Vrije Universiteit Brussel

Ultrasound Imaging From Sparse RF Samples Using System Point Spread FunctionsSchretter, Colas; Bundervoet, Shaun; Blinder, David; Dooms, Ann; D'Hooge, Jan; Schelkens,PeterPublished in:IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control

DOI:10.1109/TUFFC.2017.2772916

Publication date:2018

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Document Version:Final published version

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Citation for published version (APA):Schretter, C., Bundervoet, S., Blinder, D., Dooms, A., D'Hooge, J., & Schelkens, P. (2018). Ultrasound ImagingFrom Sparse RF Samples Using System Point Spread Functions. IEEE Transactions on Ultrasonics,Ferroelectrics and Frequency Control, 65(3), 316-326. https://doi.org/10.1109/TUFFC.2017.2772916

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https://doi.org/10.1109/TUFFC.2017.2772916https://cris.vub.be/portal/en/publications/ultrasound-imaging-from-sparse-rf-samples-using-system-point-spread-functions(99fb667c-a1a3-4cff-802c-a571e0c3d758).htmlhttps://doi.org/10.1109/TUFFC.2017.2772916

316 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

Ultrasound Imaging From Sparse RF SamplesUsing System Point Spread Functions

Colas Schretter , Member, IEEE, Shaun Bundervoet, David Blinder, Student Member, IEEE,Ann Dooms, Member, IEEE, Jan D’hooge, Member, IEEE, and Peter Schelkens, Member, IEEE

Abstract— Upcoming phased-array 2-D sensors will soonenable fast high-definition 3-D ultrasound imaging. Currently,the communication of raw radio-frequency (RF) channel datafrom the probe to the computer for digital beamforming is abottleneck. For reducing the amount of transferred data samples,this paper investigates the design of an adapted sparse samplingtechnique for image reconstruction inspired by the compressedsensing framework. Echo responses from isolated points aregenerated using a physically based simulation of ultrasoundwave propagation through tissues. These point spread functionsform a dictionary of shift-variant bent waves, which depend onthe specific sound excitation and acquisition protocols. Speckledultrasound images can be approximately decomposed in thisdictionary where sparsity is enforced at the system matrix design.The Moore–Penrose pseudoinverse is precomputed and used atthe reconstruction stage for fast minimum-norm recovery fromnonuniform pseudorandom sampled raw RF data. Results onsimulated and acquired phantoms demonstrate the benefits of anoptimized basis function design for high-quality B-mode imagerecovery from few RF channel data samples.

Index Terms— Compressed sensing (CS), physically basedsimulation, point spread function (PSF), ultrasound imaging.

I. INTRODUCTION

ULTRASOUND echographic modalities are a ubiquitouschoice for nonionizing real-time medical imaging. Thetruly interactive nature of handheld devices provides the abilityto naturally capture motion. Tremendous benefits have beendemonstrated for various clinical applications and, in partic-ular, for imaging structural heart diseases in interventionalcardiology [1]. Phased-array ultrasound probes for cardiacimaging already allow 3-D data acquisitions at acceptableframe rates [2], and latest advances in high-resolution solid-state detectors enable measurements of strain in smaller struc-tures for orthopedic applications [3]. Organs of interests are

Manuscript received July 6, 2017; accepted November 9, 2017. Date ofpublication November 9, 2017; date of current version March 1, 2018.This work was supported in part by the Agency for Innovation by Scienceand Technology in Flanders (IWT) and the iMinds 3-DUS ICON Projectunder Grant 131813, and in part by the European Research Council throughthe European Union’s Seventh Framework Programme (FP7/2007-2013)/ERCunder Grant 617779 (INTERFERE). (Corresponding author: Colas Schretter.)

C. Schretter, D. Blinder, and P. Schelkens are with the Department ofElectronics and Informatics, Vrije Universiteit Brussel, B-1050 Brussels,Belgium, and also with imec, B-3001 Leuven, Belgium (e-mail:cschrett@vub.ac.be).

S. Bundervoet and A. Dooms are with the Digital MathematicsResearch Group, Department of Mathematics, Vrije Universiteit Brussel,B-1050 Brussels, Belgium.

J. D’hooge is with the Department of Cardiovasular Sciences, University ofLeuven, Medical Imaging Research Center, University Hospitals LeuvenGasthuisberg, B-3000 Leuven, Belgium.

Digital Object Identifier 10.1109/TUFFC.2017.2772916

bundles of thin fiber structures, such as tendons, which are cur-rently difficult to see with large low-frequency transducers [4].Speckle tracking on such high-definition image sequences issuccessful in capturing subtle local deformation patterns [5].

In high-definition 3-D ultrasound imaging, a large data setof received multichannel echo responses should be recordedfor reconstructing each image line. Hardware-driven solu-tions exist where the full raw radio-frequency (RF) dataare microbeamformed with dedicated electronic circuits on amatrix probe [6] or more recently, a low-consumption FPGA isprogrammed for beamforming up to 32×32 channels [7], [8].For larger phased arrays, the necessary data bandwidth isincreasing with the surface area of the transducer and nowexceeds available resources for larger phased-array sensors.

For overcoming the data transfer challenge with a lighterhardware design on the probe, this paper investigates a sparseregularization prior for approximating RF channel data fromfew scattered measurements only. The full data are representedby linear combinations of a representative set of simulatedsystem point spread functions (PSFs). These smooth functionsform a dictionary of shift-variant bent waves that are tailoredto the specific geometry and acquisition protocol. The leastsquares method is then used at the reconstruction stage forrobust recovery of complete RF channel data.

Several image reconstruction solutions have been recentlyproposed in the ultrasound literature for tackling image recov-ery from only a subset of measurements. A promising lineof works that we build upon is leveraging the compressedsensing (CS) framework [9], [10] by exploiting sparsity forregularizing the solution space. Regularization allows for dras-tic reduction of the amount of recorded data with competitiveimage quality. The book of Eldar and Kutyniok [11] gives aview of CS in applicative imaging contexts.

Prior works in applying CS to ultrasound imaging havebeen very successful. The CREATIS team investigated thepossibility of reconstructing the 2-D RF channel data cor-responding to the set of raw RF signals collected by thereceiver elements [12], [13]. They introduce the use of waveatoms [14] as a way to sparsely represent the RF channel data.This tight frame exhibits nice properties when representing theoscillatory patterns presents in this type of data. These basisfunctions, however, are general and do not take the advan-tage of the geometry and physical limits that generated thedata. Using such prior information is beneficial by implicitlylimiting the possible solution space [15].

This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see ht.tp://creativecommons.org/licenses/by/3.0/

https://orcid.org/0000-0002-3938-3227

SCHRETTER et al.: ULTRASOUND IMAGING FROM SPARSE RF SAMPLES USING SYSTEM PSFs 317

A statistical Bayesian view of the CS paradigm hasbeen experimented to mitigate the effect of measurementnoise [16], [17]. Distributed sampling has also been pro-posed in order to combine the complementary informationfrom adjacent single RF channel signals [18]. Advancedsparse regularizations using the least absolute shrinkage andselection operator have been designed at the beamformingstage [19], [20]. The work of David et al. [21] demonstratesthat sparse time-domain compressive beamforming can beapplied as well to cardiac imaging.

Beside the compressive sensing framework, alternative sub-Nyquist sampling techniques are practical systems for process-ing analog signals whose variations are assumed to followa finite rate of innovation bound. Dubbed “Xampling”[22], [23], this methodology has been applied for 2-D [24]and 3-D imaging [25]. The RF signal was modeled as the sumof multiple reflected pulses by a finite amount of scatterers.They assumed that all these reflected pulses are similar tothe transmitted pulse but weighted according to the scattererstrength and distance to the transducer.

A more accurate modeling of the pressure wave propagationis performed by Schiffner and Schmitz [26]. In their work, theymodel the scattering by a planar ultrasound wave caused byinhomogeneities in the adiabatic compressibility and densityof the tissue. The scattered field is calculated by integrating allspherical waves emanating from the scatterers. The problemis linearized by assuming all relevant scatterers lie on a fixedgrid. This approach was used to reconstruct images due to aplane excitation from the transducer using CS.

In the latest work of Bodnariuc et al. [27], they introduce aforward model for planar wave excitations where RF data aremodeled as the superposition of reflected Gaussian modulatedplane wave from spheres located on a grid. The obtained linearsystem matrix is thus sparse and can efficiently be invertedusing regularized regression. Our solution also relies on ad hocdictionaries of similar PSFs. The aim of reaching ultrafastimaging protocols is especially challenging as only planaror divergent waves may be used instead of scanning modesthat focus the insonification beam [28], [29]. The approachbypasses explicit reconstruction of RF channels and mapsdirectly data to beamformed image lines [30].

Ultimately, two essential ingredients are needed for imagereconstruction with CS: the choice of specific sparsifyingbasis functions and the design of an adapted noncoherentundersampling strategy [31]. In this paper, instead of usingparametric mathematical models, shift-variant adapted wave-forms are generated from accurate PSFs that are calculatedby a physically based simulation. Our undersampling operatoruses pseudorandom samples drawn according to a smoothimportance probability density function [32] that is built fromthe system’s spatial sensitivity function.

The remainder of this paper is as follows: the specificationsand geometry of our acquisition system and our sparse RFchannel data decomposition model are described in Section II.Section III introduces physically based system PSFs as dictio-nary elements, along with our importance-weighted randomsampling strategy. Experimental results on numerical and realphantoms are shown in Section IV. Finally, we conclude andpropose tracks for future works in Section V.

II. SYSTEM MODEL AND IMAGE FORMATION

Starting from the acquisition side, a solid-state digitalultrasound probe can only transmit few selected samples fromthe full receiving RF channels on the host computer that in turnrecovers an approximation of the complete data. Subsequently,the standard image formation pipeline is followed. This con-sists of a delay-and-sum beamforming, envelope detection,and log-transform for mapping sound pressure into imageintensities. The full RF channels recovery step interfacesseamlessly after subsampling before beamforming and high-frequency demodulation. For each line, a full RF matrix mustbe reconstructed, thereby explaining the huge data reductionfactor from raw recordings to the lower definition imagedisplay.

A. RF Channel Data ModelAt the heart of our image reconstruction system, we took

inspiration from the compressive sensing paradigm that pro-vides a robust generic framework for the reconstruction ofa full RF signal f ∈ RN using only a small number ofmeasurements b ∈ RM , where M � N . In the generalsetting, these measurements are collected by correlating thesignal f with M sensing vectors φi , where i ∈ [1 . . .M].The measurements thus take the form bi = � f, φi �, which isgenerally written in matrix notation as b = � f . Vectors φiare the rows of the sensing matrix �. This sensing operatoris implemented on the raw data receiving stage, at the read-out electronics of the transducer, and possibilities are limitedby hardware specifications. We consider, in this paper, simplerandom selections of samples.

We model RF channel matrices by a vector of coefficientsx ∈ RL with the following linear decomposition:

f = �x (1)that consist, e.g., of an expansion into functions ψ j that arecommonly chosen as Fourier basis functions or orthonormalwavelets. The choice of basis functions is crucial for efficientrecovery of f from few samples and should be specificallydesigned for the imaging domain at hand. As we will see inSection III, we propose to use ad hoc basis functions basedon a physical model of sound transport in uniform tissues.

In CS, the signal f is supposed to be sparse in the base � ,meaning that only a small number S ≤ L of coefficients xare nonzero. In this paper, however, we do not aim atexact reconstruction of sparse signals; but instead, we usethe approximation regime and search for the minimum of aregularized minimum energy objective function.

B. RF Channel Data ReconstructionWe consider a large linear system of equations

b = � �︸︷︷︸

A

f︷︸︸︷

x (2)

for which a solution vector x is sought from the input data b.Different regularization or approximation strategies must beadopted depending on the amount of available data M = |b|,relatively to the size of the solution L = |x |.

318 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

When just enough data in b are uniquely defining a solu-tion x , then the system matrix A = �� is exactly invertible,and the solution is x = A−1 b. However, more often inpractice, an insufficient amount of measurements is availableto reconstruct a unique solution. The system matrix A is wide(fat) in this case, and regularization priors are needed to selecta unique solution among all possible matching observations.

In this paper, we will first constrain the solution space x tofew representative speckle functions as elements of � . Whenmore data are collected than possible solution elements, thenthe system matrix A is tall (skinny), and no exact solutionexists if all measurements are linearly independent. A standardway to find an approximate solution is selecting the values ofx as to minimize the integrated squared residual differencebetween the reconstructed data Ax and observations b. Theapproximation problem is thus

argminx

||b − A x ||2 (3)which can be solved with ordinary least squares methods.

Starting from M � L RF data samples b, we recoversolution coefficients x with least squares (minimum l2-normresiduals) regression, as follows:

x = [A� A]−1 A�b (4)where the square Gram matrix [A� A] depends only on themodest number of basis functions in � and can thereforebe analytically invertible. The inversion of the whole left-inverse system is actually implemented with Moore–Penrosepseudoinversion. After coefficient recovery, a synthesis of thecomplete RF channel matrix f̂ is given by the product f̂ =�x .

C. Beamforming and Envelope Detection

In order to visualize the final B-mode image, the above-mentioned reconstruction problem is solved in series for everyof the j ∈ [1 . . . Nl ] image lines. A delay-and-sum beamform-ing operator B averages RF channel entries over all channelst ∈ [1 . . . Nc] for producing a single RF line. Each 2-D RFchannel matrix counts N = Ns×Nc data samples, and rows aredenoted f j = [ f j1 . . . f jNc ]. This weighted sum accounts forthe whole incident waveform on the transducer and increasesboth the contrast and the signal-to-noise ratio (SNR).

We obtain therefore the beamformed RF lines

l j = B f j =Nc∑

t=1h(t) · f jt ∈ RNs (5)

where the weights h(t) are obtained from the Hanning smoothapodization window

h(t) = 12

(

1 − cos(

2π(t − 1)Nc − 1

))

. (6)

The RF line vectors l j still contain high-frequency infor-mation originating from the carrier pulse frequency.

The following steps consist in extracting the brightnessmode (B-mode) image profiles by means of an envelopedetection using the Hilbert transform H; hence, l jH = |H(l j )|.Finally, for perceiving the full signal dynamic range, we apply

Fig. 1. Relation between the space-variant sensitivity function of theultrasound system and the geometrical curvature of localized PSF signatures.Beamshaping yields a nonuniform insonification energy, which emphasizesthe focus point. The echo responses also get scattered and attenuated morestrongly further away than the focus point.

a log compression to a decibel (dB) scale and a linear rescalingto the range of a 7-b gray-scale image

i jdB = 20 · log10(

l jH/max(|lH|))

(7)

then i j = 127 + i jdB · (127/60) (8)where the constant max(|lH|) stands for the global maximumabsolute value after envelope detections, taken from the refer-ence full-data beamforming. The vector i j is the j th verticalline in the B-mode image I = [i1 . . . i Nl ] ∈ RNs×Nl .

III. PHYSICALLY BASED PSFS AND DATA SUBSAMPLING

We model the final B-mode image as being composedof a linear combination of independent overlapping imageelements. Therefore, a suitable model for the underlying rawRF channel data should ideally be designed starting from thesystem PSFs of the imaging device. For instance, these PSFscould either be modeled parametrically, or directly acquiredusing a calibration phantom of isolated point scatterers inwater. In this paper, we computed PSFs using the ultrasoundsimulation package Field II [33], [34].

Fig. 1 shows three of these generated noise-free basis func-tions corresponding to three locations in the image domain,where vertical shifts correspond to varying depth and hori-zontal shifts that are parallel to the 1-D probe array situatedat the top. As we can see, the geometric curvature of thewavefronts depends heavily on the location along the depth

SCHRETTER et al.: ULTRASOUND IMAGING FROM SPARSE RF SAMPLES USING SYSTEM PSFs 319

axis, and the PSFs capture these discriminative signal featuresthat are independent of the imaged sample.

Every scatterer in the physical world is generating a PSFin the measured data space, and therefore, a continuum ofresponse functions exists. Since the number of visualizablespeckles in the B-mode image depends on the intrinsic pixe-lated image definition, the set of visually important scatterersis limited. The RF channel data images generated by their PSFare, however, much more dense as the carrier ultrasound pulsemodulation is present in the RF data.

The decomposition of raw RF measurement into a small setof PSFs provides a natural sparse data representation modelthat is driven by physics. More importantly, when using aphysically based design approach, the collection of PSFs canbe tailored to the specific probe geometry and depends on boththe transmission and receiving stages of the imaging protocol.

A. Dictionary Construction

As mentioned above, f ∈ RN is typically representedusing a sparse collection of coefficients from a (over-)completedictionary � , i.e., L � N [35]. Although this approach ismathematically sound, it does present some computationalchallenges when trying to reconstruct a signal where L is verylarge. Therefore, we propose to only project acquired sampleson a smaller set of dictionary elements, i.e., L � M , whichare still able to capture the most relevant signal content.

Sparsity is, therefore, enforced at the construction stageof the system matrix. This does not only allow for fasterreconstruction without numerical solver, using a subsequentlyprecomputed Moore–Penrose pseudoinverse, but also for dic-tionary elements having a much larger support, which in turnallows for a further reduction in the number of samples neededdue to the increased support of basis functions. The goal isthus shifted from exact signal recovery to recovery of all visualinformation implicitly present in the RF data channels.

To find a spatial representation of RF channel data, wejointly treat all RF channels, meaning that the same physicalrepresentation is used for recovering all RF channels inparallel. As such, we treat the echo response as originatingfrom point scatterers disposed over a discrete 2-D grid ofscatterers located in the image plane. When imaging a singleslice, the spatial sensitivity function decays rapidly with thedistance to this 2-D cross section, and only in-plane scatterersare effectively excited.

For representing RF channels, we define a 2-D grid con-taining Nx × Nz scatterers rxz having coordinates[(

x − Nx + 12

)

�x , Sz + z λ2

]

with

{

x ∈ [1 . . . Nx ]z ∈ [1 . . . Nz ] (9)

with the lateral element pitch �x , the distance offset Sz fromthe probe surface to the first imaged row and the wavelengthof the excitation pulse λ = c/ f0 with the speed of sound intissue c = 1540 m/s, and the emission impulse frequency f0.

The grid dimensions Nx and Nz depend on the lateralresolution of the transducer and the image depth of field,respectively. In the experimental simulation case for instance,

Fig. 2. Uniform versus importance-weighted random sampling patterns forthe full RF channels data. As the beamforming step uses a Hanning windowfor progressively discarding distant channels from the central line profile, weconcentrate the data collection on the more important samples.

we consider a linear scan sequence using a 1-D array trans-ducer with a lateral definition of Nx = 64, which is equalto twice the number of transmitting elements while the axialprecision depends on the excitation pulse frequency such thatNz = 60 mm · 2 f0/c = 273 for f0 = 3.5 MHz.

In the following steps, the RF channel data fxz are collectedusing Field II for each point scatterer rxz separately, by jointlysetting the focus at (0, 0, 60) mm. This approach allows us tosee how each point reacts separately to insonification by thetransducer. By grouping all these responses in a dictionary,it is possible to trace distinct structures in RF channel f j tosimilar features in one or more of the PSFs fxz , because eachof them is linked to a spatial location (x, 0, z). The columnsof the matrix � are obtained by lexicographically reorderingthe PSFs fxz .

B. Dictionary Pruning

Finally, we evaluate the energy of each echo responses andreject grid vertices if their energy falls below a prescribedthreshold. This step is essential for shaping the dictionary tothe most important spatial PSFs. The decomposition of rawRF measurements as a sum of PSFs provides a natural datarepresentation model that is driven by physics. As mentionedbefore, dictionary elements can be tailored to the specificprobe geometry and depend on both the transmission andreceiving stages of the imaging protocol.

In linear array ultrasound systems, a designed excitationpulse is used for focusing all acoustic energies along a line.One can calculate the amount of energy received by the pointscatterer rxz by integrating over time

E(rxz) =∫

t|p(rxz, t)|2 (10)

where the associated PSF response is

p(rxz, t) = ρ0 δvn(t)δt

∗ h(rxz, t) (11)

320 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

where p(r, t) denotes the pressure at location r for time t giventhe medium density ρ0, the transducer surface vibration vn(t),and the spatial impulse response h(x, t), which characterizesthe 3-D extent of the insonification field [36].

All in one, given a mimimum absolute integrated energybound T > 0, the restricted set of PSFs is

� = {p(rxz, t) if E(rxz) > T }. (12)Such thresholding of the PSF grid reduces the size of the

dictionary. This trimming is based on the fact that not everypoint on the spatial representation grid will be insonifiedwith the same intensity. Consequently, not every coefficientin the data domain is equally important when representing theRF data. Some points on the grid will even not receive anyacoustic energy at all and can be rejected, since their influenceon data representation is void.

C. RF Channel Data Subsampling

For generating the sampling matrix �, we compared bothuniform and importance sampling strategies, using the sameHanning weighting function as used in the receiving beam-forming stage. Fig. 2 compares the two uniform and impor-tance sampling masks. For adaptive sampling, we generatedfirst uniformly distributed points that were warped according tothe density function, using a single-step inversion method [32].From experiments, we got systematically better results with theimportance sampling strategy that smoothly discards samplesin function of their lateral distance to the imaged line. Thisnonuniform importance sampling strategy is consistent withthe weighting of RF channels that is used in the beamformingstep for constructing RF lines from full RF channels.

In experiments, we compare the images reconstructed withthe same sampling realization such that solely the effects ofdictionary size and sparse matrix approximation are compared.We averaged over many realizations to confirm the repro-ducibility. A sparse matrix approximation refers to the classicalsparse matrix data structure where only nonzero elements areencoded, along with the locations inside the matrix.

IV. EXPERIMENTS

For our evaluations, we reproduced the experimental setupof Liebgott et al. [13]. Fig. 3 shows the specific imaginggeometry considered in the simulation case and the corre-spondence of spatial positions to pixel indices in the resultingB-mode image space. Parameters of the acquisition protocolsare summarized in Table I. We used the conventional numeri-cal cyst phantom shown in Fig. 3 (left) and generated ground-truth RF channel data. The impact of subsampling on theB-mode image reconstructions was then evaluated. In thesecond phase, we reconstructed their acquired RF channelsdata set using a Prosonic L14-5W/60 linear probe on a CIRS054GS general purpose phantom.

A. Numerical Phantom and Simulations

The numerical cyst phantom contains three columns of fivealigned cylinders: five aligned high-scattering point sources,

Fig. 3. Geometry of the considered ultrasound imaging system. Left: probeacquires a subset of measurements. Right: reconstruction problem is solvedon the remote computing unit for recovering raw channel matrices at eachlines of the B-mode image approximation.

TABLE I

EXPERIMENTAL PARAMETERS

five higher scattering regions, and five corresponding regionsof diameters ranging from 2 to 6 mm containing no scatterers.

The tissue was represented as a discrete set of 105 pointscatterers uniformly filling a volume of size 50 × 10 ×60 mm. For creating a background of speckles, each scattererwas accompanied by a normally distributed random acousticimpedance denoting the reflectivity to incoming pressurewaves due to perturbations in density and propagation velocity.

Five cysts were placed with center x-coordinates at 10 mmand z-coordinates at 10 mm apart starting at 40 mm. Their radiidecreased from 6 to 2 mm. Their placement was achieved bysetting as for all scatterers p inside the affected region to zero.Conversely, tenfold higher intensity areas were placed withx-coordinates located at −5 mm with increasing radii. Wethen generated noise-free ground-truth RF channel data fromthe numerical phantom, using the Field II simulation package.

A 2-D slice was imaged from the phantom, coinciding withthe surface [−20, 20] mm × [0] × [30, 60] mm. For this, wesimulated the RF channel data produced by a linear transducer

SCHRETTER et al.: ULTRASOUND IMAGING FROM SPARSE RF SAMPLES USING SYSTEM PSFs 321

Fig. 4. Spatial system sensitivity function in the image field of view beginning at 30 mm from the transducer surface until 90 mm deep in tissues. Row-wisenormalizations and thresholding yield a limited selection of potentially most important PSFs for representing RF channel data.

Fig. 5. Projection of full RF data on dictionaries of selected system’s PSFs. A limited dictionary size is sufficient for approximating the full data, since notevery possible point location has equal importance on the received echo signals in RF data space.

containing Ne = 192 piezo-electric elements. In total, Nl = 50RF channels f j ∈ RNs×Nc are recorded for each image linewhere Nc = 64 is the number of receiving channels whileNs = 60 mm · 2 fs/c = 1948 is the number of samplestaken per channel. This last quantity depends on the samplingfrequency fs and the depth range of the phantom.

The transducer was placed Sz = 30 mm away from thebeginning of the B-mode image support, and the focus distancewas 60 mm. Each element transmits a delayed acoustic pulseensuring maximal constructive interference at the focus point.The echo response recorded by the Nc receive elements overtime constitutes the RF channel data f j for line j ∈ [1 . . . Nl ].B. Reconstructions From Simulated Data

For building the dictionary of PSFs, we first created a regu-lar grid of Nx = 64 times Nz = 273 isolated point scatterers.

The covered imaging area was 31.4 mm wide, according to thelateral pixel pitch of the probe. The depth range was 60 mm.Point scatterers are spaced 0.49 mm in azimuth directionand 0.22 mm in depth. This in turn corresponds to thewidth + spacing (kerf) of probe elements and half the wave-length of the transmitted signal.

The raw RF channel data were collected separately usingthe Field II simulator for each of these point scatterers, withthe same insonification and receiving beamforming proto-cols as the one used for the reference B-mode image. Weexperimented various thresholding constants for selecting thefinal dictionary size, as shown in Fig. 4. For an acceptableapproximation quality, we set the thresholding constant to 75%of the maximum value in the row-wise normalized spatialsensitivity function image. The effect of projecting the fullRF channel data on dictionary elements is shown in Fig. 5.

322 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

Fig. 6. Reconstructions from subsampled RF channels data using a thresholding factor of 75% for building the dictionary of basis functions. Residual imageartifacts appear progressively with increasing subsampling rates. A good recovery of speckles is preserved when using up to about 5% of recorded samples.

Given the full data f , we solve the linear system

���x = �� f ⇐⇒ x = [���]−1�� f. (13)As we can see, a fair image fidelity is reached with a modestset of representative PSFs in the dictionary.

Then, we conducted evaluations for various subsamplingrates, as seen in Fig. 6. Since we recorded 64 RF channelsinstead of 128, a sampling rate of 40% corresponds to the sameabsolute number of samples than the 80% subsampling casein the publication of Liebgott et al. [13]. Only unstructuredchanges are visible on the corresponding difference images. At10% of the samples, we observe a relative drop of 2.77 dB interms of SNR, while the speckles are still visually preserved.At much higher downsampling rates, the quality of the B-modeimage degrades with prominent artifacts that are especiallyvisible in the nonechoic regions.

We evaluated the wallclock reconstruction times as a func-tion of the approximation tolerance that was used for spar-sifying the complete system matrix. The tolerance parameteris a numerical relative approximation threshold that is used

to snap very small matrix elements (in absolute value) toexactly zero. This allows for very fast sparse matrix-vectorproduct computation. This system reconstructs the RF linesfrom samples and is build from the concatenation of thepseudoinverse in (4) for coefficient recovery, the full dictionary� for RF channels synthesis, and the beamforming summationoperator that was encoded as a sparse linear system filled withHanning window weights. All in one, the complete chain fromdata sampling b = � f to beamformed RF lines l = B�xis retrieved by concatenating recovery and synthesis with thebeamforming linear operator B . Hence, the complete end-to-end chain expands to

l = B�[A�A]−1 A�b with A = ��. (14)Timings are summarized in Table II, and the relative

quality loss in comparison to a nonsparsified dictionary isshown in Fig. 7. With an approximation tolerance of 0.1%,the synthesis quality drops slightly for a remaining sparsematrix storage of only 26 MB. A key motivation for imple-menting the beamforming operator in the system matrix is

SCHRETTER et al.: ULTRASOUND IMAGING FROM SPARSE RF SAMPLES USING SYSTEM PSFs 323

Fig. 7. Impact of system matrix sparsification on the fidelity of fully beamformed RF line synthesis for the representative case of 10% samples. A slightrelative approximation tolerance results in significant savings in computational costs, with modest relative impact on the residual image.

that only the final beamformed RF lines should then beexplicitly synthesized.

These timings were recorded on a regular workstationwith a 3.4-GHz Intel Core i5 CPU and 8 GB of RAM,using the MATLAB implementation of sparse matrix-vectorproducts in double precision floating points. The runningtime is memory bound, and further accelerations are probablypossible by storing the matrix on fixed-point precision integersand parallelizing the reconstruction of individual image line.Furthermore, the reconstruction timings are the sum of 50independent lines that are processed sequentially. Therefore,a sliding window pipelining of each reconstruction task ispossible by exploiting more computing cores in parallel usingthe same sparse version of the system matrix stored in sharedmemory. Such an implementation will allow for real-timeimaging with even higher definition ultrasound probes.

C. Reconstructions From Acquired Data

Finally, we reconstructed the real physical phantom acqui-sition data set, and the resulting images are shown in Fig. 8.We used an approximation tolerance of 1% and reduced that

TABLE II

SPARSE INVERSION AND DATA SYNTHESIS FROM 10% SUBSAMPLING

way the system matrix storage size from 210 MB for a denserepresentation to only 9.78-MB sparse representation. The sizeof the system matrix was smaller than in the simulations,but only 32 instead of 64 RF channels were used in thereceiving beamforming stage, leading to a fourfold reductionin dictionary matrix size. We subsampled the 32 remainingdata channels after delay rectification and Hanning windowapodization. Since the data space was smaller than in oursimulations, reconstructing the 192 image lines, includingbeamformed and envelope detection, took only 0.315 s, usingthe approximate sparse system matrix.

324 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

Fig. 8. Comparison of the reference B-mode image and the projection from full data on a dictionary of 4760 PSFs corresponding to a relative thresholdof 75% from the maximum row-wise normalized spatial sensitivity function. The representative reconstruction from 20% is still closely resemblingthe full-data result with more severe artifacts in the near-field region. As a baseline, the plain delay and sum beamforming from sparse samples areshown.

TABLE III

QUANTIFICATIONS IN THE LESION VERSUS BACKGROUND ROIS

For this experimental use case, it was difficult to obtaingood agreement between the responses from our simulatedpoint scatterers response functions and the strong geometricbent that we observed in raw acquired RF channels. Therefore,artifacts are more visible in the near-field region, where weobserved a noticeable mismatch between predictions from oursimulation model and the actual rectified RF channels fromthe imaging device. However, the recovery fidelity of smallstructures is remarkable, and speckles are well developed inthe larger echoic ball, as well as in the background.

Table III summarizes quantitative evaluations comparingtwo regions of interest (ROIs) fully enclosed in the high-echoic circular region representing a lesion and a referencedisplaced region in the background. ROIs correspond to thegray and black discs in the first image of the second row inFig. 8. The contrast-to-noise recovery ratio (CNR) is evaluatedas the absolute difference between the SNRs in the lesionand background area. The CNR decreases, as expected, inthe reconstructed image from fewer samples. However, therelative loss between the projection from full data and thereconstruction from samples is modest, compared with astraightforward beamforming from input RF samples.

The discrepancy between the 20% image reconstruction andthe reference B-mode image was evaluated in dB units, afterlog compression using the same formula as in [13], wherethe authors reported a mean absolute error of 5.5 dB whenusing an overcomplete wave atoms representation of RF dataand sparsity promoting minimum l1-norm regularization prior.In comparison, our approximated image presents an averagerelative loss of 2.24 dB only while the associated sparseprecomputed system matrix allows for imaging at interactiveframe rate by discarding four out of five data samples.

V. CONCLUSION

This paper follows a continuity with previous effortsin translating inspiration from the CS framework to high-definition ultrasound imaging under strict transmission band-width constraints. The method was developed for providing apractical solution for recovering volumetric ultrasound imagesfrom large phased-array probes, within the bandwidth capacityof current counting electronics. The data processing chain iscombining a domain-specific design of basis functions, animportance-weighted random sampling strategy, and a precom-puted minimum-norm inversion solver.

We have chosen to form dictionaries of sampled apodizedPSFs. Such a design is incorporating the effects of the wholedata acquisition process from the transmission of a focusedexcitation impulse to the reception of echo measurements.However, a principle limitation of the approach is that allrelevant secondary physical effects might not be capturedsolely by simple linear combinations of PSFs. For example,the aberrations generated by the space-variant speed of sound

SCHRETTER et al.: ULTRASOUND IMAGING FROM SPARSE RF SAMPLES USING SYSTEM PSFs 325

transmission in different tissues’ densities play an importantrole in actual ultrasound measurements.

An adaptive importance sampling strategy has been usedunder the assumption that a constant data collection bandwidthis required. The sparse data sampling stems from a combina-tion of random point generations that are warped followingthe Hanning window that weights the importance of each datasample. In future work, we will investigate the potential ofbuffering samples on the probe for allocating a larger samplingbudget on the near and far regions from the image depth, wherethe reconstruction problem is more difficult to solve. Thistranslated to nonuniform residual losses in the experiments.

There is still potential in further refining the sensing stageby sharing information among several raw RF channel data.Indeed, the current imaging pipeline considers the independentreconstruction of each RF line. This segmentation allows forthe natural parallelization of the RF channels reconstruction.However, it would be interesting to make use of sampledinformation from adjacent RF lines in order to complementglobally the corpus of missing data for the whole image.

ACKNOWLEDGMENT

The authors would like to thank D. Kouamé andA. Basarab for conducting early experiments with theirquasi-random patterns. The CIRS phantom data were kindlyshared by D. Friboulet.

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326 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 65, NO. 3, MARCH 2018

Colas Schretter (M’12) received the M.Sc. degree(Hons.) in computer science and the master’s degree(DEA) in advanced studies from the Universitélibre de Bruxelles, Bruxelles, Belgium, in 2004 and2006, respectively, and the Ph.D. degree (magnacum laude) from Otto von Guericke UniversityMagdeburg, Magdeburg, Germany, in 2010.

In 2007, he moved to Germany for a scientistposition with Philips Research, Aachen, Germany.He was a Visiting Researcher with the Politecnicodi Milano, Milan, Italy. Since 2013, he has

been a Senior Research Scientist with the Department of Electronics andInformatics (ETRO), Vrije Universiteit Brussel, Brussels, Belgium.He is interested in problems in high-dimensional image formation usingregularized inverse methods, statistical estimation for large mixture models,and quasi-Monte Carlo numerical integration.

Dr. Schretter received a Marie Curie EST Fellowship for his Ph.D. thesis.He received a personal Marie Curie ERG Grant for a post-doctoral projectwith RWTH Aachen University, Aachen. He was selected for an M+Vision2012 Fellowship at the Massachusetts Institute of Technology, Cambridge,MA, USA.

Shaun Bundervoet received the master’s degree inmathematics from Vrije Universiteit Brussel, Brus-sels, Belgium, in 2012, with a focus on algebraicand statistical topics, which formed the foundationfor his research specialization in image process-ing and machine learning, where he is currentlypursuing the Ph.D. degree with the Department ofMathematics (DWIS).

David Blinder (S’12) received the B.Sc. degree inelectronics and information technology engineeringfrom Vrije Universiteit Brussel (VUB), Brussels,Belgium, and the M.Sc. degree in applied sciencesand engineering from VUB and the École Polytech-nique Fédérale de Lausanne, Lausanne, Switzerland,in 2013. He is currently pursuing the Ph.D. degreewith VUB.

His research focuses on the efficient representa-tion and compression of static and dynamic holo-grams, computer-generated holography, and inverseproblems for interference-based modalities.

Ann Dooms (M’08) received the master’s and Ph.D.degrees in mathematics from Vrije UniversiteitBrussel (VUB), Brussels, Belgium, in 2000 and2004, respectively.

She is currently a Professor with the Departmentof Mathematics (DWIS), VUB, where she leadsthe research group Digital Mathematics (DIMA).She specializes in the mathematical foundationsfor digital data acquisition, representation, analysis,communication, security, and forensics.

Dr. Dooms was elected as a member of the RoyalFlemish Young Academy of Sciences and Arts and the IEEE TechnicalCommittee in Information Forensics and Security.

Jan D’hooge (M’98) was born in Sint-Niklaas,Belgium, in 1972. He received the M.Sc. and Ph.D.degrees in physics from the University of Leuven,Leuven, Belgium, in 1994 and 1999, respectively.

He was a Post-Doctoral Researcher with the Med-ical Imaging Computing Laboratory, University ofLeuven, where he became acquainted with gen-eral problems in medical imaging, such as elasticregistration, segmentation, shape analysis, and dataacquisition problems related to other modalities (inparticular, MRI). In 2006, he was appointed to an

Associate Professorship with the Department of Cardiovascular Diseases,Medical Faculty, University of Leuven. Since 2009, he has been a part-timeVisiting Professor with the Norwegian Institute of Science and Technology,Trondheim, Norway. He has authored or co-authored over 150 peer-reviewedpapers, has contributed to eight books, and has co-edited one book. His currentresearch interests include myocardial tissue characterization, deformationimaging, and cardiac pathophysiology.

Dr. D’hooge was an elected AdCom Member of the IEEE-UFFC Societyfrom 2010 to 2012. He is a member of the Acoustical Society of Americaand the European Association of Echocardiography. He was the Chair of theUltrasound Conference of the SPIE Medical Imaging Symposium from 2008to 2011, the Technical Vice-Chair of the IEEE Ultrasonics Symposium from2008 to 2012, and the Technical Chair of the IEEE Ultrasonics Symposiumin 2014. In 1999, he received the Young Investigator Award of the BelgianSociety of Echocardiography, and in 2000, he was nominated for the YoungInvestigator Award of the European Society of Echocardiography.

Peter Schelkens (M’99) currently holds a Pro-fessorship with the Department of Electronicsand Informatics (ETRO), Vrije Universiteit Brus-sel, Brussels, Belgium. He is also a ResearchGroup Leader with imec, Leuven, Belgium. He has(co-)authored over 250 journal and conferencepublications. He is a co-editor of the booksThe JPEG 2000 Suite and Optical and DigitalImage Processing (Wiley, 2009 and 2011). Hiscurrent research interests include multidimensionalsignal processing, while especially focusing oncross-disciplinary research.

Dr. Schelkens is a member of the ISO/IEC JTC1/SC29/WG1 (JPEG)Standardization Committee. In 2013, he received an EU ERC ConsolidatorGrant focusing on digital holography. He is an Associate Editor of the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGYand Signal Processing: Image Communications.

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