Coherent Interferometry Algorithms for Photoacoustic Imaging
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Ammari, Habib et al. “Coherent Interferometry Algorithms for
Photoacoustic Imaging.” SIAM Journal on Numerical Analysis
50.5 (2012): 2259–2280. © 2012, Society for Industrial and Applied
Mathematics
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http://dx.doi.org/10.1137/100814275
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Society for Industrial and Applied Mathematics
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http://hdl.handle.net/1721.1/77909
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COHERENT INTERFEROMETRY ALGORITHMS FOR
PHOTOACOUSTIC IMAGING∗
HABIB AMMARI†, ELIE BRETIN‡, JOSSELIN GARNIER§,AND VINCENT JUGNON¶
Abstract. The aim of this paper is to develop new coherent interferometry (CINT) algorithms to correct the effect of an unknown cluttered sound speed (random fluctuations around a known constant) on photoacoustic images. By back-propagating the correlations between the preprocessed pressure measurements, we show that we are able to provide statistically stable photoacoustic images. The preprocessing is exactly in the same way as when we use the circular or the line Radon inversion to obtain photoacoustic images. Moreover, we provide a detailed stability and resolution analysis of the new CINT–Radon algorithms. We also present numerical results to illustrate their performance and to compare them with Kirchhoff–Radon migration functions.
Key words. imaging, migration, interferometry, random media, resolution, stability AMS subject classifications. 65R32, 44A12, 35R60
DOI. 10.1137/100814275
1. Introduction. In photoacoustic imaging, optical energy absorption causes thermoelastic expansion of the tissue, which leads to the propagation of a pressure wave. This signal is measured by transducers distributed on the boundary of the object, which in turn is used for imaging optical properties of the object.
In pure optical imaging, optical scattering in soft tissues degrades spatial reso-lution significantly with depth. Pure optical imaging is very sensitive to optical ab-sorption but can only provide a spatial resolution of the order of 1 cm at cm depths. Pure conventional ultrasound imaging is based on the detection of the mechanical properties (acoustic impedance) in biological soft tissues. It can provide good spatial resolution because of its millimetric wavelength and weak scattering at MHz frequen-cies. The major contribution of photoacoustic imaging is to provide images of optical contrasts (based on the optical absorption) with the resolution of ultrasound [30]. Potential applications include breast cancer and vascular disease.
In photoacoustic imaging, the absorbed energy density is related to the optical absorption coefficient distribution through a model for light propagation such as the diffusion approximation or the radiative transfer equation. Although the problem of reconstructing the absorption coefficient from the absorbed energy is nonlinear, efficient techniques can be designed, especially in the context of small absorbers [2]. The absorbed optical energy density is the initial condition in the acoustic wave equation governing the pressure. If the medium is acoustically homogeneous and has the same acoustic properties as the free space, then the boundary of the object plays no
∗Received by the editors November 9, 2010; accepted for publication (in revised form) July 10,
2012; published electronically September 13, 2012. This work was supported by the ERC Advanced Grant project MULTIMOD–267184.
http://www.siam.org/journals/sinum/50-5/81427.html
†Department of Mathematics and Applications, Ecole Normale Sup´erieure, 45 Rue d’Ulm, 75005
Paris, France (habib.ammari@ens.fr).
‡Centre de Math´ematiques Appliqu´ees, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau,
France (bretin@cmap.polytechnique.fr).
§Laboratoire de Probabilit´es et Mod`eles Al´eatoires & Laboratoire Jacques-Louis Lions, site
Chevaleret, case 7012, Universit´e Paris VII, 75205 Paris Cedex 13, France (garnier@math.jussieu.fr).
¶Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139-4307 (vjugnon@math.mit.edu). 2259
role and the absorbed energy density can be reconstructed from measurements of the pressure wave by inverting a spherical or a circular Radon transform [22, 23, 19, 20]. Recently, we have been interested in reconstructing initial conditions for the wave equation with constant sound speed in a bounded domain. In [1, 3], we developed a variety of inversion approaches which can be extended to the case of variable but known sound speed and can correct for the effect of attenuation on image reconstruc-tions. However, the situation of interest for medical applications is the case where the sound speed is perturbed by an unknown clutter noise. This means that the speed of sound of the medium is randomly fluctuating around a known value. In this situa-tion, 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]. Coher-ent interferometry (CINT) was introduced and analyzed in [7, 8]. While classical meth-ods back-propagate the recorded signals directly, CINT is an array imaging method that first computes cross-correlations of the recorded signals over appropriately cho-sen space-frequency windows and then back-propagates the local cross-correlations. As shown in [7, 8, 9, 10], CINT deals well with partial loss of coherence in cluttered environments.
In the present paper, combining the CINT method for imaging in clutter to-gether with a reconstruction approach for extended targets by Radon inversions, we propose CINT–Radon algorithms for photoacoustic imaging in the presence of ran-dom fluctuations of the sound speed. We show that these new algorithms provide statistically stable photoacoustic images. We provide a detailed analysis for their sta-bility and resolution from a simple clutter noise model and numerically illustrate their performance.
The paper is organized as follows. In section 2 we formulate the inverse problem of photoacoustics and describe the clutter noise considered for the sound speed. In section 3 we recall the reconstruction using the circular Radon transform when the sound speed is constant and describe the original CINT algorithm. We then propose a new CINT approach which consists in preprocessing the data (in the same way as for the circular Radon inversion) before back-propagating their correlations. Section 4 is devoted to the stability analysis of this new algorithm. Section 5 adapts the results presented in sections 3 and 4 to the case of a bounded domain. We make a parallel between the filtered back-projection of thecircular Radon inversion in free space and of theline Radon inversion when we have boundary conditions. Both algorithms end with a back-propagation step. We propose to back-propagate the correlations between the (preprocessed) data in the same way as in section 3. The paper ends with a short discussion.
2. Problem formulation. In photoacoustics, laser pulses are focused into bi-ological tissues. The delivered energy is absorbed and converted into heat, which generates thermoelastic expansion, which in turn gives rise to the propagation of an ultrasonic pressure wave p(x, t) from a source term p0(x):
(2.1) ⎧ ⎪ ⎨ ⎪ ⎩ ∂2p ∂t2(x, t) − c(x) 2Δp(x, t) = 0, p(x, 0) = p0(x), ∂p ∂t(x, 0) = 0.
The pressure field is measured at the surface of a domain Ω that contains the support of p0. The imaging problem is to reconstruct the initial value of the pressure p0 in
a search domain from measurements of p on the whole boundary ∂Ω of a domain Ω. Most of the reconstruction algorithms assume constant (or known) sound speed. However, in real applications, the sound speed is not perfectly known. It seems more relevant to consider that it fluctuates randomly around a known distribution. For simplicity, we consider the model with random fluctuations around a constant that we normalize to one: (2.2) 1 c(x)2 = 1 + σcμ x xc ,
where μ is a zero-mean stationary random process, xc is the correlation length of the fluctuations of c(x), and σc is their standard deviation.
Throughout this paper, we restrict ourselves to the two-dimensional case and assume that Ω is the unit disk with center at 0 and radius X0 = 1. The two-dimensional case is of particular 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
(3.1) f (ˆy, ω) = R f (y, t)eiωtdt, f (y, t) = 1 2π R ˆ f (y, ω)e−iωtdω.
In free space, it is possible to link the measurements of the pressure waves p(y, t) on the boundary ∂Ω to the circular Radon transform of the initial condition p0(x) as follows (see, for instance, [14, p. 682] and [15]):
(3.2) RΩ[p0](y, r) = W[p](y, r), y ∈ ∂Ω, r ∈ R+,
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− t2dt, y ∈ ∂Ω, r ∈ R +. HereS1denotes the unit circle and dσ(θ) is the surface measure on S1.
Since Ω is assumed to be the unit disk with center at 0 and radius X0 = 1, it follows that in order to find p0 we can use the following exact inversion formula [16, 24]:
(3.3) p0(x) = 1
4π2R
ΩBW[p](x), x ∈ Ω,
where RΩ (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 ˆ
f (y, ω)e−iω|x−y|dωdσ(y), x ∈ Ω,
with dσ(y) being the surface measure on ∂Ω and B a filter defined by
B[g](y, t) = 2 0 d2g dr2(y, r) ln(|r 2− t2|) dr, y ∈ ∂Ω.
Note that (3.2) and (3.3) only hold for constant sound speed (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 time variable. Hence, we introduce the Kirchhoff–Radon migration imaging function (3.4) IKRM(x) = 1 4π2R ΩBW[p](x) = 1 2π ∂Ω dσ(y) R dω ˆq(x, ω)e−iω|x−y|, where (3.5) q = 1 4π2BW[p] is the preprocessed data.
A second imaging function is to simply back-propagate the raw data (without any preprocessing) [5]: (3.6) IKM(x) = RΩ[p](x) = 1 2π ∂Ω dσ(y) R dω ˆp(y, ω)e−iω|x−y|, x ∈ Ω.
Here, the subscript KM stands for Kirchhoff migration. As will be seen later, this simplified function is sufficient for localizing point sources in homogeneous media but may fail for imaging extended targets or when in the presence of clutter noise.
When the sound speed varies as in (2.2), the phases of the measured waves ˆp(ω,y)
are shifted with respect to the deterministic, unperturbed phase because of the un-known clutter. When the data are numerically back-propagated in the homogeneous medium with speed of propagation equal to one, the phase terms do not compensate each other in (3.6) when x is a source location, which results in instability and loss of resolution. To correct this effect, the idea of the original CINT algorithm is to back-propagate the space and frequency correlations between the data [8]:
(3.7) ICI(x) = (2π)1 2 ∂Ω×∂Ω, |y2−y1|≤Xd dσ(y1)dσ(y2) R×R, |ω2−ω1|≤Ωd dω1dω2 × ˆp(y1, ω1)e−iω1|x−y1|p(ˆy2, ω2)eiω2|x−y2|.
The CINT function is close to the KM function. In fact, if Ωd → ∞ and Xd → ∞, thenICI is the square of the Kirchhoff migration function:
ICI(x) = |IKM(x)|2.
However the cutoff parameters Ωd and Xd play a crucial role. When one writes the CINT function 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 data in a time interval scaled by 1/Ωdand by superposing the back-propagated local
correlations over 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 locationsy1,y2the random phase shifts of the data ˆp(y1, ω1), ˆp(y2, ω2) are similar (say, correlated) so they cancel each other in the product ˆp(y1, ω1)ˆp(y2, ω2). We then say that the data ˆp(y1, ω1) and ˆp(y2, ω2) are coherent. In such a case, the back-propagation of this product in the homogeneous medium should be stable. The purpose of the CINT imaging function is to keep in (3.7) the pairs (y1, ω1) and (y2, ω2) for which the data ˆp(y1, ω1) and ˆp(y2, ω2) are coherent and to remove the pairs that do not bring information. It then appears intuitive that the cutoff parameters Xd, respectively, Ωd, should be of the order of the spatial (respectively, frequency) correlation radius of the recorded data, and we will confirm this intuition in the following.
As will be shown later,ICIis quite efficient in localizing point sources in cluttered media but not in finding the true value of p0 (up to a square). Moreover, when the support of the initial pressure p0is extended,ICImay fail in recovering a good photoa-coustic image. We propose two things. First, in order to avoid numerical oscillatory effects, we replace the sharp cutoffs in the integral by Gaussian convolutions. Then instead of taking the correlations between the back-propagated raw data, we prepro-cess them like we do for the Radon inversion. We thus get the following CINT–Radon imaging function: (3.8) ICIR(x) = 1 (2π)2 ∂Ω×∂Ω dσ(y1)dσ(y2) R×R dω1dω2e− (ω2−ω1)2 2Ω2d e−|y1−y2|22X2d
× q(y1, ω1)e−iω1|x−y1| q(y2, ω2)eiω2|x−y2|,
where q is given by (3.5). Note again that when Ωd → ∞ and Xd → ∞, ICIR is the square of the Kirchhoff–Radon migration function:
ICIR(x) = |IKRM(x)|2.
The purpose of the CINT–Radon imaging functionICIR is to keep in (3.8) the pairs (y1, ω1) and (y2, ω2) for which the preprocessed data ˆq(y1, ω1) and ˆq(y2, ω2) are co-herent and to remove 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 case provided that the measurements are taken on a sphere ∂Ω.
Note that because of the preprocessing step (3.5), which is in the same way as when one inverts the Radon transform in (3.3), we called, by abuse of language, the imaging functionsIKRMandICIR 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 cutoff parameters Ωd and Xd in the CINT–Radon imaging function. This analysis is based on arguments 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 their resolution 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 a medium 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 and large correlation length (compared to the typical wavelength λ0) [28, 25]. More exactly, the random travel time model is valid when the correlation length xc and the standard deviation σc of the fluctuations of the wave speed c(x) satisfy conditions [6]
(4.1) xc X0, σc2 x 3 c X03, σ 2 c X03 x3c λ20 σ2cxcX0 1 .
Here X0 is the radius of Ω (taken equal to one for simplicity). In these conditions the geometric optics 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. Therefore there is an error ν(x, y) between the theoretical travel time τ0(x, y) =
|y − x| with the background velocity equal to one and the real travel time τ(x, y),
where y is a point of the surface of the observation disk ∂Ω and x is a point of the search domain (the domain in which we try to recover p0):
τ (x, y) = τ0(x, y) + ν(x, y).
The random process ν(x, y) is given by
(4.2) ν(x, y) =σc|y − x| 2 1 0 μx + (y − x)s xc ds.
This is the integral of the fluctuations of 1/c along the unperturbed, straight ray from
y to x. Assuming that the search domain is relatively small we can assume that ν
depends only on the sensor position y and we can neglect the variations of ν with respect tox. This is perfectly correct if we analyze the expectations and the variances of the imaging functions for a fixed test pointx. (Then the search domain is just one point.) This is still correct if we analyze the covariance of the imaging function for a pair of test pointsx and x that are close to each other (closer than the correlation radius xc of the clutter noise).
Note finally that the random travel time model can be used to model another type of noisy data which arises when the medium is homogeneous but the positions of the sensors are poorly characterized.
We assume that the random process μ is a random process with Gaussian statis-tics, mean zero, and covariance function:
E σcμ x xc σcμ x xc = σ2cexp −|x − x|2 2x2c .
Here E stands for the expectation (mean value). Gaussian statistics means that for any finite 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:
(4.3) E[ν(y)ν(y)] = τc2ψ |y − y| xc , ψ(r) = 1 r r 0 exp −s2 2 ds,
where
(4.4) τc2=√2πσ2c/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),
where its variations with respect to x are neglected, has mean zero and covariance function E[ν(y)ν(y)] = σc2|y − x| |y− x| 4 1 0 ds 1 0 dsexp −|s(y − x) − s(y− x)|2 2x2c = σ 2 c|y − x| |y− x| 4 1 0 ds 1−s −s d˜s 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)] = √ 2πσ2c|y − x| 4 1 0 ds exp − s2 2x2c |y − y|2− x − y |x − y|· (y − y) 2 .
Ifx is close to the center of the disk Ω, then y − y is approximately orthogonal to
x − y and we obtain (4.3).
Using Gaussian statistics it is straightforward to compute the moments E[eiων(y)] = exp
−ω2τc2 2 , (4.6)
E[eiων(y)−iων(y)] = exp
−(ω− ω)2τc2 2 − ωω τ2 c 1− ψ |y − y| xc .
If, additionally, we assume that ω, ω ω0 with
(4.7) ω0τc 1,
then using a second order Taylor expansion of ψ and the assumption|y − y| xc, we arrive at
E[eiων(y)−iων(y)] exp −(ω− ω)2τc2 2 − |y − y|2 2Xc2 (4.8) with (4.9) Xc2= 3x 2 c ω02τc2.
Note that the typical wavelength λ0= 2π/ω0.
From (4.8) we can see that τc−1 is the frequency coherence radius and Xc is the spatial coherence radius of the recorded signals.
In the following sections it will be convenient to introduce a typical frequency
ω0. Since we deal with a wave equation with initial conditions (2.1) and the speed of propagation is one, the typical wavelength for this problem is of the order of the diameter of the support of the function p0when it is smooth or of the order of the scale of variations of the function p0when it is oscillating. Since the speed of propagation is one, the typical frequency ω0is of the order of the reciprocal of the typical wavelength. Moreover, the bandwidth (i.e., the spectral width of the data) is also of the order of the typical frequency.
4.2. Kirchhoff–Radon migration. Recall that the Kirchhoff–Radon migra-tion funcmigra-tion is IKRM(x) =2π1 ∂Ω dσ(y) R dω ˆq(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 preprocessed data
(4.11) q(0)= 1 4π2BW[p (0)] is (4.12) IKRM(0) (x) = 1 2π ∂Ω dσ(y) R dω ˆq(0)(y, ω)e−iω|y−x|= ∂Ω dσ(y)q(0)(y, |y − x|),
and it is equal to the initial condition p0(x). Here and below we denote by p(0) and
q(0) the unperturbed data obtained when the medium is homogeneous (μ≡ 0). We consider the random travel time model to describe the recorded data set [29]: (4.13) q(y, t) = q(0)(y, t − ν(y)).
This is a rough approximate model that accounts only for random time delays, but calculations can be carried out in a rather simple manner that illustrate the dual role of the cutoff parameters 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 ν being 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 following holds.
Theorem 4.2. Let IKRM be defined by (4.10). Consider a travel time model for
the effect of clutter noise and assume that conditions (4.1) and (4.7) are satisfied. We have E[IKRM(x)] = 1 2π ∂Ω dσ(y) R dω ˆq(0)(y, ω) exp −ω2τc2 2 e−iω|y−x| exp −ω20τc2 2 IKRM(0) (x) (4.14)
with A B iff A = B(1 + o(1)). Here, τc, given by (4.4), is the variance of the fluctuations of the travel times.
Theorem 4.2 shows that the mean function undergoes a strong damping compared to the unperturbed functionIKRM(0) (x). This phenomenon is standard when studying wave propagation 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, the typical wavelength λ0= 2π/ω0).
The statistics of the fluctuations can be characterized by the covariance. Using (4.8) we have EIKRM(x)IKRM(x) = 1 (2π)2 ∂Ω×∂Ω dσ(y1)dσ(y2) R×R dω1dω2qˆ(0)(y1, ω1)ˆq(0)(y2, ω2) × exp −(ω1− ω2)2τc2 2 − |y1− y2|2 2Xc2 e−iω1|y1−x|eiω2|y2−x|.
The variance that characterizes the amplitude of the fluctuations is VarIKRM(x):=E|IKRM(x)|2−EIKRM(x)2.
We assume that Xc X0, that is, the correlation radius is smaller than the propa-gation distance. 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− ωa|x−Yx−Yaa|2 2Xc−2 −(τa− |Ya− x|)2 2τc2 ⎞ ⎠ , whereWq is the Wigner transform of q(0),
Wq(Ya, ωa;κa, τa) = R dha Y⊥ a dσ(ya)ˆq(0) Ya+y2a, ωa+ ha 2 ˆ q(0) Ya−y2a, ωa− ha 2 (4.15)
× eiκa·ya−ihaτa,
andYa⊥ is defined by (4.16) Ya⊥=ya∈ R2,Ya−ya 2 ∈ ∂Ω and Ya+ ya 2 ∈ ∂Ω .
If we consider the regime in which τc−1 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|ˆq(0)(Ya, ωa)|2, and thus (4.17) E|IKRM(x)|2 Xcτc−1 ∂Ω dσ(y) R dtq(0)(y, t)2.
Combining (4.17) together with (4.12) yields (4.18) E|IKRM(x)|2 IKRM(0) (x)2 1 ω0τc Xc X0 .
Define the signal-to-noise ratio (SNR) by
(4.19) SNRKRM= |E[IKRM(x)]|
Var(IKRM(x))1/2.
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 if ω0τc 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 coherence is maintained as long as exp(iω2ν(y2)− iω1ν(y1)) is close to one. From (4.8) this requires that |ω1− ω2| < τc−1 and|y1− y2| < Xc. We can therefore anticipate that the cutoff parameters Xd and Ωd should be related to the coherence parameters Xc and τc−1. In the following we study the role of the cutoff parameters Xd and Ωd for resolution and stability of the CINT–Radon function. For doing so, we compute the expectation and variance of the imaging functionICIR.
Using (4.8), we have E[ICIR(x)] = (2π)1 2 R×R dω1dω2 ∂Ω×∂Ω dσ(y1)dσ(y2)ˆq(0)(y1, ω1)ˆq(0)(y2, ω2) (4.20)
× e−iω1|y1−x|eiω2|y2−x|exp−(ω1− ω2)2 2 τc2+ 1 Ω2d −|y1− y2|2 2 1 Xd2 + 1 Xc2 ,
where q(0) is the perfect preprocessed 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 of clutter noise and assume that conditions (4.1) and (4.7) are satisfied. Let Xc be defined by (4.9) with xc being the correlation length of the clutter noise, τc the variance of the fluctuations of the travel times, and ω0 the typical frequency. If Xc X0, where X0 is the radius of Ω, then we have
E[ICIR(x)] = s 1 (2π)3( 1 Xd2 +X1c2 )12(τc2+Ω12 d )12 ∂Ω dσ(Ya) R dωa Y⊥ a dσ(κa) R dτa (4.21) × Wq(Ya, ωa;κa, τa) exp ⎛ ⎝−κa− ωa|x−Yx−Yaa|2 2( 1 Xd2+X1c2) −(τa− |Ya− x|)2 2(τc2+Ω12 d ) ⎞ ⎠ ,
whereWq 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– Radon function is a smoothed version of the Wigner transform of the preprocessed data. It selects a band of directions (κa) and time delays (τa) that are centered around the direction and the time delay between the search pointx and the point Ya of the sensor array.
We observe that
• if Ωd > τc−1 and Xd > Xc, then the cutoff parameters Ωd and Xd have
no influence on the coherent part of the function which does not depend on (Ωd, Xd).
• if Ωd < τc−1 and Xd < Xc, then the cutoff parameters Ωd and Xd have an
influence and reduce the resolution of the coherent part of the function (they enhance the smoothing of the Wigner transform).
We next compute the covariance of the CINT–Radon function in the regime Ωd< τc−1 and Xd < Xc. Let us consider four points (yj)j=1,...,4 and four frequencies (ωj)j=1,...,4 in the bandwidth. Using the Gaussian property of the process ν(y) and the fact that ψ(0) = 1, we have
Eeiω1ν(y1)−iω2ν(y2)eiω3ν(y3)−iω4ν(y4)= exp ⎧ ⎨ ⎩− τc2 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), andya,yb = O(Xd), then Eeiω1ν(y1)−iω2ν(y2)eiω3ν(y3)−iω4ν(y4) exp
% −τc2 2 h2a+ h2b− 2hahbψ |Ya− Yb| xc &
(note that Xd xc since Xd≤ Xc and Xc xc) and therefore EICIR(x)ICIR(x)
= 1 (2π)4 ∂Ω dσ(Ya) R dha Y⊥ a dσ(ya) ∂Ω dσ(Yb) R dhb Y⊥ b dσ(yb) × ˆQ(Ya, ha,ya;x) ˆQ(Yb, hb,yb;x) × exp −(h2a+ h2b) 2 1 Ω2d + τ 2 c −(|ya|2+|yb|2) 2Xd2 × exp τc2hahbψ |Ya− Yb| xc .
Using (4.20) and the fact Ωd< τc−1 and Xd< Xc, we arrive at E[ICIR(x)] = 1 (2π)2 ∂Ω dσ(Ya) R dha Y⊥ a dσ(ya) ˆQ(Ya, ha,ya;x) exp − h2a 2Ω2d − |ya|2 2Xd2 so that
EICIR(x)ICIR(x) EICIR(x)EICIR(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 follows that if Ωd < τc−1 ω0 and Xd< Xc X0, then we have
SNRCIR 1.
To conclude, we notice that the values of the parameters Xd Xc and Ωd τc−1 achieve a good trade-off between resolution and stability. When taking smaller values Ωd < τc−1 and Xd < Xc one increases the SNR but one also reduces the resolution. In practice, these parameters are difficult to estimate directly from the data, so it is better to determine them adaptively by optimizing over Ωd and Xdthe quality of the resulting image. 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) = (2π)1 2 R×R dω1dω2 × ∂Ω dσ(y)e− (ω1−ω2)2 2Ω2d q(ˆy, ω 1)e−iω1|y−x|q(ˆy, ω2)eiω2|y−x|. (4.22)
This case could correspond to the situation in which Xc is very small, which means that the signals recorded by different sensors are so noisy that they are independent from each other.
It is easy to see that for any Ωd (4.22) is equal to
ICIR(x) = √Ωd 2π ∂Ω dσ(Ya) R dtq(Ya,|Ya− x| + t)2exp −Ω2dt2 2 .
Moreover, if Ωd is large, then (4.22) is (approximately) equivalent to the incoherent matched field function
(4.23) ICIR(x)
∂Ω
dσ(Ya)q(Ya,|Ya− x|)2.
This function is called “incoherent” because we take out the phase of the data by looking at the square modulus only.
(ii) If we take Xd→ ∞, then the CINT function has the form
ICIR(x) =(2π)1 2 R×R dω1dω2 ∂Ω×∂Ω dσ(y1)dσ(y2)e− (ω1−ω2)2 2Ω2 d
׈q(y1, ω1)e−iω1|y1−x|q(ˆy2, ω2)eiω2|y2−x|.
(4.24)
This case could correspond to the situation in which Xc is very large, which means that 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)2exp −Ω2dt2 2 .
If Ωd is large, then (4.22) is equivalent to the coherent matched field function (or square 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 the phase of the data.
5. CINT–Radon algorithm in a bounded domain. When considering pho-toacoustics in a bounded domain, we developed in [3] an approach involving the line Radon transform of the initial condition. We will consider homogeneous Dirichlet conditions: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂2p ∂t2(x, t) − c(x) 2Δp(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 the Neumann 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 the filtered back-projection algorithm
p0=RBW[∂np],
whereR 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 dω ˆf (θ, ω)e−iωx·θ, x ∈ Ω,
andB is a ramp filter
B[g](θ, s) = 1
4π
R
dω|ω|ˆg(θ, ω)e−iωs, θ ∈ S1.
Here, the hat stands for the Fourier transform (3.1) in the second (shift) variable. In the Fourier 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 the imaging function
(5.1) ICIR(x) := 1 (2π)4 S1×S1 dσ(θ1)dσ(θ2) R×R dω1dω2e− (ω2−ω1)2 2Ω2d e−|θ2−θ1|22Θ2d × BW[∂np](θ1, ω1)e−iω1x·θ1BW[∂np](θ2, ω2)eiω2x·θ2,
where Θdis an angular cutoff parameter and plays the same role as Xdin the previous sections. Using the arguments developed before, it would be possible to perform a stability and resolution analysis forICIR 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 also compare 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 differential equation ∂tP = AP + BP, where P = p ∂tp , A = 0 1 Δ 0 , and B = 0 0 (c2− 1Δ 0 .
This equation is solved on the box Q = (−2, 2)2containing Ω. We use a splitting
spec-tral Fourier approach [12] coupled with a perfectly matched layer (PML) technique
[4] to simulate a free outgoing interface on ∂Q. The operator A is computed exactly in the Fourier space while the operator B is treated explicitly with a finite difference method and a PML formulation to simulate a free outgoing interface on ∂Q in the case of free space. The inverse circular Radon formula is discretized as in [16]. Realizations of sound speed fluctuations are computed by the Fourier method (i.e., by filtering a discrete white noise). We first generate on a grid a white Gaussian noise. Then we filter the Gaussian noise in the Fourier domain by applying a low-pass filter. The parameters of the filter are linked to the correlation length of the clutter noise [21].
We consider two situations: point targets and extended targets. Since the case of interest in our analysis is when the correlation length of the clutter noise is comparable to or larger than the typical wavelength, the simulated clutter noise in the case of point targets can have high frequencies while in the case of extended targets we should use a clutter noise with only low frequencies. For point targets, onlyICIandIKMare used. The imaging functions ICIR andIKMRdo not improve image reconstruction.
In Figure 6.1, we consider p0to be the sum of six 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 the wave equation with zero initial data but with source term given by
df /dt× the sum of Dirac masses with
Fig. 6.1. Positions of the sensors.
f (t) = cos(2πω0t)tδωexp(−πt2δ2ω) with δω= 10, ω0= 3δω.
The six Dirac masses emit therefore pulses of the form f with f being an approxima-tion of the derivative of a Dirac mass in time at t = 0.
We use the random velocity c1, visualized in Figure 6.2. It is a realization of a Gaussian process with Gaussian covariance function. Figures 6.3 and 6.4 present the
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
Fig. 6.2. Left: random velocityc1 with high frequencies; right: random velocityc2 with low
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
Fig. 6.3. Test 1 (a): measured datap(y, t) with (right) and without (left) clutter noise. The
coordinates are the pixel indices. The abscissa is time and the ordinate is arclength on∂Ω.
50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500
Fig. 6.4. Test 1 (b): source localization using Kirchhoff migrationIKMwith (right) and without
(left) clutter noise.
pressure p(y, t) computed without and with clutter noise and the reconstruction of the source locations obtained by the Kirchhoff migration function IKM. Figures 6.3 and 6.4 illustrate that in the presence of clutter noise,IKMbecomes very instable and fails to really localize the targets. The images obtained byICI are plotted in Figure 6.5 and compared to those obtained byIKM. Note thatICI presents better stability properties when Xd and Ωdbecome small as predicted by the theory.
We now consider the case of extended targets and test the imaging functionICIR. We use the random velocity c2 shown in Figure 6.2. Reconstructions of the initial pressure obtained byIKRMare plotted in Figures 6.6, 6.7, 6.8, and 6.9 for two different configurations: the Shepp–Logan phantom and a line. These figures clearly highlight the fact that the clutter noise significantly affects the reconstruction. For example, in Figure 6.9 the whole line is not found.
50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250
Fig. 6.5. Test 1 (c): source localization using the standard CINT functionICIwith parameters
Xdand Ωdgiven byXd= 0.25, 0.5, 1 (from left to right) and Ωd= 25, 50, 100 (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
Fig. 6.6. Test 2 (a): measured data p(y, t) with (right) and without clutter noise for the
Shepp–Logan phantom. The abscissa is time and the ordinate is arclength on∂Ω.
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Fig. 6.7. Test 2 (b): reconstruction ofp0(x) using IKRMwith (right) and without (left) clutter
noise. 50 100 150 200 250 20 40 60 80 100 120 50 100 150 200 250 20 40 60 80 100 120
Fig. 6.8. Test 3 (a): measured datap(y, t) with (right) and without (left) clutter noise for a
line target. The abscissa is time and the ordinate is arclength on∂Ω.
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Fig. 6.9. Test 3 (b): reconstruction ofp0 usingIKRM with (right) and without (left) clutter
noise.
On the other hand, plots of ICIR presented in Figures 6.10 and 6.11 provide more stable reconstructions of p0(x). However, note that choosing small values of the parameters Xd and Ωd can affect the reconstruction in the sense thatICIR becomes very different from the expected value, p20, when Xd and Ωd tend to zero. This is a manifestation of the trade-off between resolution and stability discussed in section 4. In the case of a bounded domain, we consider the low frequency cluttered speed of Figure 6.2 on a square medium with homogeneous Dirichlet conditions. We illustrate the performance of ICIR on the Shepp–Logan phantom. Figure 6.12 shows the re-construction using the inverse Radon transform algorithm in the case with boundary conditions. We notice that the outer interface appears twice.
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Fig. 6.10. Test 2 (c): source localization using ICIRwith Xd and Ωd given byXd= 0.5, 1, 2
(from left to right) and Ωd= 50, 100, 200 (from top to bottom).
20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120
Fig. 6.11. Test 3 (c): source localization using ICIRwith Xd and Ωd given byXd= 0.5, 1, 2
(from left to right) and Ωd= 50, 100, 200 (from top to bottom).
In fact, ICIR can correct this effect. Figure 6.13 shows results with a colormap saturated at 80% of their maximum values for different values of Θd and Ωd. Here,
Fig. 6.12. Reconstruction of an extended target using line Radon transform in the case of
imposed boundary conditions.
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Fig. 6.13. Extended target reconstruction with boundary conditions usingICIR with Θd and
Ωdgiven 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.
Θd and Ωd are the cutoff parameters in (5.1) and (3.8), respectively. The imaging function ICIR can get the outer interface correctly. Moreover, if the colormap is saturated appropriately (here at 80%), ICIR reconstructs as well the inside of the target with the good contrast.
7. Conclusion. In this paper we have introduced new CINT–Radon type imag-ing functions in order to correct the effect on photoacoustic images of random fluc-tuations of the background sound speed around a known constant value. We have provided a stability and resolution analysis of the proposed algorithms and found the values of the cutoff parameters which achieve a good trade-off between resolution and stability. We have presented numerical reconstructions of both small and extended targets and compared our algorithms with Kirchhoff–Radon migration functions. The CINT–Radon imaging functions give better reconstruction than Kirchhoff–Radon mi-gration, especially for extended targets in the presence of a low-frequency clutter noise.
Appendix 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)ˆq(0)(y1, ω1)ˆq(0)(y2, ω2)
×e−iω1|y1−x|eiω2|y2−x|exp−(ω1− ω2)2 2Ω2d − |y1− y2|2 2Xd2 . (A.1) If Xd X0, then we have I(x) XdΩd (2π)3 ∂Ω dσ(Ya) Rdωa Y⊥ a dσ(κa) Rdτa ×Wq(Ya, ωa;κa, τa) exp ⎛ ⎝−Xd2κa− ωa|x−Yx−Yaa|2 2 − Ω2d(τa− |Ya− x|)2 2 ⎞ ⎠ , (A.2)
where Wq is the Wigner transform of q(0) defined by (4.15) and Ya⊥ is defined by
(4.16).
Proof. Let Ya⊥ be defined by (4.16). Assume that Xd is much smaller than
X0(= 1). Fory1,y2∈ ∂Ω such that |y1− y2| < Xd, letYa ∈ ∂Ω be the point on ∂Ω equidistant fromy1andy2and satisfying|Ya− y1| < Xd. LetYa+ya
2 (respectively,
Ya−y2a) be the orthogonal projection ofy1 (respectively,y2) on the tangent line to
∂Ω atYa. Using the change of variables
ω1= ωa+ha 2 , ω2= ωa− ha 2 , y1=Ya+ ya 2 , y2=Ya− ya 2 , the functionI(x) can be approximated as
I(x) 1 (2π)2 ∂Ω dσ(Ya) R dha Y⊥ a dσ(ya) ˆQ(Ya, ha,ya;x) × exp − h2a 2Ω2d − |ya|2 2Xd2 with ˆ Q(Ya, ha,ya;x) = R dωaqˆ(0) Ya+y2a, ωa+ ha 2 ˆ q(0) Ya−y2a, ωa− ha 2
×e−i(ωa+ha2 )|Ya+ya2 −x|ei(ωa−ha2)|Ya−ya2 −x|.
Since y1− y2 is approximately orthogonal to (y1+y2)/2 when y1,y2 ∈ ∂Ω and |y1− y2| ≤ Xd, Y⊥ a {ya∈ R2, Ya· ya= 0} and ˆ Q(Ya, ha,ya;x) R dωaqˆ(0) Ya+y2a, ωa+ ha 2 ˆ q(0) Ya−y2a, ωa− ha 2
×eiωa|x−Ya|x−Ya ·ya−iha|x−Ya|
1 (2π)2 R dωa R dτa Y⊥ a dσ(κa)Wq(Ya, ωa;κa, τa)
×ei(ωa|x−Ya|x−Ya −κa)·ya+i(τa−|x−Ya|)ha,
whereWq 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 ⎛ ⎝−Xd2κa− ωa x−Y a |x−Ya|− ( Ya |Ya|· x−Ya |x−Ya|) Ya |Ya|) 2 2 − Ω2d(τa− |Ya− x|)2 2 ⎞ ⎠ . Since for any s
Wq(Ya, ωa;κa+ sYa, τa) =Wq(Ya, ωa;κa, τa),
we obtain the desired result.
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