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Coherent Interferometry Algorithms for Photoacoustic
Imaging∗
Habib Ammari† Elie Bretin‡ Josselin Garnier§ Vincent Jugnon¶
June 21, 2012
Abstract
The aim of this paper is to develop new Coherent Interferometry (CINT) algorithmsto correct the effect of an unknown cluttered sound speed (random fluctuations arounda known constant) on photoacoustic images. By back-propagating the correlationsbetween the pre-processed pressure measurements, we show that we are able to providestatistically stable photo-acoustic images. The pre-processing is exactly in the same wayas when we use the circular or the line Radon inversion to obtain photo-acoustic images.Moreover, we provide a detailed stability and resolution analysis of the new CINT-Radon algorithms. We also present numerical results to illustrate their performanceand to compare them with Kirchhoff-Radon migration functions.
AMS subject classifications. 65R32, 44A12, 35R60
Key words. Imaging, migration, interferometry, random media, resolution, stability
1 Introduction
In photoacoustic imaging, optical energy absorption causes thermo-elastic expansion of thetissue, which leads to the propagation of a pressure wave. This signal is measured bytransducers distributed on the boundary of the object, which in turn is used for imagingoptical properties of the object.
In pure optical imaging, optical scattering in soft tissues degrades spatial resolution sig-nificantly with depth. Pure optical imaging is very sensitive to optical absorption but canonly provide a spatial resolution of the order of 1 cm at cm depths. Pure conventional ultra-sound imaging is based on the detection of the mechanical properties (acoustic impedance) inbiological soft tissues. It can provide good spatial resolution because of its millimetric wave-length and weak scattering at MHz frequencies. The major contribution of photo-acoustic
∗This work was supported by the ERC Advanced Grant Project MULTIMOD–267184.†Department of Mathematics and Applications, Ecole Normale Superieure, 45 Rue d’Ulm, 75005 Paris,
France ([email protected]).‡Centre de Mathematiques Appliquees, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France
([email protected]).§Laboratoire de Probabilites et Modeles Aleatoires & Laboratoire Jacques-Louis Lions, site Chevaleret,
case 7012, Universite Paris VII, 75205 Paris Cedex 13, France ([email protected]).¶Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cam-
bridge, MA 02139-4307, USA ([email protected]).
1
imaging is to provide images of optical contrasts (based on the optical absorption) withthe resolution of ultrasound [30]. Potential applications include breast cancer and vasculardisease.
In photo-acoustic imaging, the absorbed energy density is related to the optical ab-sorption coefficient distribution through a model for light propagation such as the diffusionapproximation or the radiative transfer equation. Although the problem of reconstructingthe absorption coefficient from the absorbed energy is nonlinear, efficient techniques canbe designed, specially in the context of small absorbers [2]. The absorbed optical energydensity is the initial condition in the acoustic wave equation governing the pressure. If themedium is acoustically homogeneous and has the same acoustic properties as the free space,then the boundary of the object plays no role and the absorbed energy density can be re-constructed from measurements of the pressure wave by inverting a spherical or a circularRadon transform [22, 23, 19, 20].
Recently, we have been interested in reconstructing initial conditions for the wave equa-tion with constant sound speed in a bounded domain. In [1, 3], we developed a variety ofinversion approaches which can be extended to the case of variable but known sound speedand can correct for the effect of attenuation on image reconstructions. However, the situa-tion of interest for medical applications is the case where the sound speed is perturbed byan unknown clutter noise. This means that the speed of sound of the medium is randomlyfluctuating around a known value. In this situation, waves undergo partial coherence loss[18] and the designed algorithms assuming a constant sound speed may fail.
Interferometric methods for imaging have been considered in [13, 26, 27]. Coherentinterferometry (CINT) was introduced and analyzed in [7, 8]. While classical methods back-propagate the recorded signals directly, CINT is an array imaging method that first computescross-correlations of the recorded signals over appropriately chosen space-frequency windowsand then back-propagates the local cross-correlations. As shown in [7, 8, 9, 10], CINT dealswell with partial loss of coherence in cluttered environments.
In the present paper, combining the CINT method for imaging in clutter together with areconstruction approach for extended targets by Radon inversions, we propose CINT-Radonalgorithms for photoacoustic imaging in the presence of random fluctuations of the soundspeed. We show that these new algorithms provide statistically stable photoacoustic images.We provide a detailed analysis for their stability and resolution from a simple clutter noisemodel and numerically illustrate their performance.
The paper is organized as follows. In Section 2 we formulate the inverse problem ofphotoacoustics and describe the clutter noise considered for the sound speed. In Section3 we recall the reconstruction using the circular Radon transform when the sound speedis constant and describe the original CINT algorithm. We then propose a new CINT ap-proach which consists in pre-processing the data (in the same way as for the circular Radoninversion) before back-propagating their correlations. Section 4 is devoted to the stabilityanalysis of this new algorithm. Section 5 adapts the results presented in Sections 3 and 4to the case of a bounded domain. We make a parallel between the filtered back-projectionof the circular Radon inversion in free space and of the line Radon inversion when we haveboundary conditions. Both algorithms end with a back-propagation step. We propose toback-propagate the correlations between the (pre-processed) data in the same way as inSection 3. The paper ends with a short discussion.
2
2 Problem Formulation
In photoacoustics, laser pulses are focused into biological tissues. The delivered energy isabsorbed and converted into heat, which generates thermo-elastic expansion, which in turngives rise to the propagation of an ultrasonic pressure wave p(x, t) from a source term p0(x):
∂2p
∂t2(x, t) − c(x)2∆p(x, t) = 0,
p(x, 0) = p0(x),∂p
∂t(x, 0) = 0.
(2.1)
The pressure field is measured at the surface of a domain Ω that contains the supportof p0. The imaging problem is to reconstruct the initial value of the pressure p0 in asearch domain from measurements of p on the whole boundary ∂Ω of a domain Ω. Most ofthe reconstruction algorithms assume constant (or known) sound speed. However, in realapplications, the sound speed is not perfectly known. It seems more relevant to considerthat it fluctuates randomly around a known distribution. For simplicity, we consider themodel with random fluctuations around a constant that we normalize to one:
1
c(x)2= 1 + σcµ
( x
xc
), (2.2)
where µ is a zero-mean stationary random process, xc is the correlation length of the fluc-tuations of c(x) and σc is their standard deviation.
Throughout this paper, we restrict ourselves to the two-dimensional case and assumethat Ω is the unit disk with center at 0 and radius X0 = 1. The two-dimensional case is ofparticular interest in photoacoustic imaging with integrating line detectors [11].
3 Imaging Algorithms
We define the Fourier transform with respect to the second variable (time-variable) by
f(y, ω) =
∫
R
f(y, t)eiωt dt, f(y, t) =1
2π
∫
R
f(y, ω)e−iωt dω. (3.1)
In free space, it is possible to link the measurements of the pressure waves p(y, t) on theboundary ∂Ω to the circular Radon transform of the initial condition p0(x) as follows (see,for instance, [14, p.682] and [15]):
RΩ[p0](y, r) = W[p](y, r), y ∈ ∂Ω, r ∈ R+, (3.2)
where the circular Radon transform is defined by
RΩ[p0](y, r) :=
∫
S1
rp0(y + rθ)dσ(θ), y ∈ ∂Ω, r ∈ R+,
and
W[p](y, r) := 4r
∫ r
0
p(y, t)√r2 − t2
dt, y ∈ ∂Ω, r ∈ R+.
Here S1 denotes the unit circle and dσ(θ) is the surface measure on S1.
3
Since Ω is assumed to be the unit disk with center at 0 and radius X0 = 1, it followsthat, in order to find p0, we can use the following exact inversion formula [16]:
p0(x) =1
4π2R?
ΩBW[p](x), x ∈ Ω, (3.3)
where R?Ω (the formal adjoint of the circular Radon transform) is a back-projection operator
given by
R?Ω[f ](x) =
∫
∂Ω
f(y, |x − y|)dσ(y) =1
2π
∫
∂Ω
∫
R
f(y, ω)e−iω|x−y|dωdσ(y), x ∈ Ω,
with dσ(y) being the surface measure on ∂Ω, and B is a filter defined by
B[g](y, t) =
∫ 2
0
d2g
dr2(y, r) ln(|r2 − t2|) dr, y ∈ ∂Ω.
Note that (3.2) and (3.3) only hold for constant speed sound (in this case sound speed 1).Note also that (3.3) reads in the Fourier domain as
p0(x) =1
(2π)3
∫
∂Ω
∫
R
BW[p](y, ω)e−iω|x−y|dωdσ(y), x ∈ Ω.
Here and below the Fourier transform, denoted with a hat, is with respect to the timevariable. Hence, we introduce the Kirchhoff-Radon migration imaging function:
IKRM(x) =1
4π2R?
ΩBW[p](x) =1
2π
∫
∂Ω
dσ(y)
∫
R
dωq(x, ω)e−iω|x−y|, (3.4)
where
q =1
4π2BW[p] (3.5)
is the pre-processed data.A second imaging function is to simply back-propagate the raw data (without any pre-
processing) [5]:
IKM(x) = R?Ω[p](x) =
1
2π
∫
∂Ω
dσ(y)
∫
R
dωp(y, ω)e−iω|x−y|, x ∈ Ω. (3.6)
Here, the subscript KM stands for Kirchhoff Migration. As will be seen later, this simplifiedfunction is sufficient for localizing point sources in homogeneous media, but may fail forimaging extended targets and/or in the presence of clutter noise.
When the sound speed varies as in (2.2), the phases of the measured waves p(ω,y) areshifted with respect to the deterministic, unperturbed phase because of the unknown clutter.When the data are numerically back-propagated in the homogeneous medium with speedof propagation equal to one, the phase terms do not compensate each other in (3.6) whenx is a source location, which results in instability and loss of resolution. To correct thiseffect, the idea of the original CINT algorithm is to back-propagate the space and frequencycorrelations between the data [8]:
ICI(x) =1
(2π)2
∫∫
∂Ω×∂Ω, |y2−y1|≤Xd
dσ(y1)dσ(y2)
∫∫
R×R, |ω2−ω1|≤Ωd
dω1dω2
×p(y1, ω1)e−iω1|x−y1|p(y2, ω2)e
iω2|x−y2|.
(3.7)
4
The CINT function is close to the KM function. In fact, if Ωd → ∞ and Xd → ∞, then ICI
is the square of the Kirchhoff migration function:
ICI(x) = |IKM(x)|2.
However the cut-off parameters Ωd and Xd play a crucial role. When one writes the CINTfunction in the time domain:
ICI(x) =
∫∫
∂Ω×∂Ω, |y2−y1|≤Xd
dσ(y1)dσ(y2)
×Ωd
π
∫
R
dt sinc(Ωdt) p(y1, |y1 − x| + t)p(y2, |y2 − x| − t),
it is clear that it forms the image by computing the local correlation of the recorded datain a time interval scaled by 1/Ωd and by superposing the back-propagated local correlationsover pairs of receivers that are not further apart than Xd. The idea that motivates the form(3.7) of the CINT function is that, at nearby frequencies ω1, ω2 and nearby locations y1,y2
the random phase shifts of the data p(y1, ω1), p(y2, ω2) are similar (say, correlated) so theycancel each other in the product p(y1, ω1)p(y2, ω2). We then say that the data p(y1, ω1)and p(y2, ω2) are coherent. In such a case, the back-propagation of this product in thehomogeneous medium should be stable. The purpose of the CINT imaging function is tokeep in (3.7) the pairs (y1, ω1) and (y2, ω2) for which the data p(y1, ω1) and p(y2, ω2) arecoherent and to remove the pairs that do not bring information. It then appears intuitivethat the cut-off parameters Xd, resp. Ωd, should be of the order of the spatial (resp.frequency) correlation radius of the recorded data, and we will confirm this intuition in thefollowing.
As will be shown later, ICI is quite efficient in localizing point sources in cluttered mediabut not in finding the true value of p0 (up to a square). Moreover, when the support ofthe initial pressure p0 is extended, ICI may fail in recovering a good photoacoustic image.We propose two things. First, in order to avoid numerical oscillatory effects, we replacethe sharp cut-offs in the integral by Gaussian convolutions. Then instead of taking thecorrelations between the back-propagated raw data, we pre-process them like we do for theRadon inversion. We thus get the following CINT-Radon imaging function:
ICIR(x) =1
(2π)2
∫∫
∂Ω×∂Ω
dσ(y1)dσ(y2)
∫∫
R×R
dω1dω2e−
(ω2−ω1)2
2Ω2d e
−|y1−y2|2
2X2d
×q(y1, ω1)e−iω1|x−y1|q(y2, ω2)e
iω2|x−y2|,
(3.8)
where q is given by (3.5). Note again that, when Ωd → ∞ and Xd → ∞, ICIR is the squareof the Kirchhoff-Radon migration function:
ICIR(x) = |IKRM(x)|2.
The purpose of the CINT-Radon imaging function ICIR is to keep in (3.8) the pairs (y1, ω1)and (y2, ω2) for which the pre-processed data q(y1, ω1) and q(y2, ω2) are coherent and toremove the pairs that do not bring information.
Using the exact inversion formulas in [17] for the spherical Radon transform, the CINT-Radon algorithm presented in this paper can be generalized to the three-dimensional caseprovided that the measurements are taken on a sphere ∂Ω.
5
Note that because of the pre-processing step (3.5), which is in the same way as when oneinverts the Radon transform in (3.3), we called, by abuse of language, the imaging functionsIKRM and ICIR Kirchhoff-Radon migration and CINT-Radon, respectively.
4 Stability and Resolution Analysis
In this section we perform a resolution and stability analysis that clarifies the role of the cut-off parameters Ωd and Xd in the CINT-Radon imaging function. This analysis is based onarguments quite similar to those used in [6, 7, 8, 9, 10] to study the original CINT function.As far as we know, both the introduction of the CINT-Radon imaging functions and theirresolution and stability analysis are new.
4.1 Random Travel Time Model
The random travel time model is a simple model used to describe wave propagation in amedium whose index of refraction has small fluctuations. The model is valid in the high-frequency regime when the fluctuations of the random medium have small amplitude andlarge correlation length (compared to the typical wavelength λ0) [28, 25]. More exactly, therandom travel time model is valid when the correlation length xc and the standard deviationσc of the fluctuations of the wave speed c(x) satisfy conditions [6]:
xc X0 , σ2c x3
c
X30
, σ2c
X30
x3c
λ20
σ2cxcX0
. 1 . (4.1)
Here X0 is the radius of Ω (taken equal to 1 for simplicity). In these conditions the geometricoptics approximation is valid, the perturbation of the amplitude of the wave is negligible,and the perturbation of the phase of the wave is of order one or larger (see [28, Chapter 6],[25, Chapter 1], and [6]).
We assume that conditions (4.1) for the random travel time model are satisfied. Thereforethere is an error ν(x,y) between the theoretical travel time τ0(x,y) = |y − x| with thebackground velocity equal to one and the real travel time τ(x,y) where y is a point of thesurface of the observation disk ∂Ω and x is a point of the search domain (where we plot theimage):
τ(x,y) = τ0(x,y) + ν(x,y).
The random process ν(x,y) is given by
ν(x,y) =σc|y − x|
2
∫ 1
0
µ(x + (y − x)s
xc
)ds. (4.2)
This is the integral of the fluctuations of 1/c along the unperturbed, straight ray from y tox. Assuming that the search domain is relatively small we can assume that ν depends onlyon the sensor position y and we can neglect the variations of ν with respect to x. This isperfectly correct if we analyze the expectations and the variances of the imaging functionsfor a fixed test point x (then the search domain is just one point). This is still correct if weanalyze the covariance of the imaging function for a pair of test points x and x′ that areclose to each other (closer than the correlation radius xc of the clutter noise).
6
Note finally that the random travel time model can be used to model another type ofnoisy data which arises when the medium is homogeneous but the positions of the sensorsare poorly characterized.
We assume that the random process µ is a random process with Gaussian statistics,mean zero, and covariance function:
E[σcµ
( x
xc
)σcµ
(x′
xc
)]= σ2
c exp(− |x − x′|2
2x2c
).
Here E stands for the expectation (mean value). Gaussian statistics means that, for anyfinite collection of points (xj)j=1,...,n in R2, the collection of random variables (µ(xj))j=1,...,n
has a multivariate normal distribution.We will need the following lemma.
Lemma 4.1 If xc X0, then ν is a random process with Gaussian statistics, mean zero,and covariance function:
E[ν(y)ν(y′)] = τ2c ψ
( |y − y′|xc
), ψ(r) =
1
r
∫ r
0
exp(− s2
2
)ds, (4.3)
whereτ2c =
√2πσ2
c/4 (4.4)
is the variance of the fluctuations of the travel times.
Proof. It follows by direct calculation from (4.2) that the random process ν(y), whereits variations with respect to x are neglected, has mean zero and covariance function
E[ν(y)ν(y′)] =σ2
c |y − x| |y′ − x|4
∫ 1
0
ds
∫ 1
0
ds′ exp[− |s(y − x) − s′(y′ − x)|2
2x2c
]
=σ2
c |y − x| |y′ − x|4
∫ 1
0
ds
∫ 1−s
−s
ds exp[− |s(y − y′) + s(x − y′)|2
2x2c
].(4.5)
Moreover, when X0 xc, we can extend the s integral in (4.5) to the entire real line.Integrating in s then gives
E[ν(y)ν(y′)] =
√2πσ2
c |y − x|4
∫ 1
0
ds exp[− s2
2x2c
(|y − y′|2 −
( x − y′
|x − y′| · (y − y′))2
)].
If x is close to the center of the disk Ω, then y − y′ is approximately orthogonal to x − y′
and we obtain (4.3). 2
Using Gaussian statistics it is straightforward to compute the moments
E[eiων(y)] = exp(− ω2τ2
c
2
), (4.6)
E[eiων(y)−iω′ν(y′)] = exp(− (ω − ω′)2τ2
c
2− ωω′τ2
c
(1 − ψ
( |y − y′|xc
))).
If, additionally, we assume that ω, ω′ ' ω0, with
ω0τc 1, (4.7)
7
then, using a second order Taylor expansion of ψ and the assumption |y − y′| xc, wearrive at
E[eiων(y)−iω′ν(y′)] ' exp(− (ω − ω′)2τ2
c
2− |y − y′|2
2X2c
), (4.8)
with
X2c =
3x2c
ω20τ
2c
. (4.9)
Note that the typical wavelength λ0 = 2π/ω0.From (4.8) we can see that τ−1
c is the frequency coherence radius and Xc is the spatialcoherence radius of the recorded signals.
In the following sections it will be convenient to introduce a typical frequency ω0. Sincewe deal with a wave equation with initial conditions (2.1) and the speed of propagation isone, the typical wavelength for this problem is of the order of the diameter of the support ofthe function p0 when it is smooth, or of the order of the scale of variations of the functionp0 when it is oscillating. Since the speed of propagation is one, the typical frequency ω0 isof the order of the reciprocal of the typical wavelength. Moreover, the bandwidth (i.e., thespectral width of the data) is also of the order of the typical frequency.
4.2 Kirchhoff-Radon Migration
Recall that the Kirchhoff-Radon migration function is
IKRM(x) =1
2π
∫
∂Ω
dσ(y)
∫
R
dωq(y, ω)e−iω|y−x| =
∫
∂Ω
dσ(y)q(y, |y − x|), (4.10)
where q = BW[p]/(4π2). The function applied to the perfect pre-processed data
q(0) =1
4π2BW[p(0)] (4.11)
is
I(0)KRM(x) =
1
2π
∫
∂Ω
dσ(y)
∫
R
dωq(0)(y, ω)e−iω|y−x| =
∫
∂Ω
dσ(y)q(0)(y, |y − x|), (4.12)
and it is equal to the initial condition p0(x). Here and below we denote by p(0) and q(0) theunperturbed data obtained when the medium is homogeneous (µ ≡ 0).
We consider the random travel time model to describe the recorded data set:
q(y, t) = q(0)(y, t− ν(y)). (4.13)
This is a rough approximate model that accounts only for random time delays, but calcula-tions can be carried out in a rather simple manner that illustrate the dual role of the cut-offparameters Ωd and Xd. Even though q arises from p via some time integration, the model(4.13) can be considered as a valid approximation since p(y, t) ' p(0)(y, t − ν(y)) with νbeing independent of the time variable t, for any t > 0.
We first consider the expectation of the function. Using (4.6) we find that the followingholds.
8
Theorem 4.2 Let IKRM be defined by (4.10). Consider a travel time model for the effectof clutter noise and assume that conditions (4.1) and (4.7) are satisfied. We have
E[IKRM(x)] =1
2π
∫
∂Ω
dσ(y)
∫
R
dωq(0)(y, ω) exp(− ω2τ2
c
2
)e−iω|y−x|
' exp(− ω2
0τ2c
2
)I(0)
KRM(x), (4.14)
with A ' B iff A = B(1 + o(1)). Here, τc, given by (4.4), is the variance of the fluctuationsof the travel times.
Theorem 4.2 shows that the mean function undergoes a strong damping compared to the
unperturbed function I(0)KRM(x). This phenomenon is standard when studying wave propa-
gation in random media and is sometimes called extinction [25]. The approximation (4.14)gives the order of magnitude depending on the typical frequency ω0 (or equivalently, thetypical wavelength λ0 = 2π/ω0)
The statistics of the fluctuations can be characterized by the covariance. Using (4.8) wehave
E[IKRM(x)IKRM(x′)
]=
1
(2π)2
∫∫
∂Ω×∂Ω
dσ(y1)dσ(y2)
∫∫
R×R
dω1dω2q(0)(y1, ω1)q(0)(y2, ω2)
× exp(− (ω1 − ω2)
2τ2c
2− |y1 − y2|2
2X2c
)e−iω1|y1−x|eiω2|y2−x′|.
The variance that characterizes the amplitude of the fluctuations is
Var(IKRM(x)
):= E
[|IKRM(x)|2
]−
∣∣E[IKRM(x)
]∣∣2.
We assume that Xc X0, that is, the correlation radius is smaller than the propagationdistance. Applying Proposition A.1, we find that
E[|IKRM(x)|2
]=
Xc
(2π)3τc
∫
∂Ω
dσ(Ya)
∫
R
dωa
∫
Y ⊥a
dσ(κa)
∫
R
dτa
×Wq(Ya, ωa;κa, τa) exp(−
∣∣κa − ωax−Ya
|x−Ya|
∣∣2
2X−2c
− (τa − |Ya − x|)22τ2
c
),
where Wq is the Wigner transform of q(0):
Wq(Ya, ωa;κa, τa) =
∫
R
dha
∫
Y ⊥a
dσ(ya)q(0)(Ya +ya
2, ωa +
ha
2)q(0)(Ya − ya
2, ωa − ha
2)
×eiκa·ya−ihaτa , (4.15)
and Y ⊥a is defined by
Y ⊥a =
ya ∈ R
2 , Ya − ya
2∈ ∂Ω and Ya +
ya
2∈ ∂Ω
. (4.16)
If we consider the regime in which τ−1c ω0 and Xc X0, then we get
E[|IKRM(x)|2
]' Xc
(2π)3τc
∫
∂Ω
dσ(Ya)
∫
R
dωa
∫
Y ⊥a
dσ(κa)
∫
R
dτaWq(Ya, ωa;κa, τa)
' Xc
2πτc
∫
∂Ω
dσ(Ya)
∫
R
dωa|q(0)(Ya, ωa)|2,
9
and thus,
E[|IKRM(x)|2
]' Xcτ
−1c
∫
∂Ω
dσ(y)
∫
R
dtq(0)(y, t)2. (4.17)
Combining (4.17) together with (4.12) yields
E[|IKRM(x)|2
]' I(0)
KRM(x)2( 1
ω0τc
)(Xc
X0
). (4.18)
Define the signal-to-noise ratio (SNR) by
SNRKRM =|E[IKRM(x)]|
Var(IKRM(x))1/2. (4.19)
Using (4.18), it follows from Theorem 4.2 that the following result holds.
Theorem 4.3 Under the same assumptions as those in Theorem 4.2, it follows that ifω0τc 1 and Xc X0, then SNRKRM is very small:
SNRKRM 1.
Here, A B iff A/B = o(1).
4.3 CINT-Radon
We consider the random travel time model (4.13). We first note that in (3.8) the coherenceis maintained as long as exp(iω2ν(y2) − iω1ν(y1)) is close to one. From (4.8) this requiresthat |ω1 − ω2| < τ−1
c and |y1 − y2| < Xc. We can therefore anticipate that the cut-offparameters Xd and Ωd should be related to the coherence parameters Xc and τ−1
c . In thefollowing we study the role of the cut-off parameters Xd and Ωd for resolution and stabilityof the CINT-Radon function. For doing so, we compute the expectation and variance of theimaging function ICIR.
Using (4.8), we have
E[ICIR(x)] =1
(2π)2
∫∫
R×R
dω1dω2
∫∫
∂Ω×∂Ω
dσ(y1)dσ(y2)q(0)(y1, ω1)q(0)(y2, ω2)
×e−iω1|y1−x|eiω2|y2−x| exp(− (ω1 − ω2)
2
2(τ2
c +1
Ω2d
) − |y1 − y2|22
(1
X2d
+1
X2c
)), (4.20)
where q(0) is the perfect pre-processed data defined by (4.11). Applying Proposition A.1,we obtain the following result.
Theorem 4.4 Let ICIR be defined by (3.8). Consider a travel time model for the effect ofclutter noise and assume that conditions (4.1) and (4.7) are satisfied. Let Xc be definedby (4.9) with xc being the correlation length of the clutter noise, τc the variance of thefluctuations of the travel times, and ω0 the typical frequency. If Xc X0, where X0 is theradius of Ω, then we have
E[ICIR(x)] =1
(2π)3( 1X2
d
+ 1X2
c
)12 (τ2
c + 1Ω2
d
)12
∫
∂Ω
dσ(Ya)
∫
R
dωa
∫
Y ⊥a
dσ(κa)
∫
R
dτa
×Wq(Ya, ωa;κa, τa) exp(−
∣∣κa − ωax−Ya
|x−Ya|
∣∣2
2( 1X2
d
+ 1X2
c
)− (τa − |Ya − x|)2
2(τ2c + 1
Ω2d
)
), (4.21)
10
where Wq is the Wigner transform of q(0) defined by (4.15).
Formula (4.21) shows that the coherent part (i.e., the expectation) of the CINT-Radonfunction is a smoothed version of the Wigner transform of the pre-processed data. It selectsa band of directions (κa) and time delays (τa) that are centered around the direction andthe time delay between the search point x and the point Ya of the sensor array.
We observe that:
- if Ωd > τ−1c and Xd > Xc then the cut-off parameters Ωd and Xd have no influence
on the coherent part of the function which does not depend on (Ωd,Xd).
- if Ωd < τ−1c and Xd < Xc then the cut-off parameters Ωd and Xd have an influence and
reduce the resolution of the coherent part of the function (they enhance the smoothingof the Wigner transform).
We next compute the covariance of the CINT-Radon function in the regime Ωd < τ−1c
and Xd < Xc. Let us consider four points (yj)j=1,...,4 and four frequencies (ωj)j=1,...,4 inthe bandwidth. Using the Gaussian property of the process ν(y) and the fact that ψ(0) = 1,we have
E[eiω1ν(y1)−iω2ν(y2)eiω3ν(y3)−iω4ν(y4)
]= exp
− τ2
c
2
[ 4∑
j=1
ω4j + 2ω1ω4ψ
( |y1 − y4|xc
)
+2ω2ω3ψ( |y2 − y3|
xc
)− 2ω1ω2ψ
( |y1 − y2|xc
)− 2ω1ω3ψ
( |y1 − y3|xc
)
−2ω2ω4ψ( |y2 − y4|
xc
)− 2ω3ω4ψ
( |y3 − y4|xc
)].
If
ω1 = ωa +ha
2, ω2 = ωa − ha
2, ω3 = ωb +
hb
2, ω4 = ωb −
hb
2,
y1 = Ya +ya
2, y2 = Ya − ya
2, y3 = Yb +
yb
2, y4 = Yb −
yb
2,
with ha, hb = O(Ωd), ωa, ωb = O(ω0), Ya,Yb = O(X0), and ya,yb = O(Xd), then
E[eiω1ν(y1)−iω2ν(y2)eiω3ν(y3)−iω4ν(y4)
]' exp
− τ2
c
2
[h2
a + h2b − 2hahbψ
( |Ya − Yb|xc
)],
(note that Xd xc since Xd ≤ Xc and Xc xc) and therefore,
E[ICIR(x)ICIR(x′)
]=
1
(2π)4
∫
∂Ω
dσ(Ya)
∫
R
dha
∫
Y ⊥a
dσ(ya)
∫
∂Ω
dσ(Yb)
∫
R
dhb
∫
Y ⊥b
dσ(yb)
×Q(Ya, ha,ya;x)Q(Yb, hb,yb;x′)
× exp[− (h2
a + h2b)
2
( 1
Ω2d
+ τ2c
)− (|ya|2 + |yb|2)
2X2d
)
× exp[τ2c hahbψ
( |Ya − Yb|xc
)].
11
Using (4.20) and the fact Ωd < τ−1c and Xd < Xc, we arrive at
E[ICIR(x)] =1
(2π)2
∫
∂Ω
dσ(Ya)
∫
R
dha
∫
Y ⊥a
dσ(ya)Q(Ya, ha,ya;x) exp(− h2
a
2Ω2d
− |ya|22X2
d
).
so thatE
[ICIR(x)ICIR(x′)
]' E
[ICIR(x)
]E
[ICIR(x′)
].
Therefore, the following theorem, where the SNR is defined analogously to (4.19), holds.
Theorem 4.5 With the same notation and assumptions as those in Theorem 4.4, it followsthat if Ωd < τ−1
c ω0 and Xd < Xc X0, then we have
SNRCIR 1.
To conclude, we notice that the values of the parameters Xd ' Xc and Ωd ' τ−1c achieve
a good trade-off between resolution and stability. When taking smaller values Ωd < τ−1c
and Xd < Xc one increases the signal-to-noise ratio but one also reduces the resolution. Inpractice, these parameters are difficult to estimate directly from the data, so it is betterto determine them adaptively, by optimizing over Ωd and Xd the quality of the resultingimage. This is exactly what is done in adaptive CINT [9].
4.4 Two Particular Cases
We now discuss the following two particular cases:
(i) If we take Xd → 0 then the CINT function has the form
ICIR(x) =1
(2π)2
∫∫
R×R
dω1dω2
∫
∂Ω
dσ(y)e−
(ω1−ω2)2
2Ω2d q(y, ω1)e
−iω1|y−x|q(y, ω2)eiω2|y−x|.
(4.22)This case could correspond to the situation in which Xc is very small, which meansthat the signals recorded by different sensors are so noisy that they are independentfrom each other.
It is easy to see that, for any Ωd, (4.22) is equal to
ICIR(x) =Ωd√2π
∫
∂Ω
dσ(Ya)
∫
R
dt∣∣q(Ya, |Ya − x| + t)
∣∣2 exp(− Ω2
dt2
2
).
Moreover, if Ωd is large, then (4.22) is (approximately) equivalent to the incoherentmatched field function
ICIR(x) '∫
∂Ω
dσ(Ya)∣∣q(Ya, |Ya − x|)
∣∣2. (4.23)
This function is called ”incoherent” because we take out the phase of the data bylooking at the square modulus only.
(ii) If we take Xd → ∞ then the CINT function has the form
ICIR(x) =1
(2π)2
∫∫
R×R
dω1dω2
∫∫
∂Ω×∂Ω
dσ(y1)dσ(y2)e−
(ω1−ω2)2
2Ω2d
×q(y1, ω1)e−iω1|y1−x|q(y2, ω2)e
iω2|y2−x|. (4.24)
12
This case could correspond to the situation in which Xc is very large, which meansthat the signals recorded by different sensors are strongly correlated with one another.This is a typical weak clutter noise case.
For any Ωd, the function (4.24) is equal to
ICIR(x) =Ωd√2π
∫
R
dt∣∣∣∫
∂Ω
dσ(Ya)q(Ya, |Ya − x| + t)∣∣∣2
exp(− Ω2
dt2
2
).
If Ωd is large, then (4.22) is equivalent to the coherent matched field function (orsquare KRM function):
ICIR(x) '∣∣∣∫
∂Ω
dσ(Ya)q(Ya, |Ya − x|)∣∣∣2
.
In contrast to (4.23), this function is called ”coherent” since we take into account thephase of the data.
5 CINT-Radon Algorithm in a Bounded Domain
When considering photoacoustics in a bounded domain, we developed in [3] an approachinvolving the line Radon transform of the initial condition. We will consider homogeneousDirichlet conditions:
∂2p
∂t2(x, t) − c(x)2∆p(x, t) = 0, x ∈ Ω,
p(x, 0) = p0(x),∂p
∂t(x, 0) = 0, x ∈ Ω,
p(y, t) = 0, y ∈ ∂Ω.
Here, Ω is not necessarily a disk. Let n denote the outward normal to ∂Ω. When c(x) = 1,we can express the line Radon transform of the initial condition p0(x) in terms of theNeumann measurements ∂np(y, t) = n(y) · ∇p(y, t) on ∂Ω × [0, T ]:
R[p0](θ, s) = W[∂np](θ, s),
where the line Radon transform is defined by
R[p0](θ, r) :=
∫
R
p0(rθ + sθ⊥)ds, θ ∈ S1, r ∈ R,
and
W[g](θ, s) :=
∫ T
0
dt
∫
∂Ω
dσ(x)g(x, t)H (x · θ + t− s) , θ ∈ S1, s ∈ R.
Here H denotes the Heaviside function. We then invert the Radon transform using thefiltered back-projection algorithm
p0 = R?BW[∂np],
where R? is the formal adjoint Radon transform:
R?[f ](x) =1
2π
∫
S1
dσ(θ)f(θ,x · θ) =1
(2π)2
∫
S1
dσ(θ)
∫
R
dωf(θ, ω)e−iωx·θ, x ∈ Ω,
13
and B is a ramp filter
B[g](θ, s) =1
4π
∫
R
dω|ω|g(θ, ω)e−iωs, θ ∈ S1.
Here, the hat stands for the Fourier transform (3.1) in the second (shift) variable. In theFourier domain, the inversion reads
p0(x) =1
(2π)2
∫
S1
dσ(θ)
∫
R
dω BW[∂np](θ, ω)e−iωx·θ, x ∈ Ω.
Therefore, a natural idea to extend the CINT imaging to bounded media is to consider theimaging function:
ICIR(x) :=1
(2π)4
∫∫
S1×S1
dσ(θ1)dσ(θ2)
∫∫
R×R
dω1dω2e−
(ω2−ω1)2
2Ω2d e
−|θ2−θ1|2
2Θ2d
× BW[∂np](θ1, ω1)e−iω1x·θ1 BW[∂np](θ2, ω2)e
iω2x·θ2 ,
(5.1)
where Θd is an angular cut-off parameter and plays the same role as Xd in the previoussections. Using the arguments developed before, it would be possible to perform a stabilityand resolution analysis for ICIR given by (5.1).
6 Numerical Illustrations
In this section we present numerical experiments to illustrate the performance of the CINT-Radon algorithms and to compare them with the Kirchhoff-Radon imaging function. We alsocompare ICI and IKM in the case where the clutter noise has high frequency components.
The wave equation (direct problem) can be rewritten as a first order partial differentialequation:
∂tP = AP + BP,
where
P =
(p∂tp
), A =
(0 1∆ 0
), and B =
(0 0
(c2 − 1)∆ 0
).
This equation is solved on the box Q = (−2, 2)2 containing Ω. We use a splitting spectralFourier approach [12] coupled with a perfectly matched layer (PML) technique [4] to sim-ulate a free outgoing interface on ∂Q. The operator A is computed exactly in the Fourierspace while the operator B is treated explicitly with a finite difference method and a PMLformulation to simulate a free outgoing interface on ∂Q in the case of free space. The inversecircular Radon formula is discretized as in [16]. Realizations of sound speed fluctuations arecomputed by the Fourier method (i.e., by filtering a discrete white noise). We first generateon a grid a white Gaussian noise. Then we filter the Gaussian noise in the Fourier domainby applying a low-pass filter. The parameters of the filter are linked to the correlation lengthof the clutter noise [21].
We consider two situations: point targets and extended targets. Since the case of interestin our analysis is when the correlation length of the clutter noise is comparable to or largerthan the typical wavelength, the simulated clutter noise in the case of point targets can havehigh frequencies while in the case of extended targets we should use a clutter noise with only
14
low frequencies. For point targets, only ICI and IKM are used. The imaging functions ICIR
and IKMR do not improve image reconstruction.In Figure 1, we consider p0 to be the sum of 6 Dirac masses. In order to solve the wave
equation with initial data p = p0 and ∂p/∂t = 0 at t = 0, we compute the solution to thewave equation with zero initial data but with source term given by df/dt× the sum of Diracmasses, with
f(t) = cos(2πω0t)tδω exp(−πt2δ2ω), with δω = 10, ω0 = 3δω.
The 6 Dirac masses emit therefore pulses of the form f , with f being an approximation ofthe derivative of a Dirac mass in time at t = 0.
We use the random velocity c1, visualized in Figure 2. It is a realization of a Gaussianprocess with Gaussian covariance function. Figures 3 and 4 present the pressure p(y, t)computed without and with clutter noise, and the reconstruction of the source locationsobtained by the Kirchhoff migration function IKM. Figures 3 and 4 illustrate that in thepresence of clutter noise, IKM becomes very instable and fails to really localize the targets.The images obtained by ICI are plotted in Figure 5 and compared to those obtained byIKM. Note that ICI presents better stability properties when Xd and Ωd become small aspredicted by the theory.
Figure 1: Positions of the sensors.
We now consider the case of extended targets and test the imaging function ICIR. We usethe random velocity c2 shown in Figure 2. Reconstructions of the initial pressure obtainedby IKRM are plotted in Figures 6, 7, 9, and 10 for two different configurations: the Shepp-Logan phantom and a line. These figures clearly highlight the fact that the clutter noisesignificantly affects the reconstruction. For example, in Figure 10 the whole line is not found.
On the other hand, plots of ICIR presented in Figures 8 and 11 provide more stablereconstructions of p0(x). However, note that choosing small values of the parameters Xd
and Ωd can affect the reconstruction in the sense that ICIR becomes very different from theexpected value, p2
0, when Xd and Ωd tend to zero. This is a manifestation of the trade-offbetween resolution and stability discussed in Section 4.
In the case of a bounded domain, we consider the low frequency cluttered speed ofFigure 2, on a square medium, with homogeneous Dirichlet conditions. We illustrate theperformance of ICIR on the Shepp-Logan phantom. Figure 12 shows the reconstruction
15
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
0.94
0.96
0.98
1
1.02
1.04
1.06
Figure 2: Left: random velocity c1 with high frequencies; right: random velocity c2 withlow frequencies.
0 0.5 1 1.5 2
50
100
150
200
250
0 0.5 1 1.5 2
50
100
150
200
250
Figure 3: Test1 (a): measured data p(y, t) with (right) and without (left) clutter noise. Thecoordinates are the pixel indices. The abscissa is time and the ordinate is arclength on ∂Ω.
Figure 4: Test1 (b): source localization using Kirchhoff migration IKM with (right) andwithout (left) clutter noise.
16
Figure 5: Test1 (c): source localization using the standard CINT function ICI, with param-eters Xd and Ωd given by Xd = 0.25, 0.5, 1 (from left to right) and Ωd = 25, 50, 100 (fromtop to bottom).
17
using the inverse Radon transform algorithm in the case with boundary conditions. Wenotice that the outer interface appears twice.
In fact, ICIR can correct this effect. Figure 13 shows results with a colormap saturatedat 80% of their maximum values for different values of Θd and Ωd. Here, Θd and Ωd arethe cut-off parameters in (5.1) and (3.8), respectively. The imaging function ICIR can getthe outer interface correctly. Moreover, if the colormap is saturated appropriately (here at80%), ICIR reconstructs as well the inside of the target with the good contrast.
50 100 150 200 250
20
40
60
80
100
120
50 100 150 200 250
20
40
60
80
100
120
Figure 6: Test2 (a): measured data p(y, t) with (right) and without clutter noise for theShepp-Logan phantom. The abscissa is time and the ordinate is arclength on ∂Ω.
Figure 7: Test2 (b): reconstruction of p0(x) using IKRM with (right) and without (left)clutter noise.
7 Conclusion
In this paper we have introduced new CINT-Radon type imaging functions in order to correctthe effect on photoacoustic images of random fluctuations of the background sound speedaround a known constant value. We have provided a stability and resolution analysis of the
18
Figure 8: Test2 (c): source localization using ICIR, with Xd and Ωd given by Xd = 0.5, 1, 2(from left to right) and Ωd = 50, 100, 200 (from top to bottom).
50 100 150 200 250
20
40
60
80
100
120
50 100 150 200 250
20
40
60
80
100
120
Figure 9: Test3 (a): measured data p(y, t) with (right) and without (left) clutter noise fora line target. The abscissa is time and the ordinate is arclength on ∂Ω.
19
Figure 10: Test3 (b): reconstruction of p0 using IKRM with (right) and without (left) clutternoise.
Figure 11: Test3 (c): source localization using ICIR, with Xd and Ωd given by Xd = 0.5, 1, 2(from left to right) and Ωd = 50, 100, 200 (from top to bottom).
20
Figure 12: Reconstruction of an extended target using line Radon transform in the case ofimposed boundary conditions.
proposed algorithms and found the values of the cut-off parameters which achieve a goodtrade-off between resolution and stability. We have presented numerical reconstructionsof both small and extended targets and compared our algorithms with Kirchhoff-Radonmigration functions. The CINT-Radon imaging functions give better reconstruction thanKirchhoff-Radon migration, specially for extended targets in the presence of a low-frequencyclutter noise.
A Calculation of the second-order moments
Proposition A.1 Let Xd,Ωd > 0. Let us consider
I(x) =1
(2π)2
∫∫
R×R
dω1dω2
∫∫
∂Ω×∂Ω
dσ(y1)dσ(y2)q(0)(y1, ω1)q(0)(y2, ω2)
×e−iω1|y1−x|eiω2|y2−x| exp(− (ω1 − ω2)
2
2Ω2d
− |y1 − y2|22X2
d
). (A.1)
If Xd X0, then we have
I(x) ' XdΩd
(2π)3
∫
∂Ω
dσ(Ya)
∫
R
dωa
∫
Y ⊥a
dσ(κa)
∫
R
dτa
×Wq(Ya, ωa;κa, τa) exp(−X2
d
∣∣κa − ωax−Ya
|x−Ya|
∣∣2
2− Ω2
d(τa − |Ya − x|)22
), (A.2)
where Wq is the Wigner transform of q(0) defined by (4.15) and Y ⊥a is defined by (4.16).
Proof. Let Y ⊥a be defined by (4.16). Using the change of variables
ω1 = ωa +ha
2, ω2 = ωa − ha
2, y1 = Ya +
ya
2, y2 = Ya − ya
2,
21
Figure 13: Extended target reconstruction with boundary conditions using ICIR, with Θd
and Ωd given by Θd = 1, 3, 6 (from top to bottom) and Ωd = 50, 100, 200 (from left to right).Colormaps are saturated at 80% of the maximum values of the images.
22
the function I(x) can be written as
I(x) =1
(2π)2
∫
∂Ω
dσ(Ya)
∫
R
dha
∫
Y ⊥a
dσ(ya)Q(Ya, ha,ya;x)
× exp(− h2
a
2Ω2d
− |ya|22X2
d
),
with
Q(Ya, ha,ya;x) =
∫
R
dωaq(0)(Ya +
ya
2, ωa +
ha
2)q(0)(Ya − ya
2, ωa − ha
2)
×e−i(ωa+ ha
2 )|Ya+ ya
2 −x|ei(ωa−ha
2 )|Ya−ya
2 −x|.
We have assumed that Xd is much smaller than X0(= 1), so that y1 − y2 is approximatelyorthogonal to (y1 + y2)/2 when y1,y2 ∈ ∂Ω and |y1 − y2| ≤ Xd. Then
Y ⊥a ' ya ∈ R
2 , Ya · ya = 0,
and
Q(Ya, ha,ya;x) '∫
R
dωaq(0)(Ya +
ya
2, ωa +
ha
2)q(0)(Ya − ya
2, ωa − ha
2)
×eiωax−Ya
|x−Ya|·ya−iha|x−Ya|
' 1
(2π)2
∫
R
dωa
∫
R
dτa
∫
Y ⊥a
dσ(κa)Wq(Ya, ωa;κa, τa)
×ei(ωax−Ya
|x−Ya|−κa)·ya+i(τa−|x−Ya|)ha ,
where Wq is the Wigner transform of q(0) defined by (4.15). Therefore, we get
I(x) ' XdΩd
(2π)3
∫
∂Ω
dσ(Ya)
∫
R
dωa
∫
Y ⊥a
dσ(κa)
∫
R
dτa
×Wq(Ya, ωa;κa, τa)
× exp(−X2
d
∣∣κa − ωa
(x−Ya
|x−Ya|− ( Ya
|Ya|· x−Ya
|x−Ya|) Ya
|Ya|
)∣∣2
2− Ω2
d(τa − |Ya − x|)22
).
Since, for any s,Wq(Ya, ωa;κa + sYa, τa) = Wq(Ya, ωa;κa, τa),
we obtain the desired result. 2
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