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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Wed May 25 22:03:08 EDT 2016
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http://hdl.handle.net/1721.1/77909
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SIAM J. NUMER. ANAL.
Vol. 50, No. 5, pp. 2259–2280
c 2012 Society for Industrial and Applied Mathematics
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 resolution significantly with depth. Pure optical imaging is very sensitive to optical absorption 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 frequencies. 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 Supérieure, 45 Rue d’Ulm, 75005
Paris, France (habib.ammari@ens.fr).
‡ Centre de Mathématiques Appliquées, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau,
France (bretin@cmap.polytechnique.fr).
§ Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, site
Chevaleret, case 7012, Université 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
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2260
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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 reconstructions. 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 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]. Coherent interferometry (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 computes cross-correlations of the recorded signals over appropriately chosen 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 together with a reconstruction approach for extended targets by Radon inversions, we
propose CINT–Radon algorithms for photoacoustic imaging in the presence of random fluctuations of the sound speed. 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 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 the circular Radon inversion in free space and
of the line 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 biological 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
⎪
⎨ ∂ p (x, t) − c(x)2 Δp(x, t) = 0,
∂t2
(2.1)
∂p
⎪
⎩p(x, 0) = p0 (x),
(x, 0) = 0.
∂t
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
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
2261
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:
x
1
= 1 + σc μ
(2.2)
,
2
c(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 twodimensional 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
1
iωt
ˆ
f (y, t)e dt,
f (y, t) =
(3.1)
f (y, ω) =
fˆ(y, ω)e−iωt dω.
2π R
R
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,
where the circular Radon transform is defined by
rp0 (y + rθ)dσ(θ),
RΩ [p0 ](y, r) :=
y ∈ ∂Ω,
S1
and
W[p](y, r) := 4r
0
r
p(y, t)
√
dt,
r 2 − t2
r ∈ R+ ,
y ∈ ∂Ω,
r ∈ R+ ,
r ∈ R+ .
Here S1 denotes 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 R BW[p](x),
4π 2 Ω
x ∈ Ω,
where RΩ (the formal adjoint of the circular Radon transform) is a back-projection
operator given by
1
RΩ [f ](x) =
f (y, |x − y|)dσ(y) =
fˆ(y, ω)e−iω|x−y| dωdσ(y), x ∈ Ω,
2π ∂Ω R
∂Ω
with dσ(y) being the surface measure on ∂Ω and B a filter defined by
2 2
d g
(y, r) ln(|r2 − t2 |) dr, y ∈ ∂Ω.
B[g](y, t) =
2
0 dr
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2262
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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
1
p0 (x) =
BW[p](y,
ω)e−iω|x−y| dωdσ(y), x ∈ Ω.
(2π)3 ∂Ω R
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
1 1
R
BW[p](x)
=
dσ(y)
dω q̂(x, ω)e−iω|x−y| ,
(3.4)
IKRM (x) =
4π 2 Ω
2π ∂Ω
R
where
(3.5)
q=
1
BW[p]
4π 2
is the preprocessed data.
A second imaging function is to simply back-propagate the raw data (without any
preprocessing) [5]:
1
(3.6)
IKM (x) = RΩ [p](x) =
dσ(y) dω p̂(y, ω)e−iω|x−y| , x ∈ Ω.
2π ∂Ω
R
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 p̂(ω, y)
are shifted with respect to the deterministic, unperturbed phase because of the unknown 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)
1
dσ(y1 )dσ(y2 )
dω1 dω2
ICI (x) =
(2π)2
∂Ω×∂Ω, |y2 −y1 |≤Xd
R×R, |ω2 −ω1 |≤Ωd
× p̂(y1 , ω1 )e−iω1 |x−y1 | p̂(y2 , ω2 )eiω2 |x−y2 | .
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 cutoff parameters Ωd and Xd play a crucial role. When one writes the
CINT function in the time domain
ICI (x) =
dσ(y1 )dσ(y2 )
∂Ω×∂Ω, |y2 −y1 |≤Xd
Ωd
×
π
R
dt sinc(Ωd t) 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/Ωd and by superposing the back-propagated local
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
2263
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 locations y1 , y2 the random phase shifts of the data p̂(y1 , ω1 ), p̂(y2 , ω2 ) are
similar (say, correlated) so they cancel each other in the product p̂(y1 , ω1 )p̂(y2 , ω2 ).
We then say that the data p̂(y1 , ω1 ) and p̂(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 p̂(y1 , ω1 ) and p̂(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, ICI is 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 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 replace the sharp cutoffs in the integral by Gaussian convolutions. Then
instead of taking the correlations between the back-propagated raw data, we preprocess them like we do for the Radon inversion. We thus get the following CINT–Radon
imaging function:
(3.8)
ICIR (x) =
1
(2π)2
dσ(y1 )dσ(y2 )
∂Ω×∂Ω
× q(y1 , ω1 )e
−iω1 |x−y1 |
q(y2 , ω2 )e
R×R
dω1 dω2 e
iω2 |x−y2 |
−
(ω2 −ω1 )2
2Ω2
d
e
−
|y1 −y2 |2
2X 2
d
,
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 function ICIR is to keep in (3.8) the pairs
(y1 , ω1 ) and (y2 , ω2 ) for which the preprocessed data q̂(y1 , ω1 ) and q̂(y2 , ω2 ) are coherent 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 threedimensional 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 functions IKRM 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 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
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2264
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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 x3c
,
X03
σc2
X03
λ2
2 0
1.
3
xc
σc xc X0
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)
σc |y − x|
ν(x, y) =
2
1
μ
0
x + (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 to x. This is perfectly correct if we analyze the expectations and the variances
of the imaging functions for a fixed test point x. (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 points x 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 statistics, mean zero, and covariance function:
x
x
|x − x |2
2
E σc μ
σc μ
= σc exp −
.
xc
xc
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:
2
|y − y |
s
1 r
2
exp −
(4.3)
E[ν(y)ν(y )] = τc ψ
,
ψ(r) =
ds,
xc
r 0
2
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
where
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(4.4)
τc2 =
√
2265
2πσc2 /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
1
σ 2 |y − x| |y − x| 1
|s(y − x) − s (y − x)|2
ds
ds exp −
E[ν(y)ν(y )] = c
4
2x2c
0
0
1−s
|s(y − y ) + s̃(x − y )|2
σc2 |y − x| |y − x| 1
ds
ds̃ exp −
(4.5)
.
=
4
2x2c
0
−s
Moreover, when X0 xc , we can extend the s̃ integral in (4.5) to the entire real line.
Integrating in s̃ then gives
E[ν(y)ν(y )]
√
2 x − y
2πσc2 |y − x| 1
s2
2
· (y − y )
ds exp − 2 |y − y | −
.
=
4
2xc
|x − y |
0
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).
Using Gaussian statistics it is straightforward to compute the moments
ω 2 τc2
iων(y)
E[e
] = exp −
(4.6)
,
2
|y − y |
(ω − ω )2 τc2
iων(y)−iω ν(y )
2
− ωω τc 1 − ψ
] = exp −
.
E[e
2
xc
If, additionally, we assume that ω, ω ω0 with
(4.7)
ω0 τc 1,
then using a second order Taylor expansion of ψ and the assumption |y − y | xc ,
we arrive at
|y − y |2
(ω − ω )2 τc2
−
(4.8)
E[eiων(y)−iω ν(y ) ] exp −
2
2Xc2
with
(4.9)
Xc2 =
3x2c
.
ω02 τc2
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 p0 when it is smooth or of the order of the scale
of variations of the function p0 when it is oscillating. Since the speed of propagation is
one, the typical frequency ω0 is 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.
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2266
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
4.2. Kirchhoff–Radon migration. Recall that the Kirchhoff–Radon migration function is
1
(4.10)
dσ(y) dω q̂(y, ω)e−iω|y−x| =
dσ(y)q(y, |y − x|),
IKRM (x) =
2π ∂Ω
R
∂Ω
where q = BW[p]/(4π 2). The function applied to the perfect preprocessed data
q (0) =
(4.11)
is
(4.12)
(0)
IKRM (x)
1
=
2π
dσ(y)
∂Ω
dω q̂
(0)
1
BW[p(0) ]
4π 2
(y, ω)e
−iω|y−x|
=
R
dσ(y)q (0) (y, |y − x|),
∂Ω
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
1
ω 2 τc2
dσ(y) dω q̂ (0) (y, ω) exp −
E[IKRM (x)] =
e−iω|y−x|
2π ∂Ω
2
R
ω02 τc2
(0)
(4.14)
exp −
IKRM (x)
2
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
(0)
to the unperturbed function IKRM (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
E IKRM (x)IKRM (x )
1
=
dσ(y1 )dσ(y2 )
dω1 dω2 q̂ (0) (y1 , ω1 )q̂ (0) (y2 , ω2 )
(2π)2
∂Ω×∂Ω
R×R
|y1 − y2 |2
(ω1 − ω2 )2 τc2
−
e−iω1 |y1 −x| eiω2 |y2 −x | .
× exp −
2
2Xc2
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
2267
The variance that characterizes the amplitude of the fluctuations is
2
Var IKRM (x) := E |IKRM (x)|2 − E IKRM (x) .
We assume that Xc X0 , that is, the correlation radius is smaller than the propagation distance. Applying Proposition A.1, we find that
Xc
2
E |IKRM (x)| dσ(Ya ) dωa
dσ(κa ) dτa
(2π)3 τc ∂Ω
R
Ya⊥
R
⎞
⎛ 2
x−Y
a κa − ωa
2
−
|Y
−
x|)
(τ
|x−Ya |
a
a
⎠,
×Wq (Ya , ωa ; κa , τa ) exp ⎝−
−
2τc2
2Xc−2
where Wq is the Wigner transform of q (0) ,
Wq (Ya , ωa ; κa , τa )
ya
ha
ya
ha
, ωa +
, ωa −
=
dha
dσ(ya )q̂ (0) Ya +
q̂ (0) Ya −
(4.15)
2
2
2
2
R
Ya⊥
× eiκa ·ya −iha τa ,
and Ya⊥ is defined by
(4.16)
ya
ya
Ya⊥ = ya ∈ R2 , Ya −
∈ ∂Ω and Ya +
∈ ∂Ω .
2
2
If we consider the regime in which τc−1 ω0 and Xc X0 , then we get
Xc
2
E |IKRM (x)| dσ(Ya ) dωa
dσ(κa ) dτa Wq (Ya , ωa ; κa , τa )
(2π)3 τc ∂Ω
R
Ya⊥
R
Xc
dσ(Ya ) dωa |q̂ (0) (Ya , ωa )|2 ,
2πτc ∂Ω
R
and thus
(4.17)
E |IKRM (x)|2 Xc τc−1
dtq (0) (y, t)2 .
dσ(y)
R
∂Ω
Combining (4.17) together with (4.12) yields
(4.18)
(0)
E |IKRM (x)|2 IKRM (x)2
1
ω0 τc
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 ω0 τc 1 and Xc X0 , then SNRKRM is very small:
SNRKRM 1.
Here, A B iff A/B = o(1).
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2268
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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(iω2 ν(y2 ) − iω1 ν(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 function ICIR .
Using (4.8), we have
(4.20)
dω1 dω2
dσ(y1 )dσ(y2 )q̂ (0) (y1 , ω1 )q̂ (0) (y2 , ω2 )
R×R
∂Ω×∂Ω
1
(ω1 − ω2 )2
1
1
|y1 − y2 |2
−iω1 |y1 −x| iω2 |y2 −x|
×e
e
exp −
+
,
τc2 + 2 −
2
Ωd
2
Xd2
Xc2
E[ICIR (x)] =
1
(2π)2
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
(4.21)
E[ICIR (x)] = s
(2π)3 ( X12 +
1
Xc2 )
1
1
2
(τc2 +
1
)
Ω2d
1
2
dσ(Ya )
∂Ω
R
dωa
Ya⊥
dσ(κa )
⎞
⎛ κa − ωa x−Ya 2
2
−
|Y
−
x|)
(τ
|x−Ya |
a
a
⎠,
× Wq (Ya , ωa ; κa , τa ) exp ⎝−
−
2( X12 + X12 )
2(τc2 + Ω12 )
d
d
c
R
dτa
d
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–
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 point x 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
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
2269
⎧
⎡
4
⎨ τ2 iω1 ν(y1 )−iω2 ν(y2 )
|y1 − y4 |
c ⎣
4
iω
ν(y
)−iω
ν(y
)
3
3
4
4
E e
= exp −
e
ωj + 2ω1 ω4 ψ
⎩ 2
xc
j=1
|y2 − y3 |
|y1 − y2 |
|y1 − y3 |
+2ω2 ω3 ψ
− 2ω1 ω2 ψ
− 2ω1 ω3 ψ
xc
xc
xc
⎤⎫
|y2 − y4 |
|y3 − y4 | ⎦⎬
− 2ω2 ω4 ψ
− 2ω3 ω4 ψ
.
⎭
xc
xc
If
ω1 = ωa +
ha
,
2
ω2 = ωa −
ha
,
2
ω3 = ωb +
hb
,
2
ω4 = ωb −
hb
,
2
y1 = Y a +
ya
,
2
y2 = Y a −
ya
,
2
y3 = Y b +
yb
,
2
y4 = Y b −
yb
2
with ha , hb = O(Ωd ), ωa , ωb = O(ω0 ), Ya , Yb = O(X0 ), and ya , yb = O(Xd ), then
% 2
&
iω1 ν(y1 )−iω2 ν(y2 )
|Ya − Yb |
τc
2
2
iω
ν(y
)−iω
ν(y
)
3
3
4
4
exp −
e
E e
ha + hb − 2ha hb ψ
2
xc
(note that Xd xc since Xd ≤ Xc and Xc xc ) and therefore
E ICIR (x)ICIR (x )
1
=
dσ(Ya ) dha
dσ(ya )
dσ(Yb ) dhb
dσ(yb )
(2π)4 ∂Ω
R
Ya⊥
∂Ω
R
Yb⊥
× Q̂(Ya , ha , ya ; x)Q̂(Yb , hb , yb ; x )
(h2a + h2b ) 1
(|ya |2 + |yb |2 )
2
+ τc −
× exp −
2
Ω2d
2Xd2
|Ya − Yb |
× exp τc2 ha hb ψ
.
xc
Using (4.20) and the fact Ωd < τc−1 and Xd < Xc , we arrive at
E[ICIR (x)]
h2a
|ya |2
1
dσ(Y
)
dh
dσ(y
)
Q̂(Y
,
h
,
y
;
x)
exp
−
−
=
a
a
a
a
a
a
(2π)2 ∂Ω
2Ω2d
2Xd2
R
Ya⊥
so that
E 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 follows that if Ωd < τc−1 ω0 and Xd < Xc X0 , then we have
SNRCIR 1.
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2270
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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 Xd the 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
1
ICIR (x) =
dω1 dω2
(2π)2
R×R
(ω −ω )2
− 1 22
2Ω
d
(4.22)
×
dσ(y)e
q̂(y, ω1 )e−iω1 |y−x| q̂(y, ω2 )eiω2 |y−x| .
∂Ω
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
Ω 2 t2 2
Ωd
ICIR (x) = √
.
dσ(Ya ) dtq(Ya , |Ya − x| + t) exp − d
2
2π ∂Ω
R
Moreover, if Ωd is large, then (4.22) is (approximately) equivalent to the
incoherent matched field function
2
dσ(Ya )q(Ya , |Ya − x|) .
(4.23)
ICIR (x) ∂Ω
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) =
(4.24)
1
(2π)2
R×R
dω1 dω2
dσ(y1 )dσ(y2 )e
−
(ω1 −ω2 )2
2Ω2
d
∂Ω×∂Ω
×q̂(y1 , ω1 )e−iω1 |y1 −x| q̂(y2 , ω2 )eiω2 |y2 −x| .
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
2
Ω 2 t2 Ωd
ICIR (x) = √
dt
dσ(Ya )q(Ya , |Ya − x| + t) exp − d
.
2
2π R
∂Ω
If Ωd is large, then (4.22) is equivalent to the coherent matched field function
(or square KRM function):
2
ICIR (x) dσ(Ya )q(Ya , |Ya − x|) .
∂Ω
In contrast to (4.23), this function is called ”coherent” since we take into
account the phase of the data.
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2271
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
5. CINT–Radon algorithm in a bounded domain. When considering photoacoustics 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:
⎧ 2
∂ p
⎪
⎪
⎪ 2 (x, t) − c(x)2 Δp(x, t) = 0, x ∈ Ω,
⎨
∂t
∂p
(x, 0) = 0, x ∈ Ω,
p(x, 0) = p0 (x),
⎪
⎪
⎪
∂t
⎩
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 ∂n p(y, t) = n(y) · ∇p(y, t) on ∂Ω × [0, T ]:
R[p0 ](θ, s) = W[∂n p](θ, s),
where the line Radon transform is defined by
R[p0 ](θ, r) :=
p0 (rθ + sθ⊥ )ds,
R
and
W[g](θ, s) :=
T
dσ(x)g(x, t)H (x · θ + t − s) ,
dt
0
θ ∈ S1 ,
r ∈ R,
θ ∈ S1 ,
s ∈ R.
∂Ω
Here H denotes the Heaviside function. We then invert the Radon transform using
the filtered back-projection algorithm
p0 = R BW[∂n p],
where R is the formal adjoint Radon transform,
1
1
dσ(θ)f (θ, x · θ) =
dσ(θ)
dω fˆ(θ, ω)e−iωx·θ ,
R [f ](x) =
2π S1
(2π)2 S1
R
and B is a ramp filter
1
B[g](θ, s) =
4π
R
dω|ω|ĝ(θ, ω)e−iωs ,
x ∈ Ω,
θ ∈ S1 .
Here, the hat stands for the Fourier transform (3.1) in the second (shift) variable. In
the Fourier domain, the inversion reads
1
−iωx·θ
p0 (x) =
dσ(θ)
dω BW[∂
, x ∈ Ω.
n p](θ, ω)e
(2π)2 S1
R
Therefore, a natural idea to extend the CINT imaging to bounded media is to consider
the imaging function
(5.1)
(ω −ω )2
|θ −θ |2
− 2 21
− 2 21
1
2Ω
2Θ
d
d
dσ(θ
)dσ(θ
)
dω
dω
e
e
ICIR (x) :=
1
2
1
2
(2π)4
1
1
S ×S
R×R
−iω1 x·θ 1 ×BW[∂
BW[∂n p](θ 2 , ω2 )eiω2 x·θ2 ,
n p](θ 1 , ω1 )e
where Θd is an angular cutoff parameter and plays the same role as Xd in the previous
sections. Using the arguments developed before, it would be possible to perform a
stability and resolution analysis for ICIR given by (5.1).
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2272
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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
∂t P = AP + BP,
where
P=
p
,
∂t p
A=
0 1
,
Δ 0
and
B=
0 0
.
(c2 − 1 Δ 0
This equation is solved on the box Q = (−2, 2)2 containing Ω. We use a splitting spectral 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, only ICI and IKM are
used. The imaging functions ICIR and IKMR do not improve image reconstruction.
In Figure 6.1, we consider p0 to 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.
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2273
COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
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f (t) = cos(2πω0 t)tδω exp(−πt2 δω2 ) with
δω = 10,
ω0 = 3δω .
The six Dirac masses emit therefore pulses of the form f with f being an approximation 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
1.05
1.06
1.04
1.03
1.04
1.02
1.02
1.01
1
1
0.99
0.98
0.98
0.97
0.96
0.96
0.95
0.94
Fig. 6.2. Left: random velocity c1 with high frequencies; right: random velocity c2 with low
frequencies.
50
50
100
100
150
150
200
200
250
250
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Fig. 6.3. Test 1 (a): measured data p(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
50
100
100
150
150
200
200
250
250
300
300
350
350
400
400
450
450
500
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 migration IKM with (right) and without
(left) clutter noise.
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2274
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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, IKM becomes very instable and
fails to really localize the targets. The images obtained by ICI are plotted in Figure
6.5 and compared to those obtained by IKM . Note that ICI presents better stability
properties when Xd and Ωd become small as predicted by the theory.
We now consider the case of extended targets and test the imaging function ICIR .
We use the random velocity c2 shown in Figure 6.2. Reconstructions of the initial
pressure obtained by IKRM are 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
50
50
100
100
100
150
150
150
200
200
200
250
250
250
50
100
150
200
250
50
100
150
200
250
50
50
50
100
100
100
150
150
150
200
200
200
250
250
50
100
150
200
100
150
200
250
50
50
100
100
100
150
150
150
200
200
200
250
100
150
200
150
200
250
50
100
150
200
250
50
100
150
200
250
250
250
50
100
250
50
250
50
50
250
50
100
150
200
250
Fig. 6.5. Test 1 (c): source localization using the standard CINT function ICI with parameters
Xd and Ωd given by Xd = 0.25, 0.5, 1 (from left to right) and Ωd = 25, 50, 100 (from top to bottom).
20
20
40
40
60
60
80
80
100
100
120
120
50
100
150
200
250
50
100
150
200
250
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.
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
20
20
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100
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120
2275
120
20
40
60
80
100
120
20
40
60
80
100
120
Fig. 6.7. Test 2 (b): reconstruction of p0 (x) using IKRM with (right) and without (left) clutter
noise.
20
20
40
40
60
60
80
80
100
100
120
120
50
100
150
200
250
50
100
150
200
250
Fig. 6.8. Test 3 (a): measured data p(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
20
40
40
60
60
80
80
100
100
120
120
20
40
60
80
100
120
20
40
60
80
100
120
Fig. 6.9. Test 3 (b): reconstruction of p0 using IKRM 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 that ICIR 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 reconstruction using the inverse Radon transform algorithm in the case with boundary
conditions. We notice that the outer interface appears twice.
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2276
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
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20
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40
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120
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120
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120
120
40
120
20
120
20
20
40
60
80
100
120
Fig. 6.10. Test 2 (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
20
20
40
40
40
60
60
60
80
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120
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120
20
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120
20
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60
80
100
120
Fig. 6.11. Test 3 (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).
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,
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2277
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
Fig. 6.12. Reconstruction of an extended target using line Radon transform in the case of
imposed boundary conditions.
0
0
0
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1
0
0.2
0.4
0.6
0.8
1
1
0
0.2
0.4
0.6
0.8
1
1
0
0
0
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1
0
0.2
0.4
0.6
0.8
1
1
0
0.2
0.4
0.6
0.8
1
1
0
0
0
0.2
0.2
0.2
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0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
1
0
0.2
0.4
0.6
0.8
1
1
0
0.2
0.4
0.6
0.8
1
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 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.
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.
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2278
H. AMMARI, E. BRETIN, J. GARNIER, AND V. JUGNON
7. Conclusion. In this paper we have introduced new CINT–Radon type imaging functions in order to correct the effect on photoacoustic images of random fluctuations 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 migration, 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
1
I(x) =
dω1 dω2
dσ(y1 )dσ(y2 )q̂ (0) (y1 , ω1 )q̂ (0) (y2 , ω2 )
(2π)2
R×R
∂Ω×∂Ω
(ω1 − ω2 )2
|y1 − y2 |2
−iω1 |y1 −x| iω2 |y2 −x|
(A.1)
.
×e
e
exp −
−
2Ω2d
2Xd2
If Xd X0 , then we have
X d Ωd
dσ(Y
)
dω
dσ(κ
)
dτa
I(x) a
a
a
(2π)3 ∂Ω
R
Ya⊥
R
⎞
⎛
x−Ya 2
2
2
Xd2 κa − ωa |x−Y
Ω
(τ
−
|Y
−
x|)
a
a
a|
⎠,
(A.2) ×Wq (Ya , ωa ; κa , τa ) exp ⎝−
− d
2
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). For y1 , y2 ∈ ∂Ω such that |y1 − y2 | < Xd , let Ya ∈ ∂Ω be the point on ∂Ω
equidistant from y1 and y2 and satisfying |Ya − y1 | < Xd . Let Ya + y2a (respectively,
Ya − y2a ) be the orthogonal projection of y1 (respectively, y2 ) on the tangent line to
∂Ω at Ya . Using the change of variables
ω1 = ωa +
ha
,
2
ω2 = ωa −
ha
,
2
y1 = Y a +
ya
,
2
y2 = Y a −
ya
,
2
the function I(x) can be approximated as
1
I(x) dσ(Y
)
dh
dσ(ya )Q̂(Ya , ha , ya ; x)
a
a
(2π)2 ∂Ω
R
Ya⊥
h2
|ya |2
× exp − a2 −
2Ωd
2Xd2
with
Q̂(Ya , ha , ya ; x) =
R
dωa q̂
(0)
×e−i(ωa +
ya
ha
ya
ha
(0)
, ωa +
, ωa −
q̂
Ya +
Ya −
2
2
2
2
ha
2
)|Ya + y2a −x| i(ωa − h2a )|Ya − y2a −x|
e
.
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COHERENT INTERFEROMETRY ALGORITHMS FOR PHOTOACOUSTICS
2279
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Since y1 − y2 is approximately orthogonal to (y1 + y2 )/2 when y1 , y2 ∈ ∂Ω and
|y1 − y2 | ≤ Xd ,
Ya⊥ {ya ∈ R2 , Ya · ya = 0}
and
ya
ha
ya
ha
, ωa +
, ωa −
dωa q̂ (0) Ya +
q̂ (0) Ya −
2
2
2
2
R
Q̂(Ya , ha , ya ; x) x−Ya
·ya −iha |x−Ya |
a|
×eiωa |x−Y
1
dωa dτa
dσ(κa )Wq (Ya , ωa ; κa , τa )
(2π)2 R
R
Ya⊥
x−Ya
×ei(ωa |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
X d Ωd
I(x) dσ(Y
)
dω
dσ(κ
)
dτa
a
a
a
(2π)3 ∂Ω
R
Ya⊥
R
×Wq (Ya , ωa ; κa , τa )
⎛
x−Ya
Ya
− ( |Y
·
Xd2 κa − ωa |x−Y
a|
a|
⎝
× exp −
2
x−Ya
Ya 2
|x−Ya | ) |Ya | )
⎞
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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