ M([0, 1]d),

where ϕl is a family of test functions defined on [0, 1]d and 〈ν, ϕl〉 =∫

[0,1]d ϕl dν. Applying Fto the measures µk := µtk for every time step k ∈ −K, . . . ,K gives

F µk = (〈µk, ϕl〉)nl=1 =

N∑i=1

wiϕl(xi + kτvi)

n

l=1

.

By construction, these measurements are composed of a vector of size n for each time samplek. We now describe how to express these measurements via an operator defined on measures onthe phase space. Consider the measure ω ∈ M(Ω) and the family of test functions ϕl,k ∈ C(Ω)given by

ω =

N∑i=1

wiδxi,vi , ϕl,k(x, v) = ϕl(x + kτv),

and the measurement operator

G :M (Ω)→ Rn×(2K+1), Gλ =(⟨λ, ϕl,k

⟩)l,k . (3.4)

With these objects we can write (F µk)k = Gω, and so the measurements are given by

y = Gω. (3.5)

The dynamical reconstruction problem is now set in the phase space, and consists in the recoveryof the sparse measure ω from the measurements (3.5). Following the approach for sparse spikerecovery, we pose this inversion as a total variation (TV) optimization problem, in which weseek to reconstruct positions and velocities simultaneously by minimizing

minλ∈M(Ω)

‖λ‖TV subject to Gλ = y, (3.6)

56


where the TV norm of a measure λ ∈ M (Ω) is defined as

‖λ‖TV := sup∫

Ω

f dλ : f ∈ C(Ω), ‖ f ‖∞ ≤ 1.

We will call (3.6) the dynamical recovery, whereas (3.2) will be called the static recovery. Theaim of this section is to determine conditions under which exact recovery holds, namely whenω is the unique minimizer of (3.6). In this way, the recovery of the measure ω from the data yreduces to a convex optimization problem.

3.2.2 The perfect low-pass case

Instead of studying the general framework outlined so far, in order to highlight the main featuresof this approach we prefer to focus on the particular case of low-frequency Fourier measure-ments, which represents a simplified model for many different applications. Thus, the theoreticalanalysis discussed below refers only to this situation, even though most parts may be extendedto the general case of a convolution operator.

The low-frequency Fourier measurements are expressed by the complex sinusoids ϕl(x) = e−2πix·l

for l ∈ Zd with ‖l‖∞ ≤ fc, where fc ∈ N is the highest available frequency for the consideredimaging system. The static measurements are given by

(F ν)l =

∫[0,1]d

e−2πix·l dν(x), l ∈ − fc, . . . , fcd .

With dynamical data, for k ∈ −K, . . . ,K, we have ϕl,k(x, v) = e2πi(x+kτv)·l and so

(Gω)l,k =

∫Ω

e−2πi(x+kτv)·l dω(x, v) =

∫Ω

e−2πi(x,v)·(l,kτl) dω(x, v). (3.7)

This expression shows that our data consist of low-frequency samples of 2d-dimensional Fouriermeasurements restricted to the d-dimensional subspaces (ξ, kτξ) ∈ R2d : ξ ∈ Rd for k ∈−K, . . . ,K. Thus, our problem is in principle harder than the one considered in [36], in whichone measures all low-frequency Fourier coefficients, and not only

(l, kτl) : l ∈ Zd, ‖l‖∞ ≤ fc, k = −K, . . . ,K, (3.8)

and different from the one considered in [46], in which the Fourier transform is sampled alonglines. For a visual representation of this restriction in the case d = 1, see Figure 3.2.

57


ξv

ξx

ξv

ξx

Figure 3.2: The allowed frequencies for the full low-frequency case (left) and in our case, when d = 1,fc = 3 and K = 2.

3.3 Exact recovery in absence of noise

3.3.1 Dual certificates

As it is standard in convex optimization, it is useful to consider the dual problem to study theexact recovery for (3.6). In order to do this, we need to introduce the concept of dual certificate.We use the notation sgnKN = η ∈ KN : |ηi| = 1 for every i = 1, . . . ,N.

Definition 3.3.1 (Dual certificate). Let T = (xi, vi)i=1,...,N ⊆ Ω be a configuration of particlesand η ∈ sgnKN . A dual certificate of the dynamical recovery problem (3.6) is a function

q(x, v) =

K∑k=−K

∑‖l‖∞≤ fc

ck,lei2πl·(x+kτv), (3.9)

where ck,l ∈ C, obeying q(xi, vi) = ηi, i ∈ 1, . . . ,N,|q(x, v)| < 1, (x, v) ∈ Ω \ T.

(3.10)

In the following, we shall say that a function of the form (3.9) has dynamical form.

In Figure 3.3, we present an example of a dual certificate for the dynamical recovery problem,which was computed using a predefined kernel as in [36].

The existence of a dual certificate guarantees exact recovery for the minimization problem (3.6).More precisely, we have the following result (for a proof, see [36, Proposition A.1]).

Proposition 3.3.1. Suppose that for every η ∈ sgnKN there exists a dual certificate for thedynamical recovery problem, and let ω be a minimizer of (3.6). Then ω = ω.

58


0 0.2 0.4 0.6 0.8

x

-10

-5

0

5

10

v

-1

-0.5

0

0.5

1

Figure 3.3: Example in d = 1 of a dual certificate for N = 5 particles (inside the red circles) with realweights. The parameters are K = 1, fc = 20 and τ = 0.025. The positions, speeds and weightswere selected at random.

With this proposition in hand, our problem reduces to finding conditions under which dual cer-tificates exist. As mentioned above, the static recovery problem was treated in [36], but theirmethodology cannot be transferred directly to our case since the static dual certificates are con-structed with all low frequency coefficients, whereas in our case we have access only to thefrequencies given by the set (3.8).

The particular structure of functions with dynamical form (3.9) allows for a simple decomposi-tion

q(x, v) =1|K|

∑k∈K

qk(x, v), qk(x, v) =∑‖l‖∞≤ fc

ck,lei2πl·(x+kτv),

where K ⊆ −K, . . . ,K is a subset of the time samples that is used to construct the dual certifi-cate. Observe that the functions qk are constant along the d-dimensional subspaces parallel to(x, v) ∈ R2d : x + kτv = 0. Thus, in principle they can be seen as functions of [0, 1]d ⊆ Rd

instead of Ω ⊆ R2d. More precisely, we write

qk(x, v) = qk(x + kτv), qk(y) =∑‖l‖∞≤ fc

ck,lei2πl·y.

Consider the values of these functions on the location of the particles at each time

γi,k := qk(xi, vi) = qk(xi + kτvi), i ∈ 1, . . . ,N, k ∈ −K, . . . ,K. (3.11)

The functions qk(x, v) are constant along the affine spaces

Li,k := (x, v) ∈ Ω : (x − xi) + kτ(v − vi) = 0,

which contain the particle i. This implies that the constants γi,k propagate along them, namely

qk(x, v) = γi,k, (x, v) ∈ Li,k.

59


← η1=γ1,1+γ1,0+γ1,−1

3•

γ 1,−1γ

1,1γ1,0

x1

v1

x

v

L1,1 L1,−1

L1,0

(a) For a single particle (x1, v1), the re-spective lines L1,k and the values γ1,k

that add up to the value η1 in (x1, v1).

x

v

x1• x2

• x3•

L1,−1 ∩ L2,0 ∩ L3,1©←

γ1,−1+γ2,0+γ3,13

©

γ1,0γ 1,−

1γ1,1

γ3,0γ 3,−

1γ3,1

γ2,0

γ2,−1γ 2,1

(b) A configuration of three static par-ticles, located in (x1, 0), (x2, 0) and(x3, 0), with the values γi,k propagat-ing along the lines Li,k.

Figure 3.4: The geometries of the problem for some simple configurations in one dimension (d = 1) withthree time measurements (K = 1) and K = −1, 0, 1

.

In other words, the values of the dual certificate on Li,k are completely determined by γi,k. More-over, by (3.10), these values must satisfy the conditions

1|K|

∑k∈K

γi,k = ηi, i ∈ 1, . . . ,N. (3.12)

In Figure 3.4 we present an example in d = 1, with three static particles (N = 3) and three timemeasurements (K = 1). The values of q are fixed by γi,k on each line Li,k, and in particular intheir points of intersection. As we shall see below, the problematic points are those where severallines (or d-dimensional affine subspaces) intersect, as in the two circled dots in the figure.

A natural way to build dual certificates for the dynamical problem is to consider dual certificatesfor the static problems at each time step in K and then to average them. In particular, we makethe choice γi,k = ηi for every i and k. More precisely, we have the following definition.

Definition 3.3.2 (Static average certificate). Take T = (xi, vi)i=1,...N ⊆ Ω. Let η ∈ sgnKN

and K ⊆ −K, . . . ,K with |K| ≥ 3. Assume that for every k ∈ K there exists a static dualcertificate, i.e. there exists qk(x) =

∑‖l‖∞≤ fc ck,lei2πl·x such that

qk(xi + kτvi) = ηi, i ∈ 1, . . . ,N,|qk(y)| < 1, y ∈ [0, 1]d \ (xi + kτvi)Ni=1.

(3.13a)

We call the function q(x, v) defined as

q(x, v) =1|K|

∑k∈K

qk(x, v) =1|K|

∑k∈K

qk(x + kτv) (3.13b)

a K-static average certificate.

60


If static dual certificates exist for every time sample k ∈ K , we can immediately build a staticaverage certificate q(x, v) by using (3.13b). By construction, it satisfies

q(xi, vi) = ηi, i ∈ 1, . . . ,N,|q(x, v)| ≤ 1, (x, v) ∈ Ω \ T.

(3.14)

This function almost satisfies (3.10), except that it may happen that |q(x, v)| = 1 for some (x, v) ∈Ω \ T . Take as example the configuration of points given in Figure 3.4b with η1 = η2 = η3 = 1and K = −1, 0, 1. The static average certificate will value 1 in each of the particles and byconstruction γi,k = 1. As a consequence, q will have value 1 also in the circled points, in which|K| = 3 lines Li,k intersect, hence breaking condition (3.10).

In order to ensure that the configuration of particles (xi, vi)i admits a dual certificate, we char-acterize these conflictive points.

Definition 3.3.3 (Ghost particles). Let (xi, vi)i=1,...,N ⊆ Ω be a configuration of particles andK = k1, . . . , km be m = |K| time samples. A point (g,w) ∈ Ω is a ghost particle if there exists aset of different indexes i1, . . . , im ∈ 1, . . . ,N such that

m⋂p=1

Lip,kp = (g,w).

To give the intuition behind the definition of ghost particles, recall that the elements of Ω repre-sent the trajectories of moving objects in [0, 1]d: given a time k, a particle (x, v) ∈ Ω describesan object located in x + kτv. From this we notice that the set Li,k represents all possible mov-ing objects in [0, 1]d that at time k would be placed at the same location as the particle i, sincex +τkv = xi +τkvi. Therefore, ghost particles can be understood as possible objects that at everytime sample share their location with a given particle. In the example presented in Figure 3.4b,the highlighted ghost point on top of the particles represents an object moving from left to right,that for k = −1, 0, and 1, would be located in x1, x2 and x3 respectively.

3.3.2 Main result

We are now ready to state the main result of this section. Several comments on the assumptionsare given after the proof.

Theorem 3.3.2. Let T = (xi, vi)i=1,...,N be a configuration of N particles, w ∈ KN and K ⊆−K, . . . ,K be such that |K| ≥ 3. Let

ω =

N∑i=1

wiδ(xi,vi) ∈ M(Ω)

be the unknown measure to be recovered. Assume that:

61


(1) for every k ∈ K and η ∈ sgnKN there exists a static dual certificate qk(x) satisfying(3.13a);

(2) and the configuration does not admit ghost particles.

Then ω is the unique solution of the dynamical recovery problem (3.6), whereG is given by (3.7).

Proof. By Proposition 3.3.1, it is sufficient to construct a dual certificate for the dynamicalrecovery problem. Thanks to assumption (1), for every η ∈ sgnKN we can build a K-staticaverage certificate q(x, v). By (3.14) and assumption (2) it is enough to prove that

|q(x, v)| = 1 =⇒ (x, v) ∈ T ∪G,

where G is the set of all the ghost particles of the configuration.

Note that for k1 , k2 and any two i, j ∈ 1, . . . ,N, the set Li,k1

⋂L j,k2 has at most 1 element. In

particular, we notice that Li,k1

⋂Li,k2 = (xi, vi). This observation will be useful below.

Suppose |q(x, v)| = 1. Since q(x, v) is defined as an average of terms qk(x, v), where each ofthem satisfies |qk(x, v)| ≤ 1, a necessary condition for |q(x, v)| = 1 is that for every k ∈ Kwe have |qk(x + kτv)| = |qk(x, v)| = 1. By definition that happens exclusively if for everyk ∈ K , (x, v) ∈ Li,k for some i ∈ 1, . . . ,N. In other words, there exists a family of indexesik ∈ 1, . . . ,N such that

(x, v) =⋂k∈K

Lik ,k.

There are two cases: if some of these indexes repeat (i.e. ik1 = ik2 , for k1 , k2), then we know that(x, v) must be equal to particle ik1 . In the case none of the indexes ik repeats, then by definition(x, v) is a ghost particle.

3.3.3 Comments on the hypotheses of Theorem 3.3.2

Let us now comment on assumptions (1) and (2), and show why these are easily satisfied. Letus start from assumption (1), namely the existence of static dual certificates.

Remark 3.3.3. Take k ∈ −K, . . . ,K. There exists a static dual certificate qk(x) satisfying(3.13a) in any of the following situations.

(a) The particles at time step kτ are sufficiently separated, namely∥∥∥(xi + kτvi) − (x j + kτv j)∥∥∥∞≥

Cd

fc, i , j, (3.15)

and fc ≥ C′d, where Cd,C′d > 0 are constants depending only on the dimension1 [36].

1Several bounds are known for these constants, for instance Cd = 2 if d = 1 and K = C, Cd = 1.87 if d = 1 andK = R, and Cd = 2.38 if d = 2 and K = R. More precise estimates may be derived, which yield slightly betterconstants.

62


(b) The weights wi are all positive and the particles at time step kτ are divided into groups,and within each group a minimum separation condition like (3.15) is satisfied [77] (thisholds in the discrete setting).

(c) The weights wi are all positive, d = 1 and fc ≥ 2N [41]. (It is remarkable that in this caseno minimum separation condition is required.)

There are reasons to believe that (c) should be sufficient also if d > 1 [81], but as far as we areaware a rigorous proof of this fact is still missing.

Let us now turn to assumption (2), and show that it is satisfied almost surely if the particles(xi, vi) are chosen uniformly at random.

Proposition 3.3.4. Assume |K| ≥ 3. Let (xi, vi)Ni=1 be independent random variables, drawnfrom absolutely continuous distributions µi supported in Ω. Then almost surely there are noghost points.

Proof. To simplify the notation denote Pi = (xi, vi) ∈ Ω and TN = PiNi=1. Set m = |K|. Let

G(Tn) denote the set of ghost particles of a configuration Tn of n particles. We now definethe potential ghost particles G(Tn−1) of a configuration Tn−1 of n − 1 particles. We say that(g,w) ∈ G(Tn−1) if there exists a particle Pn ∈ Ω such that (g,w) ∈ G(Tn−1 ∪ Pn), namely

G(Tn−1) :=⋃

Pn∈Ω

G(Tn−1 ∪ Pn).

By definition of ghost points, we have that if (g,w) ∈ G(Tn−1) then there exist m−1 distinct timesamples k1, . . . , km−1 ∈ K and m − 1 distinct particles Pi1 , . . . , Pim−1 ∈ Tn−1 such that

(g,w) ∈m−1⋂p=1

Lip,kp .

Notice that G(Tn−1) is always finite. This stems from the fact that for any k1 , k2 in K , theintersection of the sets Li,k1 , L j,k2 are singletons for any i, j ∈ 1, . . . , n. Since m ≥ 3 and Tn−1and K are finite sets, we have a finite number of sets Li,k to intersect, leading to a finite numberof elements in G(Tn−1).

Now we prove that, for any configuration of particles Tn−1 such that there are no ghost points,the set of new particles Pn that would generate a ghost point has zero measure. More precisely,if G(Tn−1) = ∅ then

µn (Pn ∈ Ω : G(Tn−1 ∪ Pn) , ∅) = 0. (3.16)

In order to prove this, notice that if G(Tn−1) = ∅ we have

G(Tn−1 ∪ Pn) ⊆ (g,w) ∈ G(Tn−1) : ∃k ∈ K , (g,w) ∈ Ln,k.

63


Thus, since (g,w) ∈ Ln,k means (g − xn) + τk(w − vn) = 0 we obtain

Pn ∈ Ω : G(Tn−1 ∪ Pn) , ∅ ⊆⋃

(g,w)∈G(Tn−1)

⋃k∈K

Pn ∈ Ω : xn + τkvn = g + τkw.

Since this is a finite union of affine subspaces of dimension d, it has zero Lebesgue measure. Bythe absolute continuity of µn, we derive (3.16).

For n ∈ 2, . . . ,N, fix a configuration of Tn−1 particles. Denoting dPi = dxidvi, fi =dµidPi

and1S = 1 if S is true and 1S = 0 if S is false, we have∫

Ω

1G(Tn),∅ fn(Pn) dPn =

∫Ω

(1G(Tn−1),∅ + 1G(Tn−1)=∅1G(Tn),∅

)fn(Pn)dPn

= 1G(Tn−1),∅,

where in the last equality we used (3.16) and that∫Ω

fn(Pn)dPn = 1. Using this property N−m+1times for n = N,N − 1, . . . ,m and setting µ = ⊗N

i=1µi, we obtain

µ((P1, . . . , PN) : G(TN) , ∅) =

∫Ω

· · ·

∫Ω

1G(TN ),∅

N∏i=1

fi(Pi)dP1 . . . dPN

=

∫Ω

· · ·

∫Ω

1G(TN−1),∅

N−1∏i=1

fi(Pi)dP1 . . . dPN−1

= · · ·

=

∫Ω

· · ·

∫Ω

1G(Tm−1),∅

m−1∏i=1

fi(Pi)dP1 . . . dPm−1

= 0,

where the last equality follows from 1G(Tm−1),∅ = 0, since there cannot be any ghost particles ifthere are more time samples than particles.

On the other hand, if ghost particles do arise, the conclusion of Theorem 3.3.2 may not be true,even if a minimum separation condition is satisfied at all time steps (so that assumption (1) issatisfied). Even though the probability of a random configuration of particles to produce ghostparticles is zero, it is worth considering it since the stability of the problem will deteriorate fornearby configurations (see Section 3.5).

In Figure 3.5 we provide an example of this case, with three time measurements (K = 1), threeparticles P1, P2 and P3 and three ghost particles G1, G2 and G3. The configuration is constructedin such a way that, at each time step, the positions of the ghost particles coincide with those ofthe physical particles, thereby producing the same measurements. In other words, we haveG(

∑i δPi) = G(

∑i δGi) and

∥∥∥∑i δPi

∥∥∥TV =

∥∥∥∑i δGi

∥∥∥TV , and so the minimization problem (3.6) has

multiple solutions.

64


•P1

•P2

•P3

G1

G2 G3

v

x

• Particle Ghost particle

k = 1

k = 0

k = −1•P3

•P1

•P2

G2

G3

G1

•P3

•P1

•P2

G1

G2

G3

•P3

•P1

•P2

G3

G1

G2

Figure 3.5: In the case of three time measurements, a configuration of points and speeds that allow mul-tiple reconstructions. The lines in the left diagram represent Li,k for each time sample k andparticle i. On the right hand side we can observe the relative position of each particle at eachtime step.

In the following result, we generalize this observation to more general configurations.

Proposition 3.3.5. Take w ∈ Rm+ , with m = |K| ≥ 3, and let (xi, vi)mi=1 ⊆ Ω be a con-

figuration of m distinct particles admitting m distinct ghost particles (g j,w j)mj=1 ⊆ Ω. Letω =

∑mi=1 wiδ(xi,vi). Suppose that for every k ∈ K and every i ∈ 1, . . . ,m there exists a unique

ghost particle (g j,w j) in the affine space Li,k, i.e.

g j − xi + kτ(w j − vi) = 0.

Then the minimization problem in (3.6) admits infinitely many solutions.

Proof. Consider the measure

h =

m∑i=1

δ(xi,vi) −

m∑j=1

δ(g j,w j) ∈ M(Ω).

First notice that h , 0. If we had h = 0, each particle would be a ghost particle. By definition ofghost particle, we would have (g,w) ∈ Li1,k1 ∩ Li2,k2 for some k1 , k2 in K , and so

(g,w), (xi1 , vi1) ∈ Li1,k1 , (g,w), (xi2 , vi2) ∈ Li2,k2 .

Since (xi1 , vi1) and (xi2 , vi2) are two different ghost particles, either Li1,k1 or Li2,k2 would containtwo ghost particles, contradicting the hypotheses.

The measure h is undetectable, since it belongs to the kernel of the operator G defined in (3.4),

65


as we now show. We readily compute

(Gh)l,k =⟨h, ϕl,k

⟩=

m∑i=1

ϕl(xi + kτvi) −m∑

j=1

ϕl(g j + kτη j) = 0.

Indeed, by our hypothesis on the ghost particles, we have that at every time sample k, for everyposition xi +kτvi ∈ Ω there exists only one ghost point such that g j +kτw j = xi +kτvi. Therefore,each term of the first sum cancels out with one term of the second sum, as desired.

For β ∈ [0,mini wi], consider the measure

ωβ = ω − βh =

m∑i=1

(wi − β)δ(xi,vi) + β

m∑j=1

δ(g j,w j).

Since Gωβ = Gω and∥∥∥ωβ∥∥∥TV =

∑i(wi − β) +

∑j β =

∑i wi = ‖ω‖TV , we obtain that ωβ is a

solution to (3.6) for every β ∈ [0,mini wi].

It is worth observing that the non-uniqueness of solutions arises also if there exists a subset ofthe particles satisfying the conditions of Proposition 3.3.5.

3.4 Other constructions of dynamical dual certificates

In this section we show that the construction of dynamical dual certificates as static average cer-tificates, although natural and efficient, is not the end of the story. In other words, exact recoverymay be guaranteed even if assumptions (1) and (2) of Theorem 3.3.2 are not satisfied. More pre-cisely, we provide alternative constructions of dynamical dual certificates for configurations thateither do not allow static dual certificates (assumption (1)) or have ghost particles (assumption(2)). In particular, the first case shows an advantage of our space-velocity model over applyingstatic reconstructions at each time sample. This aspect will also be investigated in Section 3.6below.

3.4.1 Dual certificates with no static separation condition

The following example of dual certificate is purely numerical, but shows the possibility of con-structing a dual certificate in cases in which static reconstructions are expected to fail.

The chosen configuration is presented in Figure 3.6a, where we consider the one-dimensionalcase (d = 1), five time measurements (K = 2), two static particles barely separated enough toallow a reconstruction and a third moving particle. We give positive weights wi to the staticparticles, and a negative weight to the moving one. Since this third particle is at each timemeasurement close to another particle and has a different sign, it is not possible to localize it with

66

3.4 Other constructions of dynamical dual certificates

x• •

1/ fc

←k

=−2

←k

=−1

←k

=0

←k

=1

←

k=

2

• Static particles Moving particle

(a) This diagram represents two static par-ticles and a moving particle at eachtime sample.

-1

-0.5

0

0.5

1

Static particles

Moving particle

(b) A valid dual certificate of the configu-ration, visualized in the space-velocityplane.

Figure 3.6: An example of a dynamical dual certificate whithout a static minimum separation condition.

a static reconstruction at any point. We recall that fc is the maximum imaging frequency, and 12 fc

is far below the optimal minimum separation distance (see Remark 3.3.3 and [36, Section 5]).

In Figure 3.6b we can see a dual certificate for this configuration (with fc = 20). To obtainthis dual certificate, we followed the construction done in [36] for the two dimensional case, butconsidering only the frequencies available in our setting given by the set (3.8).

3.4.2 Dual certificates in presence of ghost particles

As we saw in Figure 3.4b, in the presence of ghost points the static average dual certificate con-structed in (3.13) is not a valid dynamical dual certificate if the values ηi have a constant sign.Indeed, the static average dual certificate will have absolute value equal to 1 in the ghost parti-cles, since the values γi,k on Li,k are simply set to ηi so that 1

|K|

∑k∈K γi,k = ηi (see Figure 3.7a).

However, by making a slightly different choice for γi,k it is possible to have simultaneouslyq(xi, vi) = ηi and |q(g,w)| < 1 on every ghost particle, thereby obtaining a valid dual certificate(see Figure 3.7b).

This construction is formalized in the following result.

Proposition 3.4.1. Take d = 1, fc ≥ 128, K = 1 and ∆x ∈ [1.87/fc, 1). Consider 3 particles withlocations (−∆x, 0), (0, 0) and (∆x, 0). Then for every η ∈ sgnK3 there exists a dual certificate.

Proof. If η is not a constant vector, the static average certificate (whose existence is guaranteedby Remark 3.3.3a) given by (3.13) is a valid dual certificate, and the result is trivial. Withoutloss of generality, suppose now that η = (1, 1, 1). Set x−1 = −∆x, x0 = 0 and x1 = ∆x.

67


-0.2

0

0.2

0.4

0.6

0.8

1

Particles

Ghost part.

(a) The static average dual certificate (cor-responding to ε = 0) is not a valid dualcertificate since it has value 1 in thetwo ghost particles.

-0.2

0

0.2

0.4

0.6

0.8

1

Particles

Ghost part.

(b) The valid dual certificate constructedas a perturbation of the static averagedual certificate by taking ε = 0.08.

Figure 3.7: Dual certificates for the configuration of particles of Figure 3.4b.

We construct the dual certificate q(x, v) using the notation of §3.3.1 with K = −1, 0, 1. Moreprecisely, we write

q(x, v) =13

1∑k=−1

qk(x, v) =13

1∑k=−1

qk(x + kτv),

where the functions qk are constructed as follows. The quantities γi,k defined in (3.11) need tosatisfy (3.12), but also to keep the absolute value of q below 1 in the ghost particles. In view ofthe geometrical configuration (see Figure 3.4b), these conditions are:

γi,−1 + γi,0 + γi,1 = 3, i ∈ 1, 2, 3,

|γ1,1 + γ2,0 + γ3,−1| < 3,

|γ1,−1 + γ2,0 + γ3,1| < 3.

To take a family of solutions, for ε ∈ (0, 1) set

γi,−1 = γi,1 = 1 − ε, i ∈ 1, 3,

γi,0 = 1 + 2ε, i ∈ 1, 3,

γ2,−1 = γ2,0 = γ2,1 = 1.

For each k ∈ K , let qk(x) =∑ fc

l=− fcck,le2πilx be the low frequency trigonometric polynomial such

thatqk(xi) = γi,k, q′k(xi) = 0, i = −1, 0, 1,

constructed in [36], whose existence follows from ∆x ≥ 1.87fc

(the construction works also if|γi,k| , 1).

The final step is to ensure that q(x, v) =∑

qk(x, v) satisfies |q(x, v)| < 1 for every (x, v) ∈ Ω that

68

3.5 Stable reconstruction with noise

is not a particle. When ε = 0, qk is strictly concave in xi for every i [36]. Furthermore, the mapε 7→ qk is affine, hence every derivative of qk is continuous in ε. Therefore, since q′k(xi) = 0,the local concavity of qk is preserved for ε small and, consequently, the local maxima of qk

are the interpolation points xi. This is enough to prove that q(x, v) attains local maxima in theparticles (xi, 0) and in the ghost particles (0,∆x) and (0,−∆x). Finally, by continuity of q(x, v)with respect to ε, there exists an ε sufficiently small such that |q(x, v)| < 1 for every element inΩ that is not a particle.

The methodology presented in the proof of Proposition 3.4.1 can be applied to more generalexamples and it can be iterated to deal with the ghost particles one by one. The only requirementis to have a direction in which there are no ghost particles, so it can be used as a “sink of mass”.In the example considered, these directions are described by L1,0, L3,0, L2,−1 and L2,1.


Following the analysis done in [36] for the static case, we present a stability result for the dynam-ical problem. We will review their setting and adapt it to our case under the same hypotheses:one dimensional case (d = 1), discrete setting and a specific type of noise model with boundedtotal variation norm.

Let us consider a discrete grid Ω# with space width ∆x > 0 and velocity width ∆v > 0, namely

Ω# = ((∆x ·Z) × (∆v ·Z)) ∩Ω,

and we will assume throughout this subsection that our particles are located on the grid, i.e.T ⊆ Ω# and ω ∈ M (Ω#). We also recall from [36] the super-resolution factor in space:

S RFx =1

∆x fc,

which can be understood as the ratio between the desired space resolution and the permittedresolution given by the diffraction limit.

We consider, instead of (3.5), the following input noise model:

y = G(ω + z), ‖PGz‖TV ≤ δ,

where z ∈ M (Ω#) and PG = ∆xG∗G stands for the projection over the frequencies described

by (3.8). The reason of this projection is that all the other frequencies are filtered out by themeasurement process of G. We consider a relaxed version of the noiseless problem (3.6)

minλ∈M(Ω#)

‖λ‖TV subject to ‖Gλ − y‖1 ≤δ

∆x‖G∗‖`1→M

, (3.17)

where ‖G∗‖`1→M is the operator norm of G∗.

69


By strengthening the hypotheses of Theorem 3.3.2, we obtain the following stability result.

Theorem 3.5.1. Let fc ≥ 128, d = 1 and ∆x,∆v > 0 be such that

∆2x ≤

K(K + 1)3

τ2∆2v . (3.18)

Let T = (xi, vi)i=1,...,N ⊆ Ω# be a configuration of N particles, w ∈ KN and K = −K, . . . ,K.Let ω =

∑Ni=1 wiδ(xi,vi) ∈ M(Ω) be the unknown measure to be recovered.

Suppose that

(i) for every k ∈ K , the minimum separation condition (3.15) holds;

(ii) and for all (x, v) ∈ Ω \⋃N

i=1 Bi we have

12K + 1

K∑k=−K

min

mini∈1,...,N

|xi − x + kτ(vi − v)|2,0.16492

f 2c

≥ ∆2

x, (3.19)

where Bi = (x, v) ∈ Ω : |x − xi| + Kτ|v − vi| < 0.1649/ fc are neighborhoods of each par-ticle in T .

Let ω be a minimizer of (3.17). Then we have the following stability bound for the error:

‖ω − ω‖1 ≤ C(SRFx)2δ

for some absolute constant C > 0.

Remark 3.5.2. The stability condition (3.19) can be understood as an extension of a conditionto prevent ghost particles. As we can notice, if (x, v) is a ghost particle, the sum in the left-handside values 0.

Proof. Take η ∈ KN with |ηi| = 1. Let q(x, v) be the static average dual certificate given in(3.13), where the each static dual certificate qk is constructed as in [36] for every k ∈ K , thanksto assumption (i). With an abuse of notation, set q = qi, j = q (i∆x, j∆v). Let PT denote theprojection onto the space of vectors supported on T , namely, (PT q)i, j = qi, j if (i∆x, j∆v) ∈ T and0 otherwise.

In view of [36, Theorem 1.5], using the same proof we have the following stability bound forthe error:

‖ω − ω‖1 ≤4δ

1 − ‖PT cq‖∞. (3.20)

It remains to bound ‖PT cq‖∞ in our discrete grid, namely the values of qi, j outside the set ofparticles T . In order to do so, we shall use the following estimates from [36, Lemma 2.5]:

|qk(t)| ≤ 1 −C1 f 2c (t − (xi + kτvi))2, when |t − (xi + kτvi)| ≤ C2/ fc, (3.21)

70


and|qk(t)| ≤ 1 −C1C2

2, when min(xi,vi)∈T

|t − (xi + kτvi)| > C2/ fc, (3.22)

where C1 = 0.3353 and C2 = 0.1649.

Take (x#, v#) ∈ Ω# \ T , we want to bound the maximum of |q(x#, v#)|. We deal with two cases:when (x#, v#) is close to some particle, i.e. (x#, v#) ∈ Bi for some i, or when it is not.

Fix i such that (x#, v#) ∈ Bi, then we can write (x#, v#) = (xi + nx∆x, vi + nv∆v) with (nx, nv) ∈Z × Z \ (0, 0). From the definition of Bi, we have that we can use estimate (3.21) for everyk ∈ K . Therefore

|q(x#, v#)| =1

2K + 1

∣∣∣∣∣∣∣K∑

k=−K

qk(xi + kτvi + (nx∆x + kτnv∆v)

)∣∣∣∣∣∣∣≤

12K + 1

K∑k=−K

(1 −C1 f 2

c (nx∆x + kτnv∆v)2)

=1

2K + 1

K∑k=−K

(1 −C1 f 2

c

(n2

x∆2x + 2kτnxnv∆x∆v + k2τ2n2

v∆2v

))= 1 −

C1 f 2c

2K + 1

(((2K + 1)n2

x∆2x +

K(K + 1)(2K + 1)3

τ2n2v∆2

v

))≤ 1 −C1 f 2

c ∆2x(n2

x + n2v)

≤ 1 −C1 f 2c ∆2

x,

where we used (3.18) and that n2x + n2

v ≥ 1.

In the case where (x#, v#) < Bi for every i, we take for each k ∈ K , ik = arg mini∈1,...,N |xi − x# +

kτ(vi − v#)| and Ik = xik − x# + kτ(vik − v#). Hence, by (3.22) we have

|q(x#, v#)| =1

2K + 1

∣∣∣∣∣∣∣K∑

k=−K

qk(xik + kτvik − Ik)

∣∣∣∣∣∣∣≤ 1 −

C1 f 2c

2K + 1

K∑k=−K

I2k 1(−∞,0]

(|Ik| −

C2

fc

)+

C22

f 2c1(0,+∞)

(|Ik| −

C2

fc

)≤ 1 −

C1 f 2c

2K + 1

K∑k=−K

min(I2

k ,C22/ f 2

c)

≤ 1 −C1 f 2c ∆2

x,

where the last inequality is precisely the stability condition (3.19).

Therefore we have that ‖PT cq‖ ≤ 1 −C1 f 2c ∆2

x ≤ 1 −C/SRF2x, for some absolute constant C > 0.

By inserting this bound into (3.20), we conclude the proof.

71


3.6 Numerical simulations

3.6.1 Methods

Solving minimization problem (3.6) in all its generality is not an easy task, since it is nonlinearand infinite dimensional. It is possible to use an analogue discrete problem, where the loca-tions and velocities are fixed on a grid whose size determines the resolution we want to obtain.However, this methods becomes intractable for a fine resolution.

In this work, we seek to validate our approach using a reconstruction method in the continuum.In [36], an algorithm with solutions in the continuum is presented in the one dimensional case,but we require a method for higher dimensions. In [33], the authors develop an algorithm to solvethe following problem for any linear operator F from the space of positive punctual measures toRn:

minµ‖F µ − Y‖22 subject to ‖µ‖TV ≤ M. (3.23)

Albeit the proposed algorithm is limited to positive weights, this is a realistic expectation inthe case of many physical signals, for instance those generated by micro-bubbles in ultrafastultrasound imaging. Clearly, a minimizer of (3.23) will be a minimizer of (3.6) provided thatthe initial total variation is known.

Lemma 3.6.1. Assume that µ is the unique solution of (3.6). Then µ is the unique solution of(3.23) with M = ‖µ‖TV .

Proof. Since (3.6) admits a unique minimizer, every µ , µ such that F µ = Y verifies ‖µ‖TV >

M. Therefore, µ is the unique minimizer of (3.23).

The codes of the simulations of this chapter are available at https://github.com/panchoop/dynamic_spike_super_resolution.

3.6.2 The measurements

We consider the perfect low-pass filter described in Section 3.2.2 with d = 1 as forward mea-surement operator, where the measured Fourier frequencies are − fc, . . . , fc for some fc ∈ Nand the number of time samples are 2K + 1 with sampling rate τ. The considered parameters forthe simulations are:

fc = 20, K = 2, τ = 0.5. (3.24)

Each simulation contains a random number of particles, between 4 and 10. Furthermore eachparticle is generated uniformly in Ω and has an associated random weight taken uniformly be-tween 0.9 and 1.1. In total, we made 18, 000 simulations.

In order to validate our dynamic spike super-resolution approach, we compare dynamical andstatic reconstructions and study the stability of our setting. Dynamical reconstruction refers to

72

https://github.com/panchoop/dynamic_spike_super_resolution

https://github.com/panchoop/dynamic_spike_super_resolution


the recovery of positions and velocities in the space-velocity domain Ω ⊂ R2, whereas by staticreconstruction we mean the recovery of the positions of the particles in [0, 1] at some fixedtime.

Since we are simulating and reconstructing particles’ locations and velocities in the continuum,there are natural associated numerical errors and we require a criterion to call a reconstructioneither a success or a failure. Let ∆x,∆v,∆w > 0 be the maximal accepted errors in the recon-struction of position, velocity and weight, respectively. More precisely, if we consider a singleparticle wiδ(xi,vi) and its associated reconstruction wiδ(xi,vi), then we say that the particle wassuccessfully reconstructed if

|xi − xi| ≤ ∆x, |vi − vi| ≤ ∆v, |wi − wi| ≤ ∆w.

We consider a configuration successfully reconstructed if, for each particle, there exists a uniquesuccessfully reconstructed particle and there are no additional reconstructed particles. Thesecriteria are analogue for static reconstructions.

Notice that in the case of space and velocity, the error values scale with respect to the maximalimaging frequency ( fc in space and K fcτ in velocity), thus it is natural to consider the super-resolution factors

SRFx =1fc

1∆x

and SRFv =1

fcKτ1∆v,

which will be set appropriately in each simulation.

Let us introduce a measure of separation of a configuration of particles T = (xi, vi)i ⊆ Ω:

∆dyn(T ) =3

maxk∈−K,...,K

mini, j|xi − x j + τk(vi − v j)|, (3.25)

where3

max is the third highest element of the set. The quantity ∆dyn represents condition (1) ofTheorem 3.3.2, for any subset of measurementsK ⊆ −K, . . . ,Kwith |K| = 3. We will evaluatethe simulated reconstructions against this measure of separation, which will be scaled by 1

fc, as

the theoretical allowed resolution depends on this value (see Remark 3.3.3a). Condition (2) ofTheorem 3.3.2 on the absence of ghost particles could also be considered, but a formula for thatpurpose is extremely complicated and in view of Proposition 3.3.4 we believe it is not necessaryto include it for the following analysis.

3.6.3 Results

Comparison of dynamical and static reconstructions

For a given configuration of particles, we consider three reconstruction procedures.

• The dynamical reconstruction: we take the whole data and recover both positions andvelocities of all particles.

73


• The static reconstruction: we perform static reconstruction of the positions at each timestep independently, and we call it a success if the reconstruction is successful for at leastone time step.

• The static 3 reconstruction: we perform static reconstruction of the positions at each timestep independently, and we call it a success if the reconstruction is successful for at leastthree time steps. The rationale of this case is that with three successful static reconstruc-tions, it would be possible to use some tracking technique to reconstruct the velocitiesafterwards. This case also encodes the theoretical limit given by Theorem 3.3.2.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

∆dyn

Cor

rect

reco

ntru

ctio

nra

te

dynamic

static

static3

·1/fc

Figure 3.8: Successful reconstruction rates for the cases described in Section 3.6.3, with SRFx = SRFv =

1000 and ∆w = 0.01. In the horizontal axis we consider the measure of separation (3.25)scaled by 1/fc.

In Figure 3.8 we present the rates of successful reconstructions for each of the three cases forrandomly generated particles. We observe that the dynamical reconstruction has a much higherreconstruction rate than the static reconstructions for small values of ∆dyn, namely for configu-rations of particles that are never well-separated. This gives a big advantage of dynamical overstatic reconstructions, and shows that the assumptions of Theorem 3.3.2 are in fact too strict:in practice, successful recovery happens much more often than the current theory predicts. Forlarger values of ∆dyn, the dynamical reconstruction rate remains slightly below 1: this could beexplained by the presence of ghost particles, or by numerical issues of the minimization algo-rithm.

Robustness to noise

We now study the stability of the dynamical reconstruction method and compare it to the sta-bility for the static approaches. We consider a measurement noise model, given by a normallydistributed noise scaled by a factor α ≥ 0. More precisely, for a frequency l and time sample k,

74


our measurements are

(Gω)l,k =

N∑i=1

wie−i2πl(xi+kτvi) + α(Nl,k,1 + iNl,k,2

),

where Nl,k, j are independent standardized normal random variables. In Figure 3.9a we plotthe reconstruction rate for different values of α. To understand the noise level, we remind thatwi ∈ (0.9, 1.1). The desired super-resolution factors and the weight threshold are larger in theseexperiments, since in the presence of noise we do not expect infinitely resolved reconstruc-tions.

We observe that the dynamical reconstruction method is stable to measurement noise, and thisstability is similar to the one of the static reconstruction, as we can see in Figure 3.9b.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

∆dyn

Cor

rect

reco

ntru

ctio

nra

te

α = 0

α = 0.025

α = 0.05

α = 0.075

α = 0.1

·1/fc

(a) Successful dynamical reconstructionrates under different intensities ofnoise.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

∆dyn

Cor

rect

reco

ntru

ctio

nra

te

dynamic

static

static3

·1/fc

(b) Successful reconstruction rates for thecases described in Section 3.6.3, witha fixed noise level α = 0.075.

Figure 3.9: Reconstruction rates for the measurement noise model described in Section 3.6.3, withSRFx = SRFv = 40 and ∆w = 0.05.

Robustness to curvature of trajectory

We study how the reconstruction algorithm fares when instead of a constant velocity, we considera curved trajectory for the imaged particles. For this purpose we consider the following dynamicsfor a particle δxi,vi :

x(t) = xi + vit +a2

t2 = xi + vit(1+

a2vi

t).

We consider β = a2viτK as measure of curvature. In Figure 3.10a we present one example of a

considered curvature for a trajectory, and in Figure 3.10b we present the dynamical reconstruc-tion rates for different values of β.

75


For the presented examples, we relaxed the super resolution factors to 1, since the algorithm isreconstructing some position and velocity, but reasonably it is not located exactly in the target(xi, vi); similarly, we relaxed the weight threshold condition. Nonetheless, we notice that thestability of the method with respect to changes in the curvature of the trajectories is quite poor.This may be explained by a not optimal choice of the parameters of the minimization algorithm(which should take into account that the forward model is not exact) and by the difficulties for alinear model to capture nonlinear movements. In any case, higher order models are expected tosolve this issue and represent an interesting direction for future research.

−1 −0.5 0 0.5 1

-0.2

-0.1

0.0

0.1

0.2

Time

Spac

e

β = 0

β = 0.03

(a) Comparison of the trajectories of aparticle with speed 0.25 and curvatureβ = 0 and β = 0.03.

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

∆dyn

Cor

rect

reco

ntru

ctio

nra

teβ = 0

β = 0.0075

β = 0.015

β = 0.0225

β = 0.03

·1/fc

(b) Successful dynamical reconstructionrates for different levels of curvature.

Figure 3.10: Reconstruction rates with the curvature model described in Section 3.6.3, with SRFx =

SRFv = 1 and ∆w = 0.2.

3.7 Applications to ultrafast ultrasonography

In this section, we describe a protocol to apply our method to the problem of super-resolutedimaging of blood vessels arising from ultrafast ultrasonography, as mentioned in the Introduc-tion. The setting is the following: we have a sequence of images of a medium containing bloodvessels, in which point reflectors are randomly activated during a small time interval. Thesereflectors are moving inside blood vessels and their velocity are approximately the same as thatof blood in the blood vessels.

We assume that we can filter out clutter signal coming from other sources than these reflectors.The recorded images are then convolutions of these point sources by the point spread function(PSF). The PSF of ultrafast ultrasound imaging was derived in Chapter 2, but filtering out clut-ter signal changes the shape of the PSF, which is here approximated by a Gaussian function.However, the analysis of Chapter 2 allows for a precise derivation, which we leave for futureinvestigation.

76

3.7 Applications to ultrafast ultrasonography

0 100 200 300 400 500 600 700 800 900 1,000

0

2

4

6

Figure 3.11: Graph of the `2 norm of the simulated measurements in each frame.

3.7.1 Fully automated imaging protocol

The blood vessels are not necessarily straight lines, and we therefore restrict to a few framesduring which the movement of the point reflectors can be approximated by a straight line. Wemust also make sure that the reconstruction algorithm works in this sequence, and that reflectorsdo not appear or disappear in a chosen sequence. Given all these remarks, we propose thefollowing imaging procedure, which fully automatically produces a super-resoluted image ofblood vessels with velocities. It is composed of two steps:

Choosing reconstruction intervals. In order to apply our dynamical reconstruction algo-rithm to this problem, we have to select consecutive frames during which the particles do notappear or vanish from one frame to another. One way to ensure this condition is to select time in-tervals during which the `2 norm of the observations is constant. Particles appearing or vanishingmake the `2 norm jump, whereas close particles will make the `2 norm vary slowly. Figure 3.11illustrates the variation of `2 norm as a function of frame number (dashed line), and the selectedintervals (red solid line), for the case presented in the numerical experiments below.

Reconstructing position and velocities. We then propose to reconstruct positions and ve-locities using the algorithm presented in the previous sections, using the PSF of ultrafast ultra-sound, in each of the intervals chosen in the previous step. By aggregating all positions obtainedusing this algorithm, we obtain a super-resoluted image of the blood vessels.

3.7.2 Numerical experiments

Setting

In order to test this procedure, we generate images using a toy example, in which we have twovertical blood vessels that are slightly separated, with particles flowing upwards and downwards

77


frame 20 frame 24 frame 28

0 0.2 0.4 0.6 0.8 1

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Figure 3.12: Three different measurements with ultrafast ultrasound of the simulated particles.

in their respective vessel. The speed of the particles depends linearly on their position and theirweights are set to 1. The considered parameters are such that they are similar to the experimentalones:

• Domain size: 1 mm × 1 mm.

• Pixel size: 0.04 mm × 0.04 mm.

• Vessel separation: 0.04 mm.

• Point spread function: (x, y) 7→ e−(x2+y2)/2σ2, σ = 0.04.

• Maximum and minimum particles’ speeds: 15 mm/s, 5 mm/s.

• Sampling rate: τ = 0.002 s−1

• Total acquisition time: 2 s.

Since in ultrafast ultrasound microbubble imaging the particles are activated and deactivatedrandomly, we simulate this behavior in the following fashion. The activation of a single parti-cle is modeled as a Bernoulli random variable on each time sample, whereas the deactivationtime is modeled as a Poisson random variable. Further, we include measurement noise as inSection 3.6.3, with α = 0.01.

Results

In Figure 3.11 we present the `2 norms of the measurements at each time step. As an illustration,in Figure 3.12 we show three frames of the simulated measurements. In Figure 3.13a we presentthe B-mode image, i.e. the average signal intensity over all the frames, which shows that the res-olution does not allow for separation of the two vessels. Figure 3.13b present the reconstructedposition and velocities using the algorithm described in this work. The results show that theprocedure can reconstruct the positions and the velocities very accurately.

78

3.8 Conclusion

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.00

0.05

0.10

0.15

0.20

(a) B-mode image of the vessels, taken over 2 sec-onds of measurements.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

-10

-5

0

5

10

(b) Super resolved reconstruction of the simulated parti-cles; the colors represent the velocity in the verticaldirection.

Figure 3.13: Simulated vessels’ reconstructions.

3.8 Conclusion

In this chapter, we have introduced and studied a new framework for dynamical super-resolutionimaging, which allows for super-resolved recovery of positions and velocities of particles fromlow-frequency measurements. The presented theoretical results are validated by extensive simu-lations, related to low-frequency one-dimensional Fourier measurements and to two-dimensionalultrafast ultrasound localization microscopy.

In fact, the numerical experiments show that this approach works much better than what thecurrent theory predicts, and so there is a need of further theoretical investigation. Indeed, thearguments presented in this chapter are still based on the static reconstruction, while ideally oneshould consider the dynamical problem directly. Further, it would be nice to generalize thismethod to higher order models, in order to relax the assumption of linear trajectories.

79


80

4 Heat generation with plasmonicnanoparticles

4.1 Introduction

Our aim in this chapter is to provide a mathematical and numerical framework for analyzing pho-tothermal effects using plasmonic nanoparticles. A remarkable feature of plasmonic nanoparti-cles is that they exhibit quasi-static optical resonances, called plasmonic resonances. At or nearthese resonant frequencies, strong enhancement of scattering and absorption occurs [14, 18, 86].The plasmonic resonances are related to the spectra of the non-self adjoint Neumann-Poincaretype operators associated with the particle shapes [14, 18, 21, 22, 56, 65]. Plasmonic nanoparti-cles efficiently generate heat in the presence of electromagnetic radiation. Their biocompatibilitymakes them suitable for use in nanotherapy [25].

Nanotherapy relies on a simple mechanism. First nanoparticles become attached to tumor cellsusing selective biomolecular linkers. Then heat generated by optically-simulated plasmonicnanoparticles destroys the tumor cells [54]. In this nanomedical application, the temperatureincrease is the most important parameter [74, 83]. It depends in a highly nontrivial way on theshape, the number, and the arrangement of the nanoparticles. Moreover, it is challenging tomeasure it at the surface of the nanoparticles [54].

In this chapter, we derive an asymptotic formula for the temperature at the surface of plasmonicnanoparticles of arbitrary shapes. Our formula holds for clusters of simply connected nanopar-ticles. It allows to estimate the collective response of plasmonic nanoparticles. In particular,the heat generated by two interacting nanoparticles may be significantly higher than double theheat generated by a single nanoparticle. The more interacting nanoparticles, the stronger thetemperature increase. Our results in this chapter formally explain the experimental observationsreported in [54].

The chapter is organized as follows. In Section 4.2 we describe the mathematical setting for thephysical phenomena we are modeling. To this end, we use the Helmholtz equation to modelthe propagation of light which we couple to the heat equation. Later on, we present our mainresults in this chapter which consist on original asymptotic formulas for the inner field and thetemperature on the boundaries of the nanoparticles. In Section 4.3 we recall basic results onlayer potentials. In Section 4.4 we prove Theorems 4.2.1 and 4.2.2. These results clarify thestrong dependency of the heat generation on the geometry of the particles as it depends on theeigenvalues of the associated Neumann-Poincare operator. In Section 4.5 we present numericalexamples of the temperature at the boundary of single and multiple particles. Appendix 4.7 is

81

4 Heat generation with plasmonic nanoparticles

devoted to the asymptotic analysis of layer potentials for the Helmholtz equation in dimensiontwo. We also include an analysis of the invertibility of the single-layer potential for the Laplacianin the case of multiple particles.

4.2 Setting of the problem and the main results

In this chapter, we use the Helmholtz equation for modeling the propagation of light. This canbe thought of as a special case of Maxwell’s equations, when the incident wave ui is a trans-verse electric or transverse magnetic polarized wave. This approximation, also called paraxialapproximation [55, p. 19–20], is a good model for a laser beam which is used, in particular,in full-field optical coherence tomography. We will therefore model the propagation of a laserbeam in a host domain (tissue), hosting a nanoparticle.

Let the nanoparticle occupy a bounded domain D b R2 of class C1,α for some 0 < α < 1.Furthermore, let D = z + δB, where B is centered at the origin and |B| = O(1).

We denote by εm(x) = ε0ε′m and µm(x) = µ0µ

′m, x ∈ R2\D the permittivity and permeability of

the host medium, both of which do not depend on the frequency ω of the incident wave. Assumethat εm and µm are positive constants. Here and throughout, ε0 and µ0 are the permittivity andpermeability of vacuum.

Similarly, we denote by εc(x) and µc(x), x ∈ D, the electric permittivity and magnetic permeabil-ity of the particle, respectively. We assume on one hand that the nanoparticle is nonmagnetic,i.e., µc(x) = µ0µ

′m , and on the other hand, that εc depends on the frequency ω of the incident

wave. We also assume that εc(x) = ε0ε′c, and Re ε′c < 0, Im ε′c > 0.

The index of refraction of the medium (with the nanoparticle) is given by

n(x) =√ε′cµ′mχ(D)(x) +

√ε′mµ

′mχ(R2\D)(x),

where χ denotes the indicator function.

The scattering problem for a TE incident wave ui is modeled as follows:

∇ ·c2

n2∇u + ω2u = 0 in R2\∂D,

u+ − u− = 0 on ∂D,

1εm

∂u∂ν

∣∣∣∣∣+

−1εc

∂u∂ν

∣∣∣∣∣−

= 0 on ∂D,

us := u − ui satisfies the Sommerfeld radiation condition at infinity,

(4.1)

82

4.2 Setting of the problem and the main results

where ∂∂ν denotes the outward normal derivative and c = 1√

ε0µ0is the speed of light in vacuum.

We use the notation ∂∂ν

∣∣∣∣±

indicating

∂u∂ν

∣∣∣∣±

(x) = limt→0+∇u(x ± tν(x)) · ν(x),

with ν being the outward unit normal vector to ∂D.

The interaction of the electromagnetic waves with the medium produces a heat flow of energywhich translates into a change of temperature governed by the heat equation [24]

ρC∂τ

∂t− ∇ · γ∇τ =

ω

2πIm(ε)|u|2 in (R2\∂D) × (0,T ),

τ+ − τ− = 0 on ∂D,

γm∂τ

∂ν

∣∣∣∣∣+

− γc∂τ

∂ν

∣∣∣∣∣−

= 0 on ∂D,

τ(x, 0) = 0,

(4.2)

where ρ = ρcχ(D) + ρmχ(R2\D) is the mass density, C = Ccχ(D) + Cmχ(R2\D) is the thermalcapacity, γ = γcχ(D) + γmχ(R2\D) is the thermal conductivity, T ∈ R is the final time ofmeasurements and ε = εcχ(D) + εmχ(R2\D). The model (4.2) corresponds to a continuous-wave illumination with a monochromatic incident electromagnetic wave [26]. Here, we areinterested in the temperature evolution until the establishment of the steady-state profile. Theresults and methods of this chapter can, with slight modifications, be extended to the case of apulsed illumination, where the source term in the first equation in (4.2) depends on time too.

We further assume that ρc, ρm,Cc,Cm, γc, γm are positive constants. Note that Im(ε) = 0 inR2\Dand so, outside D, the heat equation is homogeneous.

The coupling of equations (4.1) and (4.2) describes the physics of our problem.

We remark that, in general, the index of refraction varies with temperature; hence, a solutionto the above equations would imply a dependency on time for the electric field u, which con-tradicts the time-harmonic assumption leading to model (4.1). Nevertheless, the time-scale onthe dynamics of the index of refraction is much larger than the time-scale on the dynamics ofthe interaction of the electromagnetic wave with the medium. Therefore, we will not integrate atime-varying component into the index of refraction.

Let G(·, k) be the Green function for the Helmholtz operator ∆ + k2 satisfying the Sommerfeldradiation condition. In dimension two, G is given by

G(x, k) = −i4

H(1)0 (k|x|),

where H(1)0 is the Hankel function of first kind and order 0. We denote G(x, y, k) := G(x−y, k).

83


Define the following single-layer potential and Neumann-Poincare integral operator:

SkD[ϕ](x) =

∫∂D

G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D or x ∈ R2,

and(Kk

D)∗[ϕ](x) =

∫∂D

∂G(x, y, k)∂ν(x)

ϕ(y)dσ(y), x ∈ ∂D.

Let I denote the identity operator and let SD and K∗D respectively denote the single-layer po-tential and the Neumann-Poincare operator associated to the Laplacian. Our main results in thischapter are the following.

Theorem 4.2.1. For an incident wave ui ∈ C2(R2), the solution u to (4.1), inside a plasmonicparticle occupying a domain D = z + δB, has the following asymptotic expansion as δ → 0 inL2(D):

u(x) = ui(z) +((x − z) + SD

(λεI − K∗D

)−1[ν](x))· ∇ui(z) + O

(δ2

dist(λε, σ(K∗D))

), (4.3)

where ν is the outward normal to D, σ(K∗D) denotes the spectrum of K∗D in H−12 (∂D) and

λε :=εc + εm

2(εc − εm).

Theorem 4.2.2. Let u be the solution to (4.1). The solution τ to (4.2) on the boundary ∂D of aplasmonic particle occupying the domain D = z + δB has the following asymptotic expansion asδ→ 0, uniformly in (x, t) ∈ ∂D × (0,T ):

τ(x, t) = FD(x, t, bc) −VbcD (λγI − K∗D)−1

[∂FD(·, ·, bc)

∂ν

](x, t) + O

(δ4 log δ

dist(λε, σ(K∗D))2

), (4.4)

where ν is the outward normal to D and

λγ :=γc + γm

2(γc − γm),

bc :=ρcCc

γc,

FD(x, t, bc) :=ω

2πγcIm(εc)

∫ t

0

∫D

e−|x−y|2

4bc(t−t′)

4πbc(t − t′)|u|2(y)dydt′,

VbcD [ f ](x, t) :=

∫ t

0

∫∂D

e−|x−y|2

4bc(t−t′)

4πbc(t − t′)f (y, t′)dydt′.

Remark 4.2.3. We remark that Theorem 4.2.1 and Theorem 4.2.2 are independent. A general-ization of Theorem 4.2.2 to R3 is straightforward and the same type of small volume approxi-mation can be found using the techniques presented in this chapter. In fact, in R3, the operators

84

4.3 Layer potentials for the Helmholtz equation in two dimensions

involved in the first term of the temperature small volume expansion are

FD(x, t, bc) :=ω

2πγcIm(εc)

∫ t

0

∫D

e−|x−y|2

4bc(t−t′)(4πbc(t − t′)

) 32

|E|2(y)dydt′,

VbcD [ f ](x, t) :=

∫ t

0

∫∂D

e−|x−y|2

4bc(t−t′)(4πbc(t − t′)

) 32

f (y, t′)dydt′.

Here E is the vectorial electric field as a result of Maxwell equations. A small volume expansionfor E inside the nanoparticle for the plasmonic case can be found using the same techniques asthose of [18].

Remark 4.2.4. It is worth emphasizing that the leading-order terms in (4.3) and (4.4) are oforder of δ/dist(λε, σ(K∗D)) and δ2/(dist(λε, σ(K∗D))2, respectively.

Throughout this chapter, we denote by L(E, F) the set of bounded linear applications from E toF and let L(E) := L(E, E) and let Hs(∂D) to be the standard Sobolev space of order s on ∂D.

4.3 Layer potentials for the Helmholtz equation in twodimensions

Let us recall some properties of the single-layer potential and the Neumann-Poincare integraloperator [9]:

(i) SkD : H−

12 (∂D)→ H

12 (∂D),H1

loc(R2\∂D) is bounded;

(ii) (∆ + k2)SkD[ϕ](x) = 0 for x ∈ R2\∂D, ϕ ∈ H−

12 (∂D);

(iii) (KkD)∗ : H−

12 (∂D)→ H−

12 (∂D) is compact;

(iv) SkD[ϕ], ϕ ∈ H−

12 (∂D), satisfies the Sommerfeld radiation condition at infinity;

(v)∂Sk

D[ϕ]∂ν

∣∣∣∣±

= (±12 I + (Kk

D)∗)[ϕ].

Let u be the solution of ∇ · c2

n2∇u + ω2u = 0 in R2\∂D such that u − ui satisfies the Sommerfeld

radiation condition. Then there exist unique ψ, φ ∈ H−12 (∂D) such that

u :=

ui + SkmD [ψ], x ∈ R2\D,

SkcD [φ], x ∈ D,

(4.5)

with km = ω√εmµm and kc = ω

√εcµc.

85


To satisfy the boundary transmission conditions, ψ, φ ∈ H−12 (∂D) need to satisfy the following

system of integral equations on ∂D:S

kmD [ψ] − Skc

D [φ] = −ui,

1εm

(12 I + (Kkm

D )∗)[ψ] + 1

εc

(12 I − (Kkc

D )∗)[φ] = −

1εm

∂ui

∂ν.

(4.6)

In [12], it is shown that the operator

T :(H−

12 (∂D)

)2→ H

12 (∂D) × H−

12 (∂D)

(ψ, φ) 7→(S

kmD [ψ] − Skc

D [φ],1εm

(12

I + (KkmD )∗

)[ψ] +

1εc

(12

I − (KkcD )∗

)[φ]

)is invertible.

4.4 Heat generation

In this section we rewrite equations (4.1) and (4.2) into

∇ ·c2

n2∇u + ω2u = 0 in R2\∂D,

u+ − u− = 0 on ∂D,

1εm

∂u∂ν

∣∣∣∣∣+

−1εc

∂u∂ν

∣∣∣∣∣−

= 0 on ∂D,

us := u − ui satisfies the Sommerfeld radiation condition at infinity,

ρcCc

γc

∂τ

∂t− ∆τ =

ω

2πγcIm(εc)|u|2 in D × (0,T ),

ρmCm

γm

∂τ

∂t− ∆τ = 0 in (R2\D) × (0,T ),

τ+ − τ− = 0 on ∂D,

γm∂τ

∂ν

∣∣∣∣∣+

− γc∂τ

∂ν

∣∣∣∣∣−

= 0 on ∂D,

τ(x, 0) = 0.

(4.7)

It is worth noticing that, in (4.7), we first compute u and then compute τ.

Under the assumption that the index of refraction n does not depend on the temperature, we cansolve equation (4.1) separately from equation (4.2).

Our goal is to establish a small volume expansion for the resulting temperature at the surfaceof the nanoparticule as a function of time. To do so, we first need to compute the electric field

86

4.4 Heat generation

inside the nanoparticule as a result of a plasmonic resonance. We make use of layer potentialsfor the Helmholtz equation, described in Section 4.3.

4.4.1 Small volume expansion of the inner field

We proceed in this section to prove Theorem 4.2.1.

Rescaling

Since we are working with nanoparticles, we want to rescale equation (4.6) to study the solutionfor a small volume approximation by using representation (4.5).

Recall that D = z + δB. For any x ∈ ∂D, x := x−zδ ∈ ∂B and for each function f defined on ∂D,

we introduce a corresponding function defined on ∂B as follows

η( f )(x) = f (z + δx). (4.8)

It follows thatSk

D[ϕ](x) = δSδkB [η(ϕ)](x),

(KkD)∗[ϕ](x) = (Kδk

B )∗[η(ϕ)](x),(4.9)

so system (4.6) becomesSδkmB [η(ψ)] − Sδkc

B [η(φ)] = −η(ui)δ

,

1εm

(12 I + (Kδkm

B )∗)

[η(ψ)] + 1εc

(12 I − (Kδkc

B )∗)

[η(φ)] = −1εmη

(∂ui

∂ν

).

(4.10)

Note that the system is defined on ∂B.

For δ2k2m smaller than the first Dirichlet eigenvalue of −∆ in B, Sδkm

B is invertible (see Appendix4.7). Therefore,

η(ψ) = (SδkmB )−1S

δkcB [η(φ)] − (Sδkm

B )−1[η(ui)δ

].

Hence, we have the following equation for η(φ):

AIB(δ)[η(φ)] = f I ,

87


where

AIB(δ) =

1εm

(12

I + (KδkmB )∗

)(Sδkm

B )−1SδkcB +

1εc

(12

I − (KδkcB )∗

),

f I = −1εmη

(∂ui

∂ν

)+

1εm

(12

I + (KδkmB )∗

)(Sδkm

B )−1[η(ui)δ

].

(4.11)

Proof of Theorem 4.2.1

To express the solution to (4.1) in D, asymptotically on the size of the nanoparticle δ, we makeuse of the representation (4.5). We derive an asymptotic expansion for η(φ) on δ to later computeδSδkc

B [η(φ)] and scale back to D. We divide the proof into three steps.

Step 1. We first compute asymptotic estimates ofAIB(δ) and f I .

Let H∗(∂B) be defined by (4.7.3) with D replaced by B. In L(H∗(∂B)), we have the followingasymptotic expansion as δ→ 0 (see Appendix 4.7)

(SδkmB )−1S

δkcB = PH∗0

+Uδkm(SB + Υδkc) + O(δ2 log δ),

12

I ± (KδkB )∗ =

(12

I ± K∗B

)+ O(δ2 log δ).

Let ϕ0 be an eigenfunction of K∗B associated to the eigenvalue 1/2 (see Appendix 4.7) and letUδkm be defined by (4.32) with k replaced with δkm. Then it follows that(

12

I +K∗B

)Uδkm = Uδkm .

Therefore, in L(H∗(∂B)),

AIB(δ) =

((1

2εm+

12εc

)I +

(1εm−

1εc

)K∗B

)PH∗0

+1εmUδkm(SB + Υδkc) + O(δ2 log δ),

and from the definition ofUδkm we get

AIB(δ) =

((1

2εm+

12εc

)I +

(1εm−

1εc

)K∗B

)PH∗0

+1εm

SB[ϕ0] + τδkc

SB[ϕ0] + τδkm

(·, ϕ0)H∗ϕ0 + O(δ2 log δ). (4.12)

In the same manner, in the spaceH∗(∂B),

f I =1εm

(−η

(∂ui

∂ν

)+

(12

I +K∗B

)PH∗0S−1

B

[η(ui)δ

]+Uδkm

[η(ui)δ

]+ O(δ2 log δ)

).

88

4.4 Heat generation

We can further develop f I . Indeed, for every x ∈ ∂B, a Taylor expansion yields

η

(∂ui

∂ν

)(x) = ν(x) · ∇ui(δx + z) = ν(x) · ∇ui(z) + O(δ),

η(ui)δ

(x) =ui(δx + z)

δ=

ui(z)δ

+ x · ∇ui(z) + O(δ).

The regularity of ui ensures that the previous formulas hold inH∗(∂B).

The fact that x · ∇ui(z) is harmonic in B and Lemma 4.7.7 imply that

−ν · ∇ui(z) =

(12

I − K∗B

)PH∗0S−1

B [x · ∇ui(z)]

inH∗(∂B).

Thus, inH∗(∂B),

f I =1εm

(PH∗0S−1

B [x · ∇ui(z)] +Uδkm

[ui(z)δ

+ x∇ui(z)]

+ O(δ)).

From the definition ofUδkm we get

f I =1εm

PH∗0 S−1B

[x · ∇ui(z)

]+

ui(z)ϕ0

δ(SB[ϕ0] + τδkm)−

(S−1B [x · ∇ui(z)], ϕ0)H∗ϕ0

SB[ϕ0] + τδkm

+ O(δ)

. (4.13)

Step 2. We compute (AIB(δ))−1 f I .

We begin by computing an asymptotic expansion of (AIB(δ))−1.

The operatorAI0 :=

(( 12εm

+ 12εc

)I +

( 1εm− 1

εc

)K∗B

)mapsH∗0 intoH∗0 . Hence, the operator defined

by (which appears in the expansion ofAIB(δ))

AIB,0 := AI

0PH∗0+

1εm

SB[ϕ0] + τδkc

SB[ϕ0] + τδkm

(·, ϕ0)H∗ϕ0,

is invertible of inverse

(AIB,0)−1 = (AI

0)−1PH∗0+ εm

SB[ϕ0] + τδkm

SB[ϕ0] + τδkc

(·, ϕ0)H∗ϕ0.

Therefore, we can write

(AIB)−1(δ) =

(I + (AI

B,0)−1O(δ2 log δ))−1(AI

B,0)−1.

SinceK∗B is a compact self-adjoint operator inH∗(∂B) it follows that the following lemma holds[7, 14].

89


Lemma 4.4.1. We have‖(AI

0)−1‖L(H∗(∂B)) ≤c

dist(0, σ(AI0)), (4.14)

for a constant c.

Therefore, for δ small enough, we obtain from Lemma 4.4.1 that

(AIB(δ))−1 f I =

(I + (AI

B,0)−1O(δ2 log δ))−1(AI

B,0)−1 f I

=(I + (AI

B,0)−1O(δ2 log δ))−1

ui(z)ϕ0

δ(SB[ϕ0] + τδkc)−

(S−1B [x · ∇ui(z)], ϕ0)H∗ϕ0

SB[ϕ0] + τδkc

+ (AI0)−1 1

εmPH∗0S−1

B [x · ∇ui(z)] + O(

δ

dist(0, σ(AI0))

) )=

ui(z)ϕ0


(S−1B [x · ∇ui(z)], ϕ0)H∗ϕ0

SB[ϕ0] + τδkc

+ (AI0)−1 1

εmPH∗0S−1

B [x · ∇ui(z)] + O(

δ

dist(0, σ(AI0))

).

Using the representation formula of K∗B described in Lemma 4.7.5 we can further develop thethird term in the above expression to obtain

(AI0)−1PH∗0

S−1B [x · ∇ui(z)] =

∞∑j=1

(S−1B [x · ∇ui(z)], ϕ j)H∗ϕ j( 12 +

εm2εc

)−

( εmεc− 1

)λ j

=

∞∑j=1

(S−1B [x · ∇ui(z)], ϕ j)H∗ϕ j(12 +

εm2εc

)−

( εmεc− 1

)λ j− (S−1

B [x · ∇ui(z)], ϕ j)H∗ϕ j

+PH∗0 S

−1B [x · ∇ui(z)]

= PH∗0S−1

B [x · ∇ui(z)] +

∞∑j=1

(λ j −

12

)(S−1

B [x · ∇ui(z)], ϕ j)H∗ϕ j

λ − λ j.

Using the same arguments as those in the proof of Lemma 4.7.7, we have(λ j −

12

)(S−1

B [x · ∇ui(z)], ϕ j)H∗ = (ν · ∇ui(z), ϕ j)H∗ ,

and consequently,

(AI0)−1 1

εmPH∗0S−1

B [x · ∇ui(z)] = PH∗0 S−1B [x · ∇ui(z)] + (λεI − K∗B)−1[ν] · ∇ui(z).

90

4.4 Heat generation

Therefore,

(AIB(δ))−1 f I =

ui(z)ϕ0


(S−1B [x · ∇ui(z)], ϕ0)H∗ϕ0

SB[ϕ0] + τδkc

+ PH∗0 S−1B [x · ∇ui(z)]

+ (λεI − K∗B)−1[ν] · ∇ui(z) + O(

δ

dist(0, σ(AI0))

).

Step 3. Finally, we compute η(u) = δSδkcB (AI

B(δ))−1 f I .

From Appendix 4.7, the following holds when SδkcB is viewed as an operator from H∗(∂B) onto

H(∂B):

SδkcB = SB + Υδkc + O(δ2 log δ).

In particular, we have

SδkcB [ϕ0] = SB[ϕ0] + τδkc + O(δ2 log δ).

It can be verified that the same expansion holds when viewed as an operator fromH∗(∂B) ontoL2(B).

Note that the following identity holds

−(S−1

B [x · ∇ui(z)], ϕ0)H∗ϕ0

SB[ϕ0] + τδkc

+ PH∗0 S−1B [x · ∇ui(z)] = −

Υδkc

[S−1

B [x · ∇ui(z)]]ϕ0

SB[ϕ0] + τδkc

+ S−1B [x · ∇ui(z)].

Straightforward calculations and the fact thatSB[ϕ] is harmonic in B for any ϕ ∈ H∗(∂B) yield

δSδkcB (AI

B(δ))−1 f I = ui(z) + δ(x + SB

(λεI − K∗B

)−1[ν])· ∇ui(z) + O

(δ2

dist(λε, σ(K∗B))

)in L2(B). Using Lemma 4.7.6 to scale back the estimate to D leads to the desired result.

4.4.2 Small volume expansion of the temperature

We proceed in this section to prove Theorem 4.2.2. To do so, we make use of the Laplacetransform method [40, 59, 64].

Consider equation (4.7) and define the Laplace transform of a function g(t) by

L(g)(s) =

∫ ∞

0e−stg(t)dt.

91


Taking the Laplace transform of the equations on τ in (4.7) we formally obtain the followingsystem:

sρcCc

γcτ(·, s) − ∆τ(·, s) = L(gu)(·, s) in D,

sρmCm

γmτ(·, s) − ∆τ(·, s) = 0 in R2\D,

τ+(·, s) − τ−(·, s) = 0 on ∂D,

γm∂τ

∂ν

∣∣∣∣∣+

− γc∂τ

∂ν

∣∣∣∣∣−

= 0 on ∂D,

τ(·, s) satisfies the Sommerfeld radiation condition at infinity,

(4.15)

where τ(·, s) and L(gu)(·, s) are the Laplace transforms of τ and gu := ω2πγc

Im(εc)|u|2, respec-tively, and s ∈ C\(−∞, 0].

A rigorous justification for the derivation of system (4.15) and the validity of the inverse trans-form of the solution can be found in [59].

Using layer potential techniques we have that, for any p, q ∈ H−12 (∂D), τ defined by

τ :=

−SβγmD [ p], x ∈ R2\D,

−FD(·, y, βγc) − SβγcD [q], x ∈ D,

(4.16)

satisfies the differential equations in (4.15) together with the Sommerfeld radiation condition.

Here βγm := i√

sρmCmγm

, βγc := i√

sρcCcγc

and

FD(·, βγc) :=∫

DG(·, y, βγc)L(gu)(y)dy.

To satisfy the boundary transmission conditions, p and q ∈ H−12 (∂D) should satisfy the following

system of integral equations on ∂D:−S

βγmD [ p] + S

βγcD [q] = −FD(·, βγc),

−γm(1

2 I + (KβγmD )∗

)[ p] + γc

(− 1

2 I + (KβγcD )∗

)[q] = −γc

∂FD(·, βγc)∂ν

.(4.17)

Rescaling of the equations

Recall that D = z + δB, for any x ∈ ∂D, x := x−zδ ∈ ∂B, for each function f defined on ∂D, η is

such that η( f )(x) = f (z + δx) and

SkD[ϕ](x) = δSδkB [η(ϕ)](x),

(KkD)∗[ϕ](x) = (Kδk

B )∗[η(ϕ)](x).

92

4.4 Heat generation

We can also verify that

FD(x, βγc) = δ2FB(x, δβγc),

∂FD

∂ν(x, βγc) = δ

∂FB

∂ν(x.δβγc).

Note that in the above identity, in the left-hand side we differentiate with respect to x while inthe right-hand side we differentiate with respect to x. To simplify the notation, we will use FB

to refer to FB(·, δβγc).

We rescale system (4.17) to arrive at−S

δβγmB [η( p)] + S

δβγcB [η(q)] = −δFB,

−γm(

12 I + (Kδβγm

B )∗)

[η( p)] + γc(− 1

2 I + (KδβγcB )∗

)[η(q)] = −γcδ

∂FB

∂ν.

For δ small enough, SδβγcB is invertible (see Appendix 4.7). Therefore, it follows that

η( p) = (SδβγmB )−1S

δβγcB [η(q)] + (Sδβγm

B )−1[δFB

].

Hence, we have the following equation for η(q):

AhB(δ)[η(q)] = f h,

where

AhB(δ) = −γm

(12 I + (Kδβγm

B )∗)

(SδβγmB )−1S

δβγcB + γc

(− 1

2 I + (KδβγcB )∗

),

f h = −γcδ∂FB

∂ν+ γm

(12 I + (Kδβγm

B )∗)

(SδβγmB )−1

[δFB

].

(4.18)

Proof of Theorem 4.2.2

To express the solution of (4.2) on ∂D × (0,T ), asymptotically on the size of the nanoparticle δ,we make use of the representation (4.16). We will compute an asymptotic expansion for η(q) onδ to later compute δSδβγc

B [η(q)] on ∂B, scale back to D and take Laplace inverse.

Using the asymptotic expansions of Appendix 4.7 the following asymptotic for AhB(δ) holds in

L(H∗(∂B))

AhB(δ) = Ah

0+O(δ2 log δ),

where

Ah0 = −

(12(γc + γm

)I −

(γc − γm

)K∗B

).

93


In the same manner, inH∗(∂B),

f h = −γcδ∂FB

∂ν+ γm

(12

I +K∗B

)S−1

B [δFB] + O(

δ5 log δdist(λε, σ(K∗D))2

)= −γcδ

∂FB

∂ν− γm

(12

I − K∗B

)S−1

B [δFB] + γmS−1B [δFB] + O

(δ5 log δ


).

Here the remainder comes from the fact that FB is proportional to |u|2 and therefore, it is of the

order of(

δ2

dist(λε,σ(K∗D))2

).

Note that ∆FB = η(L(gu)) − δ2β2γc

FB in B and ∆FB = 0 in R2\D. We can further verify that FB

satisfies the assumption required in Lemma 4.7.7. Thus we have(12

I − K∗B

)S−1

B [δFB] = −δ∂FB

∂ν+ Cuϕ0 + γmS

−1B [δFB] + O

(δ5


),

where Cu is a constant such that Cu = O(

δ3


).

After replacing the above in the expression of f h we find that

η(q) = (AhB(δ))−1 f h

= (λγI − K∗B)−1[δ∂FB

∂ν

]+

Cuγm

(γc − γm)(λγ − 12 )ϕ0 + O

(δ5 log δ


),

(4.19)

where

λγ =γc + γm

2(γc − γm).

Finally, inH∗(∂B),

η(τ) = − δ2FB − δSδβγcB (λγI − K∗B)−1

[∂δFB

∂ν

]−

Cuγm

(γc − γm)(λγ − 12 )δS

δβγcB [ϕ0] + O

(δ6 log δ


). (4.20)

It can be shown, from the regularity of the remainders, that the previous identity also holds inL2(∂B).

Using Holder’s inequality we can prove that

‖SδβγcB [ϕ]‖L∞(∂B) ≤ C‖ϕ‖L2(∂B),

for some constant C. Hence, we find that identity (4.20) also holds true uniformly on ∂B and

94

4.4 Heat generation

CuδSδβγc fB [ϕ0](x) = O

(δ4 log δ


), uniformly on ∂B. Scaling back to D gives

τ(x, s) = −FD(x, βγc) − SβγcD (λγI − K∗D)−1

∂FD(·, βγc)∂ν

+ O(


). (4.21)

Before we take the inverse Laplace transform to (4.21), we note that (see [64])

L(K(x, ·, bc)

)= −G(x, βγc),

where bc := ρcCcγc

and K(x, ·, bc) is the fundamental solution of the heat equation. In dimensiontwo, K is given by

K(x, t, γ) =e−

|x|24bct

4πbct.

We denote K(x, y, t, t′, bc) := K(x − y, t − t′, bc). By the properties of the Laplace transform, wehave

−FD(x, βγc) = −

∫D

G(x, y, βγc)L(gu)(y)dy = L(∫ ·

0

∫D

K(x, y, ·, t′, bc)gu(y)dydt′).

We define FD as follows

FD(x, t, bc) :=∫ t

0

∫D

K(x, y, t, t′, bc)gu(y)dydt′. (4.22)

Similarly, we have that for a function f

−

∫∂D

G(x, y, βγc)L( f )(y)dy = L(∫ ·

0

∫∂D

K(x, y, ·, t′, bc) f (y, t′)dydt′).

We defineVbcD as follows

VbcD [ f ](x, t) :=

∫ t

0

∫∂D

K(x, y, t, t′, bc) f (y, t′)dydt′. (4.23)

Finally, using Fubini’s theorem and taking Laplace inverse we find that


[∂FD(·, ·, bc)

∂ν

](x, t) + O

(δ4 log δ


),

uniformly in (x, t) ∈ ∂D × (0,T ).

4.4.3 Temperature elevation at the plasmonic resonance

Suppose that the incident wave is ui(x) = eikmd·x, where d is a unit vector. For a nanoparticleoccupying a domain D = z + δB, the inner field u solution to (4.1) is given by Theorem 4.2.1,

95


which states that, in L2(D),

u ≈ eikmd·z(1 + ikmSD(λεI − K∗D

)−1[ν] · d),

and hence

|u|2 ≈ 1 + 2km Re(iSD

(λεI − K∗D

)−1[ν] · d)

+∣∣∣∣kmSD

(λεI − K∗D

)−1[ν])· d

∣∣∣∣2. (4.24)

Using Lemma 4.7.5, we can write

SD(λεI − K∗D

)−1[ν] · d =

∞∑j=1

(ν · d, ϕ j)H∗SD[ϕ j]λε − λ j

,

and therefore, for a given plasmonic frequency ω, we have

SD(λεI − K∗D

)−1[ν] · d ≈(ν · d, ϕ j∗)H∗SD[ϕ j∗]

λε(ω) − λ j∗.

Here j∗ is such that λ j∗ = Re(λε(ω)) and the eignevalue λ j∗ is assumed to be simple. If thiswas not the case, (ν · d, ϕ j∗)H∗SD[ϕ j∗] should be replaced by the corresponding sum over anorthonormal basis of eigenfunctions for the eigenspace associated to λ j∗ .

Replacing in (4.24) we find

|u|2 ≈ 1 + 2km(ν · d, ϕ j∗)H∗SD[ϕ j∗]|λε(ω) − λ j∗ |

+ k2m

(ν · d, ϕ j∗)2H∗SD[ϕ j∗]2

|λε(ω) − λ j∗ |2 .

Thus, at a plasmonic resonance ω,

FD[gu](x, t, bc) ≈(FD[1] + 2km

(ν · d, ϕ j∗)H∗|λε(ω) − λ j∗ |

FD[SD[ϕ j∗]]

+ k2m

(ν · d, ϕ j∗)2H∗

|λε(ω) − λ j∗ |2 FD[SD[ϕ j∗]2]

(x, t, bc),

∂FD(x, t, bc)∂ν

≈

(2km

(ν · d, ϕ j∗)H∗|λε(ω) − λ j∗ |

∂FD[SD[ϕ j∗]]∂ν

+ k2m

(ν · d, ϕ j∗)2H∗

|λε(ω) − λ j∗ |2

∂FD[SD[ϕ j∗]2]∂ν

(x, t, bc).

Then, the temperature on the boundary of a nanoparticle at the plasmonic resonance can be

estimated by plugging the above approximations of FD and∂FD(x, t, bc)

∂νinto


[∂FD(·, ·, bc)

∂ν

](x, t) + O

(δ4 log δ


).

96

4.5 Numerical results

4.4.4 Temperature elevation for two close-to-touching particles

In this subsection, we let D = D1∪D2, where D1 and D2 are close-to-touching particles. Lemma4.7.7 implies that


= −(12

I − K∗D)S−1

D [FD](x, t) + O(


).

Therefore, we can write the temperature on the boundary of the nanoparticle as

τ(x, t) = FD(x, t, bc) +VbcD

(λγI − K∗D

)−1PH∗\E 1

2

[∂FD(·, ·, bc)

∂ν

](x, t)

+ O(


), (4.25)

where PH∗\E 12

is the projection into H∗\E 12: the complement in H∗(∂D) of the eigenspace

associated to the eigenvalue 12 of K∗D. This implies that, even if λγ is close to 1

2 , the quantity

(λγI −K∗D)−1PH∗\E 12[∂FD(·, ·, bc)

∂ν](x, t) will remain of order O

(δ2


), provided that the

second largest eigenvalue of K∗D is not close to 12 .

This is in general the case for smooth boundaries ∂D. Nevertheless, it turns out that the situationmay be more involved for close-to-touching particles. The following result from [31] holds.

Lemma 4.4.2. For nanoparticles with two connected close-to-touching subparts with contactof order m, a family of eigenvalues of K∗D inH∗\E 1

2approaches 1

2 as

λζn ∼

12− cnζ

1− 1m + o(ζ1− 1

m ),

where ζ is the distance between connected subparts and cn is an increasing sequence of positivenumbers.

Now, λγ ≈ 12 is the kind of situations encountered for metallic nanoparticles immersed in water

or some biological tissue. As an example, the thermal conductivity of gold is γc = 318 WmK and

that of pure water is γm = 0.6 WmK . This gives λγ ≈ 0.5019.

In view of this, the second term in (4.25) may increase considerably for some type of close-to-touching particles. Nevertheless, we stress that this is not the general case. For a more refinedanalysis, asymptotics of the eigenfunctions of K∗D should be also studied.


The numerical experiments for this work can be divided into two parts. The first one illustratesthe inner asymptotic expansion of scattered field derived in Theorem 4.2.1. The second part

97


illustrates the temperature distribution proved in Theorem 4.2.2. We consider both the singleand multiple particle cases.

The major tasks surrounding the numerical implementation of the asymptotic formulas in The-orems 4.2.1 and 4.2.2 are the numerical computations of the operators FD[·] and ∂νFD[·], whichcan be achieved by meshing the domain D and integrating semi-analytically inside the trianglesthat are close to the singularities. Towards this, we use the following formula to avoid numericaldifferentiation:


=1

2πbc

∫D

exp(−|x − y|2

4bct

)〈y − x, νx〉

|x − y|2gu(y)dy, x ∈ ∂D. (4.26)

For all the performed simulations, we consider an incident plane wave given by

ui(x) = eikmd·x,

where d = (1,1)/√

2 ∈ R2 is the illumination direction and km = 2π/750 · 109 is the frequency(in the red range). The considered nanoparticles are ellipses with semi-axes 30nm and 20nm,respectively.

It is worth noticing that the illumination direction d is relevant solely in the asymptotic formulain Theorem 4.2.1. Its role is to define the coefficients of a linear combination of both componentsof SD(λεI −K∗D)−1[v] ∈ R2. We will see from the numerical simulations that this is fundamentalif we wish to maximize the produced electromagnetic field, and therefore the generated heatinside the nanoparticles.

With respect to the asymptotic formula established in Theorem 4.2.1, besides the nanoparticle’sshape D, the sole parameter that is left is λε. For all the following simulations, we will considerthis as a free parameter that we will use to excite the eigenvalues of the Neumann-Poincareoperator and hence to generate resonances. The physical justification that allows us to do this isbased on the Drude model [7]. Whenever we mention that we approach a particular eigenvalueλ j of K∗D, we will set λε = λ j + 0.001i.

With respect to the heat equation coefficients, we use realistic values of gold for nanoparticles,and water for tissues.

4.5.1 Single-particle simulation

We consider one elliptical nanoparticle D b R2 centered at the origin, with its semi-major axisaligned with the x-axis.

98


Single-particle plasmonic resonance

Resonances are attained by exciting the eigenvalues of the Neumann-Poincare operator K∗D as-sociated to eigenfunctions which are not orthogonal in H∗(∂D) to the normal ν of ∂D. Itturns out that for some eigenfunctions of K∗D, the normal of the shape is almost orthogonal, inH∗(∂D), to them. Therefore, we cannot observe resonance for their associated eigenvalues; see[15]. In Figure 4.1 we can see values of the inner product between the eigenfunctions ofK∗D andthe components νx and νy of ν. Figure 4.1 suggests us which are the available resonant modeswith the respective strength of each coordinate. In Figure 4.2 we present the absolute value of

0 10 20 30 40 500

2

4

6

8

10

x

y

Figure 4.1: Inner product inH∗(∂D) between the eigenfunctions of K∗D and the components νx and νy ofthe normal ν to ∂D.

the inner field for the first three resonant modes, corresponding to the second, third and sixtheigenvalue of K∗D, respectively. The resonant modes are associated to normalized eigenfunc-tions of K∗D in H∗(∂D). In Figure 4.3 we decompose the inner field into the zeroth-order andthe first-order terms respectively given by ui(z) + (x − z)∇ui(z) and SD

(λεI − K∗D

)−1[ν] · ∇ui(z).Figure 4.4 shows the components of the vector SD(λεI − K∗D)−1[ν].

First resonance mode Second resonance mode Third resonance mode

Figure 4.2: Absolute value of the electromagnetic field inside the nanoparticle at the first resonant modes,being those when λε approaches the second, third and sixth eigenvalue of K∗D.

From Figure 4.3, we can see that when we excite the nanoparticle at its resonant mode, thelargest contribution to the electromagnetic field comes from the first-order term of the smallvolume expansion formula established in Theorem 4.2.1.

99


Zeroth-order component First-order component

Figure 4.3: First resonant mode of the nanoparticle decomposed in its first- and second-order term in theformula given by Theorem 4.2.1. Both images are absolute values of the respective compo-nent.

The x-component The y-component

Figure 4.4: Absolute value of the vectorial components of the first-order term for the first resonant mode.

Observing the vectorial components of the first-order term in Figure 4.4 tells us how importantis the illumination direction as the x-component is significantly stronger than the y-component.If we wish to maximize the electromagnetic field and therefore the generated heat, the recom-mended illumination direction would be around d = (1, 0)t (with t being the transpose), as it wasinitially suggested by Figure 4.1.

Single-particle surface heat generation

Considering the electromagnetic field inside the nanoparticle given by the first resonant modepresented in Figure 4.2, following the formula given by Theorem 4.2.2, we compute the gener-ated heat on the surface of the nanoparticle. In Figure 4.5 we plot the generated heat in threedimensions and present a two dimensional plot obtained by parameterizing the boundary. InFigure 4.6 we decompose the heat in its first- and second-order terms given by formula 4.2.2,

being FD(x, t, bc) and −VbcD (λγI−K∗D)−1[

∂FD(·, ·, bc)∂ν

](x, t), respectively. In Figure 4.7, we showthe evolution of the heat profile plotted in Figure 4.5 over time. More precisely, we integrate thetotal heat on the boundary and plot it as a function of time, for each component.

We can observe that the heat is not symmetric, this can be noticed from the total inner field forthe first resonance mode in Figure 4.2. The reason behind this non symmetry is because we areilluminating with direction d = (1,1)t/

√2 over an ellipse. From Figure 4.7 we can notice that the

first-order term converges, while the zeroth-order term increases logarithmically, as expected

100


3D plot of generated heat at time T = 1 2D plot of generated heat at time T = 1

1.2

2 2

1.3

10-8

10-8

00

1.4

10-14

1.5

-2-2

- - /2 0 /21

1.2

1.4

1.610

-14

Figure 4.5: At the left-hand side, we can see a three-dimensional plot of the nanoparticle heat, the redshape is a reference value to show where the nanoparticle is located. At the right-hand side wecan see a two-dimensional plot of the generated heat, where the boundary was parametrizedfollowing p(θ) = (a cos(θ), b sin(θ)), θ ∈ [−π, π], with a and b being the semi-major andsemi-minor axes, respectively.

Two-dimensional plot of the zeroth-orderterm at T = 1

Two-dimensional plot of the first-order termat T = 1

- - /2 0 /23.24

3.26

3.28

3.310

-15

- - /2 0 /20.8

1

1.2

1.410

-14

Figure 4.6: Two-dimensional plots of the zeroth- and first-order components of the heat on the boundarywhen time is equal to one. As time goes on, each point of the graph increases, but the generalshape is preserved.

Integrated heat over time Integrated zeroth-order componentof heat over time

Integrated first-order component ofheat over time

10-5

10-3

100

time

0

1.25

2.5

10-20

10-5

10-3

100

time

4.2

5

5.8

10-21

10-5

10-3

100

time

10-25

10-20

10-15

Figure 4.7: Time-logarithmic plots showing the total heat on the boundary for each component of theheat. The values were obtained for each fixed time, by integrating over the boundary thecomputed heat. From left- to right-hand side: The total heat, the zeroth-order and its first-order, according to formula given by Theorem 4.2.2. Notice that the first-order term is plottedin a log-log scale.

from the known solution of the heat equation for constant source in two dimensions that the heatincreases logarithmically.

101


4.5.2 Two particles simulation

We consider two elliptical nanoparticles D1, D2, D = D1 ∪ D2, with the same shape and orien-tation as the nanoparticle considered in the above example. The particle D1 is centered at theorigin and D2 is centered at (0, 4.1 · 10−9), resulting in a separation distance of 0.1nm betweenthe two particles.

Two particles Helmholtz resonance

Following the same analysis as the one for a single particle, in Figure 4.8 we present the innerproduct between the eigenfunctions of K∗D with each component of the normal of D. We canobserve that there are more available resonant modes. In particular we can see that when λεapproaches the 36th or 37th eigenvalues, we achieve strong resonant modes.

0 10 20 30 40 500

2

4

6

8

10

12

x

y

Figure 4.8: Inner product inH∗(∂D) between the eigenfunctions ofK∗D and each component of the normalof ∂D, νx and νy.

In Figure 4.9 we show the absolute value of the inner field for the resonant modes correspondingto the 6th, 37th and 38th eigenvalues of K∗D. Similarly to the case with one particle, the dom-inant term in the electromagnetic field for each case is the first-order term. In Figure 4.10 wedecompose the first-order term in its x-component and y-component.

As suggested by Figure 4.8, the inner product inH∗(∂D) between the 38th eigenfunction of K∗Dand νy is stronger than the one between the 38th eigenfunction of K∗D and νx. This means that ifwe wish to maximize the electromagnetic field, and therefore the generated heat, it is suggestedto consider the illumination vector d = (0, 1)t.

Two particles surface heat generation

Similarly to the analysis carried out for one particle, we now compute the generated heat forthese two particles while undergoing resonance for the resonant mode associated to the 38th

102


Resonant mode associated to the 6theigenvalue of K∗D

Resonant mode associated to the37th eigenvalue of K∗D

Resonant mode associated to the38th eigenvalue of K∗D

Figure 4.9: Absolute value of the electromagnetic field inside the nanoparticle at the resonant modes asso-ciated to the 6th, 37th and 38th resonant modes, obtained when λε approaches the respectiveeigenvalues of K∗D.

The x-component The y-component

Figure 4.10: Absolute value of the vectorial components of the first-order term for the 38th resonantmode.

eigenvalue of K∗D. In Figure 4.11 we plot generated heat in the boundary of the two nanoparti-cles. In Figure 4.12 we decompose the generated heat in its zeroth and first-order component,explicited for each of the two nanoparticles.

Similarly to the single nanoparticle case, there is no symmetry on the heat values on the bound-ary, which is due to the illumination. We have not provided the plots of the heat integrated alongthe boundary, as the conclusions are the same as the ones in the single nanoparticle case: Thetotal heat on the boundary increases logarithmically, initially on time the dominant term is thefist-order one, but as time increases the zeroth-order term becomes the predominant one.

103


3D plot of heat on ∂(D1 ∪ D2) at time T = 1.

4

6

5

6

4

7

8

10-8

2

10-12

9

2

10

10-800

11

12

-2

13

-2

Heat on ∂D1 at time T = 1

- - /2 0 /20

0.5

1

1.510

-11

Heat on ∂D2 at time T = 1

- - /2 0 /20

0.5

1

1.510

-11

Figure 4.11: Generated heat on the boundary of the nanoparticles for the 38th mode and T = 1. On theleft we can see a three dimensional view of the heat, the red shapes are referential to showthe location of the nanoparticles. On the right-hand side we can see the two dimensionalheat plots corresponding to each nanoparticle. To obtain these plots we parameterized theboundary of each nanoparticle with p(θ) = (a cos(θ), b sin(θ)) + z, θ ∈ [−π, π], where z ∈ R2

corresponds to the center of each nanoparticle. On the top we can see nanoparticle D2 andon the bottom nanoparticle D1.

4.6 Concluding remarks

In this chapter we have derived asymptotic formula for the temperature elevation due to plas-monic nanoparticles. We have considered thermal coupling within close-to-touching nanoparti-cles, where the temperature field deviates significantly from the one generated by single nanopar-ticles. Combined with the methods developed in [10, 11], our results can be used for the opticaland thermal detection and localization of plasmonic nanoparticles. As reported in [79], the de-tection and localization of nanoparticles in highly scattering media such as biological tissue re-mains a challenge. They can also be used for monitoring temperature elevation due to plasmonicnanoparticles based on the photoacoustic signal recently analyzed in [93]. Thermoacoustic sig-nals generated by nanoparticle heating can be computed numerically based on the successiveresolution of the thermal diffusion problem considered in this chapter and a thermoelastic prob-lem, taking into account the size and shape of the nanoparticle, thermoelastic and elastic prop-erties of both the particle and its environment, and the temperature-dependence of the thermalexpansion coefficient of the environment. For sufficiently high illumination fluences, this tem-perature dependence yields a nonlinear relationship between the photoacoustic amplitude and

104

4.7 Appendix: Asymptotic analysis of the single-layer potential in two dimensions

Heat zeroth component on ∂D2

- - /2 0 /23.24

3.26

3.28

3.310

-12

Heat zeroth component on ∂D1 at time T = 1

- - /2 0 /23.24

3.26

3.28

3.310

-12

Heat first component on ∂D2 at time T = 1

- - /2 0 /20

0.5

1

1.510

-11

Heat first component on ∂D1 at time T = 1

- - /2 0 /2-5

0

5

10

1510

-12

Figure 4.12: Two-dimensional plots of the zeroth and first component of the heat at time 1, for eachnanoparticle. On the left column we have the zeroth component of the heat, on the right-hand side column we have the first component of the heat. On top we show the values fornanoparticle D2, on the bottom we show the values for nanoparticle D1. We remark that thetouching point corresponds to −π/2 for D1 and π/2 for D2.

the fluence [82]. It would be very interesting to investigate this nonlinear model.

4.7 Appendix: Asymptotic analysis of the single-layer potentialin two dimensions

In this section we make an analysis of the single-layer potential SkD for small values of k, i.e,

|k| 1. We use this in Section 4.4.1 to derive expansions with respect to δ of the operatorAB(δ)and its inverse.

The results in this section were first established in [21] for a connected domain D. Here wegeneralize them to non-connected domains. The main observation is the invertibility of theoperatorAD defined by (4.27) for D non-connected.

105


4.7.1 Layer potentials for the Laplacian in two dimensions

Recall the definition of the single-layer potential and Neumann-Poincare operators for the Lapla-cian:

SD[ϕ](x) =

∫∂D

12π

log |x − y|ϕ(y)dσ(y), x ∈ ∂D,

K∗D[ϕ](x) =

∫∂D

12π

(x − y, ν(x))|x − y|

ϕ(y)dσ(y), x ∈ ∂D.

Let (·, ·)− 12 ,

12

denote the duality pairing between H−1/2(∂D) and H1/2(∂D).

In R2 the single-layer potential SD : H−1/2(∂D) → H1/2(∂D) is not, in general, invertible.Hence, −(u,SD[v])− 1

2 ,12

does not define an inner product and the symmetrization technique de-scribed in [9, subsection 2.1.4] is no longer valid.

To overcome this difficulty, we will introduce a substitute of SD, in the same way as in [21].

We first need the following lemma.

Lemma 4.7.1. Let C = ϕ ∈ H−1/2(∂D); ∃ α ∈ C, SD[ϕ] = α. We have dim(C) = 1.

Proof. It is known that

AD : H−1/2(∂D) × C → H1/2(∂D) × C

(ϕ, a) 7→(SD[ϕ] + a,

∫∂Dϕdσ

), (4.27)

is invertible [13, Theorem 2.26].

We can see that C = Π1A−1D (0,C), where Π1[(ϕ, a)] = ϕ. The invertibility of AD implies that

Ker(Π1A−1D (0, ·)) = 0. Thus, by the range theorem we have

1 = dim(Im(Π1A−1D (0, ·))) + dim(Ker(Π1A

−1D (0, ·))) = dim(Im(Π1A

−1D (0, ·))) = dim(C).

Definition 4.7.1. We call ϕ0 the unique element of C such that∫∂D ϕ0dσ = 1.

Note that for every ϕ ∈ H−1/2(∂D) we have the decomposition

ϕ = ϕ −( ∫

∂Dϕdσ

)ϕ0 +

( ∫∂Dϕdσ

)ϕ0 := ψ +

( ∫∂Dϕdσ

)ϕ0,

where we can see that (ψ, 1)− 12 ,

12

= 0. This kind of decomposition, ϕ = ψ+αϕ0, with (ψ, 1)− 12 ,

12

=

0 is unique.

Note that we can decompose H−1/2 as a direct sum of elements with zero-mean and multiples ofϕ0, H−1/2(∂D) = H−1/2

0 (∂D) ⊕ µϕ0, µ ∈ C. This allows us to define the following operator.

106


Definition 4.7.2. Let SD be the linear operator that satisfies

SD : H−1/2(∂D) → H1/2(∂D)

ϕ →

SD[ϕ] if (ϕ, 1)− 1

2 ,12

= 0,−1 if ϕ0 = ϕ.

Remark 4.7.2. When SD is invertible, SD is similar enough to keep the invertibility. When SD

is not invertible, then C = ker(SD) and the operator SD becomes an invertible alternative to SD

that images the kernel C to the space µχ(∂D), µ ∈ C.

Remark 4.7.3. SD : H−1/2(∂D)→ H1(D) follows the same definition.

Theorem 4.7.4. SD is invertible, self-adjoint and negative for (·, ·)− 12 ,

12

and satisfies the follow-

ing Calderon identity: SDK∗D = KDSD.

Proof. The invertibility is a direct consequence of Lemma 4.7.1.

Indeed, since SD is Fredholm of zero index, so is SD. Therefore, we only need the injectivity.Suppose that, ∃ ϕ , 0 such that SD[ϕ] = 0. This mean that, ∃ α , 0 ∈ C such that ϕ = αϕ0.Therefore, SD[ϕ] = αSD[ϕ0] = −α = 0, which is a contradiction. Hence ϕ = 0.

The self-adjointness comes directly form that of SD. Noticing that ϕ0 is an eigenfunction ofeigenvalue 1/2 of K∗D we get the Calderon identity from a similar one satisfied by SD: SDK

∗D =

KDSD; see [9, Lemma 2.12].

It is known that∫∂D ψSD[ψ]dσ < 0 if (ψ, 1)− 1

2 ,12

= 0 and ψ , 0, see [9, Lemma 2.10]. Therefore,

writing ϕ = ψ +( ∫

∂D ϕdσ)ϕ0, with ψ = ϕ −

( ∫∂D ϕdσ

)ϕ0, and noticing that

∫∂D ϕ0SD[ψ]dσ =∫

∂D SD[ϕ0]ψdσ = −∫∂D ψdσ = 0, we have∫

∂DϕSD[ϕ]dσ =

∫∂DψSD[ψ]dσ +

( ∫∂Dϕdσ

)2SD[ϕ0]

=

∫∂DψSD[ψ]dσ −

( ∫∂Dϕdσ

)2< 0,

if ϕ , 0.

Definition 4.7.3. We define the space H∗(∂D) as the Hilbert space resulting from endowingH−1/2(∂D) with the inner product

(u, v)H∗ := −(u, SD[v])− 12 ,

12. (4.28)

Similarly, we letH to be the Hilbert space resulting from endowing H1/2 with the inner product

(u, v)H = −(S−1D [u], v)− 1

2 ,12. (4.29)

107


If D is C1,α, we have the following result.

Lemma 4.7.5. Let D be a C1,α bounded domain of R2 and let SD be the operator introduced inDefinition 4.7.2. Then the following hold:

(i) The operator K∗D is compact self-adjoint in the Hilbert space H∗(∂D) and H∗(∂D) isequivalent to H−

12 (∂D); Similarly, the Hilbert spaceH(∂D) is equivalent to H

12 (∂D).

(ii) Let (λ j, ϕ j), j = 0, 1, 2, . . . , be the eigenvalue and normalized eigenfunction pair of K∗Dwith λ0 = 1

2 . Then, λ j ∈ (−12 ,

12 ] and λ j → 0 as j→ ∞.

(iii) The following representation formula holds: for any ϕ ∈ H−1/2(∂D),

K∗D[ϕ] =

∞∑j=0

λ j(ϕ, ϕ j)H∗ ⊗ ϕ j.

The following lemmas are needed in the proof of Theorem 4.2.1 and Theorem 4.2.2.

Lemma 4.7.6. Let D = z + δB and η be the function such that, for every ϕ ∈ H∗(∂D), η(ϕ)(x) =

ϕ(z + δx), for almost all x ∈ ∂B. Then

‖ϕ‖H∗(∂D) = δ‖η(ϕ)‖H∗(∂B).

Similarly, if for every ϕ ∈ L2(D), η(ϕ)(x) = ϕ(z + δx), for almost all x ∈ B, then

‖ϕ‖L2(D) = δ‖η(ϕ)‖L2(B).

Proof. We only prove the scaling inH∗(∂D). From the proof of Theorem 4.7.4, we have

‖ϕ‖2H∗(∂D) = −

∫∂DψSD[ψ]dσ +

( ∫∂Dϕdσ

)2,

where ψ = ϕ −( ∫

∂D ϕdσ)ϕ0. Note that (ψ, 1)− 1

2 ,12

= 0 and so, (η(ψ), χ(∂B))− 12 ,

12

= 0 as well.

By a rescaling argument we find that

‖ϕ‖2H∗(∂D)

= −δ2∫∂B

∫∂B

12π

log |δ(x − y)|η(ψ)(x)η(ψ)(y)dσ(x)dσ(y) + δ2( ∫

∂Bη(ϕ)dσ

)2

= −1

2πδ2 log(δ)

( ∫∂Bη(ψ)dσ

)2+ δ2

(−

∫∂Bη(ψ)SD[η(ψ)]dσ +

( ∫∂Bη(ϕ)dσ

)2)

= δ2‖η(ϕ)‖2H∗(∂B).

108


Lemma 4.7.7. Let g ∈ H1(D) be such that ∆g = f with f ∈ L2(D). Then, inH∗(∂D),(12

I − K∗D

)S−1

D [g] = −∂g∂ν

+ T f .

For some T f ∈ H∗(∂D) and ‖T f ‖H∗ ≤ C‖ f ‖L2(D) for a constant C.

Moreover, if g ∈ H1loc(R2), ∆g = 0 in R2\D, lim|x|→∞ g(x) = 0, then

T f = c fϕ0 + S−1D [g],

with

c f =

∫D

f (x)dx −∫∂DS−1

D [g](y)dσ(y),

where ϕ0 is given in Definition 4.7.1. Here, by an abuse of notation, we still denote by g thetrace of g on ∂D.

Proof. Let ϕ ∈ H∗(∂D). Then((12

I − K∗D

)S−1

D [g], ϕ)H∗

= −

(S−1

D [g],(12

I − KD

)SD[ϕ]

)− 1

2 ,12

= −

(S−1

D [g], SD

(12

I − K∗D

)[ϕ]

)− 1

2 ,12

= −

(g,

(12

I − K∗D

)[ϕ]

)− 1

2 ,12

= −

g,−∂SD[ϕ]∂ν

∣∣∣∣−

− 1

2 ,12

=

∫∂D

∂g∂νSD[ϕ]dσ −

∫D

(f SD[ϕ] − ∆SD[ϕ]

(g))

dx

= −

(∂g∂ν, ϕ

)H∗

−

∫D

f SD[ϕ]dx.

We have used the fact that SD[ϕ] is harmonic in D.

Consider the linear application T f [ϕ] := −∫

D f SD[ϕ]dx. We have

|T f [ϕ]| ≤ C‖ f ‖L2(D)‖SD[ϕ]‖L2(D) ≤ C f ‖SD[ϕ]‖H1(D) ≤ C f ‖SD[ϕ]‖H

12 (∂D)

≤ C f ‖ϕ‖H−12 (∂D)

.

Here we have used Holder’s inequality, a standard Sobolev embedding, the trace theorem andthe fact that SD : H−

12 (∂D)→ H

12 (∂D) is continuous. By the Riez representation theorem, there

exists v ∈ H∗(∂D) such that T f [ϕ] = (v, ϕ)H∗ ,∀ϕ ∈ H∗(∂D).

109


By abuse of notation, we still denote T f := v to make explicit the dependency on f . It followsthat

‖T f ‖2H∗

= −

∫D

f SD[T f ]dx ≤ C‖ f ‖L2(D)‖SD[T f ]‖L2(D)

≤ C‖ f ‖L2(D)‖SD[T f ]‖H1(D)

≤ C‖ f ‖L2(D)‖SD[T f ]‖H 12 (∂D)

≤ C‖ f ‖L2(D)‖T f ‖H∗ .

We now show that inH∗0 (∂D), T f = S−1D [g].

Indeed, let ϕ ∈ H∗0 (∂D); then(S−1

D [g], ϕ)H∗

= −(S−1

D [g], SD[ϕ])− 1

2 ,12

= − (g, ϕ)− 1

2 ,12

= −

g, ∂SD[ϕ]∂ν

∣∣∣∣+−∂SD[ϕ]∂ν

∣∣∣∣−

− 1

2 ,12

=

∫∂D

∂g∂νSD[ϕ]dσ −

∫∂D

∂g∂νSD[ϕ]dσ +

∫∂B∞

∂g∂νSD[ϕ]dσ

−

∫∂B∞

g∂SD[ϕ]∂ν

dσ −∫R2

(f SD[ϕ] − ∆SD[ϕ]

(g))

dx

= −

∫D

f SD[ϕ]dx.

Here we have used the assumption on g, the fact that SD[ϕ] is harmonic in D andR2\D and that

for ϕ ∈ H∗0 (∂D) we have SD[ϕ](x) = O( 1|x| ) and

∂SD[ϕ]∂ν

(x) = O( 1|x| ) for |x| → ∞.

Therefore,

T f = (T f − S−1D [g], ϕ0)H∗ϕ0 + S−1

D [g].

Finally, recaling the definition of ϕ0 given in Definition 4.7.1 we obtain that

(T f − S−1D [g], ϕ0)H∗ =

∫D

f (x)dx −∫∂DS−1

D [g](y)dσ(y).

110


4.7.2 Asymptotic expansions

Let us now consider the single-layer potential for the Helmholtz equation in R2 given by

SkD[ϕ](x) =

∫∂D

G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D,

where G(x, y, k) = −i4

H(1)0 (k|x− y|) and H(1)

0 is the Hankel function of first kind and order 0. Wehave, for k 1,

−i4

H(1)0 (k|x − y|) =

12π

log |x − y| + τk +

∞∑j=1

(b j log k|x − y| + c j)(k|x − y|)2 j,

where

τk =1

2π(log k + γe − log 2) −

i4, b j =

(−1) j

2π1

22 j( j!)2 , c j = −b j

γe − log 2 −iπ2−

j∑n=1

1n

,and γe is the Euler constant. Thus, we get

SkD = Sk

D +

∞∑j=1

(k2 j log k

)S

(1)D, j +

∞∑j=1

k2 jS(2)D, j, (4.30)

where

SkD[ϕ](x) = SD[ϕ](x) + τk

∫∂Dϕdσ,

S(1)D, j[ϕ](x) =

∫∂D

b j|x − y|2 jϕ(y)dσ(y),

S(2)D, j[ϕ](x) =

∫∂D|x − y|2 j(b j log |x − y| + c j)ϕ(y)dσ(y).

Lemma 4.7.8. The norms ‖S(1)D, j‖L(H∗(∂D),H(∂D)) and ‖S(2)

D, j‖L(H∗(∂D),H(∂D)) are uniformly boundedwith respect to j. Moreover, the series in (4.30) is convergent in L(H∗(∂D),H(∂D)) for k < 1.

Observe that(SD − SD

)[ϕ] =

(SD − SD

)[PH∗0 [ϕ] + (ϕ, ϕ0)H∗ϕ0] = (ϕ, ϕ0)H∗ (SD[ϕ0] + 1) .

Then it follows that

SkD[ϕ] = SD[ϕ] + (ϕ, ϕ0)H∗ (SD[ϕ0] + 1) + τk

∫∂DPH∗0

[ϕ] + (ϕ, ϕ0)H∗ϕ0dσ = SD[ϕ] + Υk[ϕ],

111


whereΥk[ϕ] = (ϕ, ϕ0)H∗ (SD[ϕ0] + 1 + τk) . (4.31)

Therefore, we arrive at the following result.

Lemma 4.7.9. For k small enough, SkD : H∗(∂D)→ H(∂D) is invertible.

Proof. Υk is clearly a compact operator. Since SD is invertible, the invertibility of SkD is

equivalent to that of SkDS−1D = I + ΥkS

−1D . By the Fredholm alternative, we only need to prove

the injectivity of I + ΥkS−1D .

Since ∀ v ∈ H1/2(∂D), ΥkS−1D [v] ∈ C, for

(I + ΥkS

−1D

)[v] = 0, we need to show that v =

SD[αϕ0] = −α ∈ C.We have (

I + ΥkS−1D

)SD[αϕ0] = α(SD[ϕ0] + τk) = 0 iff SD[ϕ0] = −τk or α = 0.

Since we can always find a small enough k such that SD[ϕ0] , −τk, we need α = 0. This yieldsthe stated result.

Lemma 4.7.10. For k small enough, the operator SkD : H∗(∂D)→ H(∂D) is invertible.

Proof. The operator SkD − S

kD : H∗(∂D) → H(∂D) is a compact operator. Because Sk

D isinvertible for k small enough, by the Fredholm alternative only the injectivity of Sk

D is necessary.It is indeed known that Sk

D is injective for k2 not an eigenvalue of −∆ with Dirichlet boundaryconditions on ∂D. Therefore, for k2 smaller than the first Dirichlet eigenvalue in D, Sk

D isinvertible.

Lemma 4.7.11. The following asymptotic expansion holds for k small enough:

(SkD)−1 = PH∗0

S−1D +Uk − k2 log kPH∗0 S

−1D S

(1)D,1PH

∗0S−1

D + O(k2)

with

Uk = −(S−1

D [·], ϕ0)H∗

SD[ϕ0] + τkϕ0. (4.32)

Note thatUk = O(1/ log k).

Proof. We can write (4.30) asSk

D = SkD + Gk,

where Gk = k2(log k)S(1)D,1 + O(k2). From Lemma 4.7.9 and Lemma 4.7.10 we get the identity

(SkD)−1 =

(I + (Sk

D)−1Gk)−1

(SkD)−1.

112


Hence, we have(Sk

D)−1 =(S−1

D SkD

)−1︸︷︷︸Λ−1

k

S−1D .

Here,

Λk = I − (·, ϕ0)H∗(SD[ϕ0] + 1 + τk)ϕ0

= PH∗0− (·, ϕ0)H∗(SD[ϕ0] + τk)ϕ0.

Then,

Λ−1k = PH∗0 − (·, ϕ0)H∗

1SD[ϕ0] + τk

ϕ0,

and therefore,

(SkD)−1 = PH∗0 S

−1D −

(S−1D [·], ϕ0)H∗

SD[ϕ0] + τkϕ0.

It is clear that ‖(SkD)−1‖L(H(∂D),H∗(∂D)) is bounded for k small. Since ||Gk||L(H(∂D),H∗(∂D)) → 0 as

k → 0, for k small enough, we can write

(SkD)−1 = (Sk

D)−1 − (SkD)−1Gk(Sk

D)−1 + O(k4(log k)2

),

which yields the desired result.

We now consider the expansion of the boundary integral operator (KkD)∗ as k → 0. We have

(KkD)∗ = K∗D +

∞∑j=1

(k2 j log k

)K

(1)D, j +

∞∑j=1

k2 jK(2)D, j, (4.33)

where

K(1)D, j[ϕ](x) =

∫∂D

b j∂|x − y|2 j

∂ν(x)ϕ(y)dσ(y),

K(2)D, j[ϕ](x) =

∫∂D

∂(|x − y|2 j(b j log |x − y| + c j)

)∂ν(x)

ϕ(y)dσ(y).

Lemma 4.7.12. The norms ‖K (1)D, j‖L(H∗(∂D),H∗(∂D)) and ‖K (2)

D, j‖L(H∗(∂D),H∗(∂D)) are uniformly boundedfor j ≥ 1. Moreover, the series in (4.33) is convergent in L(H∗(∂D),H∗(∂D)).

113


114

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Curriculum Vitae

Personal details

Name Francisco Romero Hinrichsen

Date of birth December 1st, 1989

Place of birth Santiago, Chile

Citizenship Chilean

Education

PhD studies in Mathematics 2015–2018ETH Zurich Zurich, Switzerland

Master in Mathematics 2015Universite Paris-Sud Paris, France

Mathematical engineering degree 2008–2014Universidad de Chile Santiago, Chile

Publications

Dynamic spike super-resolution and applications to ultrafast ultrasound imaging.Giovanni S. Alberti, Habib Ammari, Francisco Romero and Timothe Wintz.arXiv:1803.03251, submitted to the SIAM Journal on Imaging Sciences, 2018

Heat generation with plasmonic nanoparticles.Habib Ammari, Francisco Romero and Matias Ruiz.SIAM Journal on Multiscale Modeling and Simulation, 2018

A Signal separation technique for sub-cellular imaging using optical coherence tomogra-phy. Habib Ammari, Francisco Romero and Cong Shi.SIAM Journal on Multiscale Modeling and Simulation, 2017

123

References

Mathematical analysis of ultrafast ultrasound imaging.Giovanni S. Alberti, Habib Ammari, Francisco Romero and Timothe Wintz.SIAM Journal on Applied Mathematics, 2017

Simultaneous source and attenuation reconstruction in SPECT using ballistic and singlescattering data. Matias Courdurier, Francois Monard, Axel Osses and Francisco Romero.Inverse Problems, 2015

Presentations

SIAM conference on imaging science Bologna, ItalyDynamic super-resolution with applications to ultrafast ultrasound June 2018

Equadiff 2017 Bratislava, SlovakiaHeat generation with Plasmonic Nanoparticles July 2017

Applied Inverse Problems Hangzhou, ChinaHeat generation with Plasmonic Nanoparticles May 2017

124

Documents

Mathematics of medical imaging with dynamic data