Full Field Inversion in Photoacoustic Tomography with Variable Sound Speed

Abstract

To accelerate photoacoustic data acquisition, in [R. Nuster, G. Zangerl, M. Haltmeier, G. Paltauf (2010). Full field detection in photoacoustic tomography. Optics express, 18(6), 6288–6299] a novel measurement and reconstruction approach has been proposed, where the measured data consist of projections of the full 3D acoustic pressure distribution at a certain time instant T. Existing reconstruction algorithms for this kind of setup assume a constant speed of sound. This assumption is not always met in practice and thus can lead to erroneous reconstructions. In this paper, we present a two-step reconstruction method for full field detection photoacoustic tomography that takes variable speed of sound into account. In the first step, by applying the inverse Radon transform, the pressure distribution at the measurement time is reconstructed point-wise from the projection data. In the second step, a final time wave inversion problem is solved where the initial pressure distribution is recovered from the known pressure distribution at time T. We derive an iterative solution approach for the final time wave inversion problem and compute the required adjoint operator. Moreover, as the main result of this paper, we derive its uniqueness and stability. Our numerical results demonstrate that the proposed reconstruction scheme is fast and stable, and that ignoring sound speed variations significantly degrades the reconstruction.

1. Introduction

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high spatial resolution of ultrasound and high contrast of optical tomography [1,2,3,4,5]. In PAT, a semitransparent sample is illuminated by a short laser pulse. As a result, parts of the optical energy are absorbed inside the sample. This causes an initial pressure distribution and a subsequent acoustic pressure wave. The pressure wave is detected outside the investigated object and used to recover an image of the interior. In standard PAT, the induced pressure is measured on a detection surface as a function of time. In the case of constant sound speed and when the observation surface exhibits a special geometry (planar, cylindrical, spherical), the initial pressure distribution can be recovered by closed-form inversion formulas; see [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] and references therein. While some algorithms take the size and other properties of the detectors into account [21,22,23,24], most algorithms assume that the acoustic pressure is known point-wise on a detection surface for all times in the measurement interval. Due to the finite width of the commonly used piezoelectric elements this assumption is only approximately satisfied. One approach to address this issue uses the concept of integrating detectors [25]. Integrating detectors measure integrals of the acoustic pressure over planes, lines or circles and several inversion methods have been derived in the literature [26,27,28,29].

Inspired by the concept of integrating line detectors, a full field detection and reconstruction method that uses projections of the full acoustic field surrounding the sample has been developed in [30,31]. The pressure field within the depth of field is imaged by the use of an optical phase contrast imaging system with a CCD camera. In this way, one obtains 2D projections of the pressure field at a time instant T. Reconstructing 2D projections of the initial pressure has been investigated in [32,33]. In [30,31] we go one step further and show that projection data from different directions allow for a full 3D reconstruction of the initial pressure by Radon or Fourier transform techniques. We refer to this approach as full field detection photoacoustic tomography (FFD-PAT). A photograph of the FFD-PAT setup developed in Graz is shown in Figure 1. For practical aspects and a detailed description of the working principle of the FFD-PAT detection system we refer to the original works [30,31]. Existing image reconstruction methods for FFD-PAT data assume a constant speed of sound. However, there are relevant cases when the assumption of constant speed of sound is inaccurate [34,35]. For example, it is known that acoustic properties vary within female human breasts. Consequently, for accurate image reconstruction, variable speed of sound has to be incorporated in the wave propagation model. Iterative methods are capable to deal with this assumption. In the case of standard PAT, such methods have been studied in [36,37,38,39,40,41] for spatially variable speed of sound, that is smooth and bounded from below. Moreover, it is assumed to satisfy the so-called non-trapping condition, which means that the supremum of the lengths of all geodesics connecting any two points inside the volume enclosed by the measurement surface S is finite. Under these assumptions, it is known that the initial pressure can be stably reconstructed from pressure data given on $S\times [0,T]$.

In this paper, for the first time, we study image reconstruction in FFD-PAT with a spatially variable speed of sound. We will give a precise mathematical formulation of FFD-PAT and describe the inverse problem we are dealing with (see Section 2). For its solution, we propose a two-step process. In the first step, the acoustic pressure at time T is reconstructed pointwise from the FFD data. In the second step, we recover the desired initial pressure distribution from the known valued of the pressure at time T. The first step can be approximated by inverting the well-known Radon transform. The second step consists in a final time wave inversion problem with spatially varying speed of sound. To the best of our knowledge, the latter has not been addressed in the literature so far. We develop iterative reconstruction methods based on an explicit computation of an adjoint problem. As main theoretical results of this paper, we establish uniqueness and stability of the final time wave inversion problem (see Section 3). In particular, this implies linear convergence for the proposed iterative reconstruction methods. In Section 4, we present numerical results demonstrating that the proposed algorithm is efficient and stable. Moreover, the presented numerical results clearly highlight the importance of taking sound speed variations into account in FFD-PAT image reconstruction.

2. Full Field Detection Photoacoustic Tomography

In this section, we describe a mathematical model for FFD-PAT including the variable sound speed case, and state the inverse problem under consideration. Additionally, we outline the proposed two-step reconstruction procedure and formulate the final time inverse problem.

2.1. Mathematical Model

In the case of variable sound speed, acoustic wave propagation in PAT is commonly described by the initial value problem [34,38,39,41]

$$\begin{array}{ccc}\hfill p_{tt}(\mathbf{x},t)-c^2\left(\mathbf{x}\right)\Delta p(\mathbf{x},t)& =0,\hfill & (\mathbf{x},t)\in {\mathbb{R}}^3\times {\mathbb{R}}_{>0}\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill p(\mathbf{x},0)& =f\left(\mathbf{x}\right),\hfill & \mathbf{x}\in {\mathbb{R}}^3\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill p_t(\mathbf{x},0)& =0,\hfill & \mathbf{x}\in {\mathbb{R}}^3.\hfill \end{array}$$

(3)

here $c\left(\mathbf{x}\right)>0$ is the sound speed at location $\mathbf{x}\in {\mathbb{R}}^3$ and $f\in C_0^{\infty }\left({\mathbb{R}}^3\right)$ is the initial pressure distribution that encodes the inner structure of the object. It is assumed that the object is contained inside the ball $B(0,R)=\left\{\mathbf{x}\in {\mathbb{R}}^3\mid ∥\mathbf{x}∥<R\right\}$, of radius R centered at the origin, and that the sound speed is smooth, positive and has the constant value $c_0$ outside $B(0,R)$. We denote by $C_0^{\infty }\left(B(0,R)\right)$ the set of all smooth functions $f:{\mathbb{R}}^3\to \mathbb{R}$ that vanish outside $B(0,R)$.

In FFD-PAT, linear projections (integrals along straight lines) of the 3D pressure field $p(·,T)$ for a fixed time $T>0$ are recorded. As illustrated in Figure 1, this can be implemented using a special phase contrast method and a CCD-camera that records full field projections of the pressure field [30,31]. The linear projections are collected for rotation angles $\theta \in [0,\pi ]$ around the $e_3=(0,0,1)$ axis and are given by

$$\begin{array}{c}\hfill g_R(\theta ,s,z)={\int }_{\mathbb{R}}p(scos\left(\theta \right)-tsin\left(\theta \right),ssin\left(\theta \right)+tcos\left(\theta \right),z,T)dt,\end{array}$$

(4)

for $(\theta ,s,z)\in M_R\left\{(\theta ,s,z)\in [0,\pi ]\times {\mathbb{R}}^2\mid s^2+z^2\ge R^2\right\}$. Here $M_R$ determines the set of admissible projections and the defining condition $s^2+z^2\ge R^2$ means that in practice only pressure integrals over those lines are recorded, which do not intersect the possible support $B(0,R)$ of the imaged object; compare Figure 2.

2.2. Description of the Inverse Problem

In order to describe the inverse problem of FFD-PAT in a compact way, we introduce some further notation. First, we define the operator

$${\mathbf{W}}_T:C_0^{\infty }\left(B(0,R)\right)\to C_0^{\infty }\left({\mathbb{R}}^3\right):f\mapsto p(·,T),$$

(5)

where p denotes the solution of (1)–(3). The operator ${\mathbf{W}}_T$ maps the initial data f to the solution (full field) of the wave Equations (1)–(3) at the given measurement time $T>0$. Second, we define the X-ray transform

$$\begin{array}{c}\mathbf{X}:C_0^{\infty }\left({\mathbb{R}}^3\right)\to C_0^{\infty }\left([0,\pi ]\times {\mathbb{R}}^2\right)\hfill \\ \left(\mathbf{X}h\right)(\theta ,s,z){\int }_{\mathbb{R}}h(scos\left(\theta \right)-tsin\left(\theta \right),ssin\left(\theta \right)+tcos\left(\theta \right),z)dt\hfill \end{array}$$

(6)

Note that for any fixed $z\in \mathbb{R}$, the function $\left(\mathbf{X}h\right)(·,z)$ is the Radon transform of $h(·,z)$ in the horizontal plane ${\mathbb{R}}^2\times \left\{z\right\}$. Finally, we define the restricted X-ray transform

$${\mathbf{X}}_R:C_0^{\infty }\left({\mathbb{R}}^3\right)\to C^{\infty }\left(M_R\right){:h\mapsto \left(\mathbf{X}h\right)|}_{M_R}.$$

(7)

For $\left|z\right|\le R$, $\left({\mathbf{X}}_{R}h\right)(·,z)$ is the exterior Radon transform of $h(·,z)$ and consist of integrals over lines not intersecting the disc $\left\{(x,y)\in {\mathbb{R}}^2\mid x^2+y^2<R^2-z^2\right\}$; compare Figure 2. Otherwise, $\left({\mathbf{X}}_{R}h\right)(·,z)$ coincides with the standard Radon transform of $h(·,z)$.

Using the operator notation introduced above we can write the inverse problem of FFD-PAT in the form

$$\begin{array}{c}\hfill \mathrm{Recover}f\mathrm{from}\mathrm{data}{g}_R={\mathbf{X}}_R{\mathbf{W}}_{T}f+\mathrm{noise}.\end{array}$$

(8)

Evaluation of ${\mathbf{X}}_R{\mathbf{W}}_{T}f$ is referred to as the forward problem in FFD-PAT. In this paper, we study the solution of the inverse problem (8).

2.3. Two-Step Reconstruction

One possible approach to solve the inverse problem of FFD-PAT is to directly recover f from data in (8) via iterative methods. Typically, each iteration step requires the evaluation of the forward operator ${\mathbf{X}}_R{\mathbf{W}}_T$ and its adjoint ${\left({\mathbf{X}}_R{\mathbf{W}}_T\right)}^{*}={\mathbf{W}}_T^{*}{\mathbf{X}}_R^{*}$. In this paper, we consider a two-step approach where we first invert ${\mathbf{X}}_R$ via a direct method and then use an iterative method to invert ${\mathbf{W}}_T$. This avoids repeated and time consuming evaluation of ${\mathbf{X}}_R$ and its adjoint. The proposed reconstruction method consists of the following two steps.

■

Inverse Radon transform: Assume that projection data $g_R={\mathbf{X}}_R{\mathbf{W}}_{T}f$ are given and consider the zero extension $g:[0,\pi ]\times {\mathbb{R}}^2\to \mathbb{R}$ by $g(\theta ,x,z)=g_R(\theta ,x,z)$ for $(\theta ,x,z)\in M_R$ and $g(\theta ,x,z)=0$ otherwise. We then define an approximation to ${\mathbf{W}}_{T}f$ by applying an inversion formula of the Radon transform in planes ${\mathbb{R}}^2\times \left\{z\right\}$. Using the well-known filtered backprojection inversion formula for the Radon transform [42] yields

$${\mathbf{W}}_{T}f(x,y,z)\simeq {\mathbf{X}}^{♯}g(x,y,z)\frac{1}{2{\pi }^2}{\int }_0^{\pi }\mathrm{P}.\mathrm{V}.{\int }_{\mathbb{R}}\frac{\left({\partial }_{s}g\right)(\theta ,s,z)ds}{(xcos(\theta )+ysin(\theta \left)\right)-s}d\theta ,$$

where $\mathrm{P}.\mathrm{V}.$ denotes the principal value integral.

■

Final time wave inversion: For the second step, assume that an approximation $h\simeq {\mathbf{W}}_{T}f$ to the 3D acoustic field at time T is given. Recovering the initial pressure then yields the final time wave inversion problem

$$\mathrm{Recover}f\mathrm{from}\mathrm{data}h={\mathbf{W}}_{T}f+\mathrm{noise}.$$

(9)

To the best of our knowledge, the final time wave inversion problem (9) has not been considered so far and its investigation will be the main theoretical focus of this work.

For solving the wave inversion problem (9), we propose iterative solution methods that are described in Section 3. As main theoretical results, we derive its uniqueness and stability.

Another possible approach for solving the first step would be to work with the exterior Radon transform [43,44,45]. Instead, we invert the standard Radon transform after setting the missing values of $\mathbf{X}{\mathbf{W}}_{T}f$ to zero. We observe numerically that the missing values are indeed close to zero for large enough measurement time T. Although we do not have a rigorous mathematical proof for this claim, it is supported by at least two facts. First, in the case of a non-trapping sound speed, the known decay estimate [46] for the solution p of (1)–(3) with initial data f supported in $B(0,R)$ states

$$\left|\frac{{\partial }^{k+\left|m\right|}p(\mathbf{x},t)}{{\partial }_t^k{\partial }_{\mathbf{x}}^{\left|m\right|}}\right|\le \gamma e^{-\beta t}{∥f∥}_2\mathrm{for}(\mathbf{x},t)\in B(0,R)\times (T,\infty )$$

(10)

for all $(k,m)\in {\mathbb{N}}^2$. Here $\beta >0$ is a constant only depending on $c\left(\mathbf{x}\right)$ and T, and $\gamma$ is a constant depending on $B(0,R)$. Second, in the case of constant sound speed, the X-ray transform reduces (1)–(3) to a 2D wave equation with initial data $\mathbf{X}f$ supported in a disc of radius R. Thus, in the constant sound speed case, $\mathbf{X}\left(p\right(·,t\left)\right)$ describes 2D propagation which rapidly decays inside the disc. Formulating and proving precise statements in such a direction, however, is an interesting unsolved problem.

3. Final Time Wave Inversion Problem for Variable Sound Speed

In this section we study the final time wave inversion problem (9), where the forward operator ${\mathbf{W}}_T:f\mapsto p(·,T)$ is defined by (5). According to standard results for the wave equation [47], the forward operator extends to a bounded linear operator ${\mathbf{W}}_T:L^2\left(B(0,R)\right)\to L^2\left({\mathbb{R}}^3\right)$. Below we establish uniqueness and stability results and derive an iterative reconstruction algorithm. Note that for constant sound speed, recovering the function f from the solution at time T of (1) with initial data $(0,f)$ instead of $(f,0)$, is equivalent to the inversion from spherical means with fixed radius. Uniqueness and an inversion method for this problem has been obtained in the classical book of Fritz John [48]. However, neither for the case of initial data $(f,0)$ nor in the variable sound speed case we are aware of any similar results.

3.1. Uniqueness and Stability Theorem

The following theorem is the main theoretical result of this paper and states that the final time wave inversion problem (9) has a unique solution that stably depends on the right-hand side.

Theorem 1 (Uniqueness and stability of inverting W_T).

For a non-trapping speed of sound, the operator ${\mathbf{W}}_T:L^2\left(B(0,R)\right)\to L^2\left({\mathbb{R}}^3\right)$ is injective and bounded from below.

The proof of Theorem 1 is presented in the following Section 3.2. See [19,41,49] for related results and methods in the case of standard PAT data. Theorem 1 states that

$$binf\left\{\frac{{∥{\mathbf{W}}_{T}f∥}_{L^2\left({\mathbb{R}}^3\right)}}{{∥f∥}_{L^2\left(B(0,R)\right)}}|f\in L^2\left(B(0,R)\right)\right\}>0.$$

(11)

In particular, ${\mathbf{W}}_T:L^2\left(B(0,R)\right)\to Ran\left({\mathbf{W}}_T\right)$ has a bounded inverse, where $Ran\left({\mathbf{W}}_T\right)$ denotes the range of ${\mathbf{W}}_T$. The latter implies that gradient based iterative methods for (9) converge linearly, similar to the case of standard PAT data; see [38].

3.2. Proof of Theorem 1

Let c be a non-trapping sound speed. We first prove two crucial Lemmas, from which Theorem 1 follows rather quickly.

Lemma 1.

Assume that ${\mathbf{W}}_{T}f=0$, then $f=0$. That is, ${\mathbf{W}}_T$ is injective.

Proof. 

Let p denote the solution of (1)–(3). We will construct a solution $\overline{p}$ of the wave equation which is periodic in time with period $4T$ such that $\overline{p}=p$ on ${\mathbb{R}}^3\times [0,T]$. Once this is done, we obtain $f=\overline{p}(·,0)=\overline{p}(·,4^{n}T)$ for any n. Using (10), we arrive at

$$\forall \mathbf{x}\in {\mathbb{R}}^3:f\left(\mathbf{x}\right)=\underset{n\to \infty }{lim}\overline{p}(\mathbf{x},4^{n}T)=0.$$

It remains to construct the above-mentioned solution $\overline{p}$ of the wave equation. The idea is to properly reflect the solution p in the time variable t through the time moments $t=T,2T,\cdots ,$ as follows. We first construct $\overline{p}$ on $[0,2T]$ by the odd reflection of p through the moment $t=T$: $\overline{p}(·,T)=p(·,T)$ for $t\in [0,T]$ and $\overline{p}(·,T)=-p(·,2T-t)$ for all $t\in [T,2T]$. Since $p(·,T)=0$ on ${\mathbb{R}}^3$, we obtain that $\overline{p}$ and ${\overline{p}}_t$ are continuous at $t=T$. Therefore, p is continuous on $[0,2T]$ and solves the wave equation on that interval. Next note that ${\overline{p}}_t(·,2T)=-{\overline{p}}_t(·,0)=0$ on ${\mathbb{R}}^3$. By the even reflection through $t=2T$: $\overline{p}(·,T)=\overline{p}(·,4T-t)$ for all $t\in [2T,4T]$, we obtain that $\overline{p}$ is a solution of the wave equation in $[0,4T]$. Finally, we extend the solution by periodicity with period $4T$. Noting that $\overline{p}(·,0)=\overline{p}(·,4T)$ and ${\overline{p}}_t(·,0)={\overline{p}}_t(·,4T)=0$, we obtain that $\overline{p}$ and ${\overline{p}}_t$ are continuous for all time and $\overline{p}$ satisfies the wave equation in ${\mathbb{R}}^3\times {\mathbb{R}}_{+}$. This finishes our proof. □

Lemma 2.

There is a constant C and a pseudo-differential operator ${\mathbf{K}}_T$ of order at most $-1$ such that

$$\forall f\in L^2\left(B(0,R)\right):{\parallel f\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le 2\left(\parallel {\mathbf{W}}_T{f\parallel }_{L^2\left({\mathbb{R}}^3\right)}+{\parallel {\mathbf{K}}_{T}f\parallel }_{L^2\left(B(0,R)\right)}\right).$$

Proof. 

Let p denote the solution of wave Equations (1)–(3) and recall the parametrix formula $p(\mathbf{y},t)=\frac{1}{{\left(2\pi \right)}^3}{\sum }_{\sigma =\pm }{\int }_{{\mathbb{R}}^3}{a}_{\sigma }(\mathbf{y},t,\xi )e^{i{\varphi }_{\pm }(\mathbf{y},T,\xi )}\widehat{f}\left(\xi \right)d\xi ={\sum }_{\sigma =\pm }{p}_{\sigma }(\mathbf{y},t)$; see [47]. Here, the phase function ${\varphi }_{\pm }$ solves the eikonal equation ${\partial }_t{\varphi }_{\pm }(\mathbf{y},t,\xi )\pm c\left(\mathbf{y}\right)\left|{\nabla }_{\mathbf{y}}{\varphi }_{\pm }(\mathbf{y},t,\xi )\right|=0$ for $(\mathbf{y},t)\in {\mathbb{R}}^3\times {\mathbb{R}}_{+}$ with the initial condition ${\varphi }_{\pm }(\mathbf{x},0,\xi )=\mathbf{x}·\xi$. The amplitude function is a classical symbol $a_{\pm }(\mathbf{y},t,\xi )={\sum }_{k=0}^{\infty }{a}_{-k,\pm }(\mathbf{y},t,\xi )$, where $a_{-k}$ is homogeneous of order $-k$ in $\xi$. Its leading term $a_{0,\pm }$ satisfies the transport equation

$$\left({\partial }_t{\varphi }_{\pm }(\mathbf{y},t,\xi ){\partial }_t-c^2\left(\mathbf{y}\right){\nabla }_{\mathbf{y}}{\varphi }_{\pm }(\mathbf{y},t,\xi )·{\nabla }_{\mathbf{y}}+C_{\pm }(\mathbf{y},t,\xi )\right)a_{0,\pm }(\mathbf{y},t,\xi )=0,$$

(12)

with the initial condition $a_{\pm ,0}(\mathbf{x},0,\xi )=1/2$, see [41]. Here, $C(\mathbf{y},\xi ,t)$ only depends on the sound speed c and the phase function ${\varphi }_{\pm }$. Let us denote by ${\gamma }_{\mathbf{x},\xi }$ the unit speed geodesics originating at $\mathbf{x}$ along the direction $\xi$. Then, ${\gamma }_{\mathbf{x},\xi }$ is a characteristics curve of the above transport equation; that is, (12) reduces to a homogeneous ODE on each geodesic curve.

We then write

$$\begin{array}{ccc}\hfill {\mathbf{W}}_T\left(f\right)\left(\mathbf{y}\right)& =& \frac{1}{{\left(2\pi \right)}^3}\sum _{\sigma =\pm }{\int }_{{\mathbb{R}}^3}{a}_{\sigma }(\mathbf{y},T,\xi )e^{i{\varphi }_{\pm }(\mathbf{y},T,\xi )}\widehat{f}\left(\xi \right)d\xi =\sum _{\sigma =\pm }{\mathbf{W}}_{\sigma }\left(f\right)\left(\mathbf{y}\right).\hfill \end{array}$$

Each operator ${\mathbf{W}}_{\pm }$ is a Fourier integral operator (FIO) with the canonical relation given by the pairs $({\mathbf{y}}_{\pm },\lambda {\eta }_{\pm };\mathbf{x},\lambda \xi )$ for any $\lambda \in \mathbb{R}$, $\xi ,\eta$ unit vectors, ${\mathbf{y}}_{\pm }={\gamma }_{\mathbf{x},\xi }(\pm T)$, and ${\eta }_{\pm }={\dot{\gamma }}_{\mathbf{x},\xi }(\pm T)$. Let ${\mathbb{R}}^3$ be equipped with the metrics $c^{-2}\left(\mathbf{x}\right)d{\mathbf{x}}^2$. Then, $({\mathbf{y}}_{\pm },{\eta }_{\pm })$ is obtained by translating $(\mathbf{x},\xi )$ on the geodesic ${\gamma }_{\mathbf{x},\pm \xi }$ by the distance T. From the initial condition of ${\varphi }_{\pm }$ and $a_{0,\pm }$ we see that, up to lower order terms,

$$p_{-}(\mathbf{x},0)=p_{+}(\mathbf{x},0)=\frac{1}{2}f\left(\mathbf{x}\right).$$

(13)

Heuristically, under Equations (1)–(3), each singularity of f at $(\mathbf{x},\xi )$ is broken into two equal parts. They propagate along the geodesic ${\gamma }_{\mathbf{x},\xi }$ in the opposite directions $\pm \xi$ to generate a singularity of ${\mathbf{W}}_T\left(f\right)$ at $({\mathbf{y}}_{\pm },{\eta }_{\pm })$.

From the standard theory of FIOs (see [50]), the adjoint ${\mathbf{W}}_{\pm }^{*}$ translates $({\mathbf{y}}_{\pm },{\eta }_{\pm })$ back to $(\mathbf{x},\xi )$ and ${\mathbf{W}}_{\pm }^{*}{\mathbf{W}}_{\pm }$ is a pseudo differential operator. On the other hand, ${\mathbf{W}}_{\mp }^{*}{\mathbf{W}}_{\pm }$ is a FIO whose canonical relation consists of the pairs $(\mathbf{y},\eta ;\mathbf{x},\xi )$ given by $\mathbf{y}={\gamma }_{\mathbf{x},\xi }(\pm 2T)$, and $\eta ={\dot{\gamma }}_{\mathbf{x},\xi }(\pm 2T)$. That is, ${\mathbf{W}}_{\pm }^{*}{\mathbf{W}}_{\mp }$ is an infinitely smoothing operator on B. Therefore, microlocally, we can write ${\mathbf{W}}_T^{*}{\mathbf{W}}_{T}f={\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}\left(f\right)+{\mathbf{W}}_{-}^{*}{\mathbf{W}}_{-}\left(f\right).$ We will show that the principal symbol ${\theta }_{\pm }(\mathbf{x},\xi )$ of ${\mathbf{W}}_{\pm }^{*}{\mathbf{W}}_{\pm }$ satisfies ${\theta }_{\pm }(\mathbf{x},\xi )=1/4$. This result can be intuitively understood as follows. Let us consider ${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}$ and a singularity of f at $(\mathbf{x},\xi )$. Under Equations (1)–(3), half of this singularity propagates into the direction $\xi$ (corresponding to the function $p_{+}$). At the moment $t=T$, it is transformed to a singularity of ${\mathbf{W}}_{+}\left(f\right)=p_{+}\left(T\right)$ at $({\mathbf{y}}_{+},{\eta }_{+})$. Under the adjoint Equation (15), half of this singularity propagates back to $(\mathbf{x},\xi )$ at $t=0$ to generate a singularity of ${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}\left(f\right)$. It is natural to believe that this recovered singularity is $1/4$ of the original singularity of f (due to twice splitting, as described). The proof below verifies this intuition.

Indeed, denote by $q_{+}$ the solution of the time-reversed wave equation, e.g., Equation (15), with the initial condition given by $g_{+}={\mathbf{W}}_{+}\left(f\right)$. Then, by definition (see Theorem 2) ${\mathbf{W}}_T^{*}{g}_{+}=q_{+}{(·,0)|}_B$. The solution $q_{+}$ can be decomposed into the sum $q_{+}=q_0+q_1$. Here, $q_0,q_1$, up to smooth terms, are solutions of the wave equations in ${\mathbb{R}}^3\times (0,T)$ and satisfy $q_0(·,0)={\mathbf{W}}_{+}^{*}\left(g_{+}\right)$, $q_1(·,0)={\mathbf{W}}_{-}^{*}\left(g_{+}\right)$. We are only concerned with $q_0$ since it defines ${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}f=q_0(·,0)$. We can write

$$q_0(\mathbf{y},t)=\frac{1}{{\left(2\pi \right)}^3}{\int }_{{\mathbb{R}}^3}b(\mathbf{y},t,\xi )e^{i{\varphi }_{+}(\mathbf{y},t,\xi )}\widehat{f}\left(\xi \right)d\xi .$$

(14)

Let $b_0$ be the principal part of b. Then, the principal symbol ${\theta }_{+}$ of ${\mathbf{W}}_{+}^{*}{\mathbf{W}}_{+}$ is given by ${\theta }_{+}(\mathbf{x},\xi )=b_0(\mathbf{x},0,\xi )$. We note that $b_0$ satisfies the same equation as $a_{0,+}$ (see (12)). Therefore, on each bicharacteristic curve the ratio $b_0/a_{0,+}$ is constant which implies $b_0(\mathbf{x},0,\xi )=a_{+,0}(\mathbf{x},0,\xi )b_0({\mathbf{y}}_{+},T,{\eta }_{+})/a_{+,0}({\mathbf{y}}_{+},T,{\eta }_{+})$. Similar to the argument below Equation (13), up to lower order terms, we have

$$q_0({\mathbf{y}}_{+},T)=\frac{1}{2}{g}_{+}({\mathbf{y}}_{+},T)=\frac{1}{{\left(2\pi \right)}^3}{\int }_{{\mathbb{R}}^3}\frac{1}{2}{a}_{+}({\mathbf{y}}_{+},T,\xi )e^{i{\varphi }_{+}({\mathbf{y}}_{+},T,\xi )}\widehat{f}\left(\xi \right)d\xi .$$

This and Equation (14) implies that $b_0({\mathbf{y}}_{+},T,\xi )=a_{+,0}({\mathbf{y}}_{+},T,\xi )/2$. Therefore, we obtain $b_0(\mathbf{x},0,\xi )=a_{+,0}(\mathbf{x},0,\xi )/2=1/4$. Combining with a similar argument for ${\mathbf{W}}_{-}^{*}{\mathbf{W}}_{-}$, we obtain that the principal symbol of ${\mathbf{W}}_T^{*}{\mathbf{W}}_T$ is $\theta (\mathbf{x},\xi )=1/2$. That is, ${\mathbf{W}}_T^{*}{\mathbf{W}}_T=\mathbf{I}/2+{\mathbf{K}}_T$, where ${\mathbf{K}}_T$ is a pseudodifferential operator of order at most $-1$ and $\mathbf{I}$ is the identity. We have $({\mathbf{W}}_{T}f,{\mathbf{W}}_{T}f)=({\mathbf{W}}_T^{*}{\mathbf{W}}_{T}f,f)=(f,f)/2+({\mathbf{K}}_{T}f,f)$ and therefore conclude ${\parallel f\parallel }_{L^2}^2\le 2(\parallel {\mathbf{W}}_T{f\parallel }_{L^2}^2+\parallel {\mathbf{K}}_{T}f{\parallel }_{L^2}^2)$. □

We are now ready to prove Theorem 1.

Proof of Theorem 1.

Recall from Lemma 2 that ${\parallel f\parallel }_{L^2\left(B(0,R)\right)}\le 2(\parallel {\mathbf{W}}_T{f\parallel }_{L^2\left({\mathbb{R}}^3\right)}+\parallel {\mathbf{K}}_{T}f{\parallel }_{L^2\left(B(0,R)\right)})$ where ${\mathbf{K}}_T$ is a pseudo-differential operator of order at most $-1$. Since ${\mathbf{K}}_T$ is compact and ${\mathbf{W}}_T$ is injective, applying ([51] Theorem V.3.1), we obtain ${\parallel f\parallel }_{L^2\left(B(0,R)\right)}\le C{\parallel {\mathbf{W}}_{T}f\parallel }_{L^2\left({\mathbb{R}}^3\right)}$ for some constant $C\in (0,\infty )$. This finishes our proof. □

3.3. Continuous Adjoint Operator

Iterative methods for solving (9) require knowledge of the adjoint operator of ${\mathbf{W}}_T$. In this subsection, we compute the continuous adjoint operator ${\mathbf{W}}_T^{*}:L^2\left({\mathbb{R}}^3\right)\to L^2\left(B(0,R)\right)$ and prove that it is again given by the solution of a wave equation. More precisely, we have the following result.

Theorem 2.

Let $g\in C_0^{\infty }\left({\mathbb{R}}^3\right)$, consider the time reversed final state problem for the wave equation,

$$\begin{array}{ccc}\hfill q_{tt}(\mathbf{x},t)-c^2\left(\mathbf{x}\right)\Delta q(\mathbf{x},t)& =0,\hfill & (\mathbf{x},t)\in {\mathbb{R}}^3\times (-\infty ,T)\hfill \\ \hfill q(\mathbf{x},T)& =g\left(\mathbf{x}\right),\hfill & \mathbf{x}\in {\mathbb{R}}^3\hfill \\ \hfill q_t(\mathbf{x},T)& =0\hfill & \mathbf{x}\in {\mathbb{R}}^3,\hfill \end{array}$$

(15)

and let ${\chi }_{B(0,R)}$ denote the indicator function of $B(0,R)$. Then, ${\mathbf{W}}_T^{*}g={\chi }_{B(0,R)}q(·,0)$.

Proof. 

It is clearly sufficient to show ${\mathbf{W}}_T^{*}g={\chi }_{B(0,R)}{u}_t(·,0)$, where u solves the wave equation $u_{tt}-c^2\Delta c=0$ on ${\mathbb{R}}^3\times (-\infty ,T)$, with the final state conditions $(u(·,T),u_t(·,T))=(0,f)$. Using the weak formulation (similar to [38]) for the wave equation, shows that ${\int }_0^T{\int }_{{\mathbb{R}}^3}{c}^{-2}\left(\mathbf{x}\right)u_{tt}(\mathbf{x},t)v(\mathbf{x},t)d\mathbf{x}dt+{\int }_0^T{\int }_{{\mathbb{R}}^3}\nabla u(\mathbf{x},t)·\nabla v\left(\mathbf{x},t\right)d\mathbf{x}dt=0$ for $v\in C_0^{\infty }\left({\mathbb{R}}^3\right)$. Two times integration by parts, rearranging terms and using the final state conditions for u yields ${\int }_{{\mathbb{R}}^3}{c}^{-2}\left(\mathbf{x}\right)[f\left(\mathbf{x}\right)v(\mathbf{x},T)-u_t(\mathbf{x},0)v(\mathbf{x},0)+u(\mathbf{x},0)v_t(\mathbf{x},0)]d\mathbf{x}={\int }_0^T{\int }_{{\mathbb{R}}^3}u(\mathbf{x},t)\left[c^{-2}\left(\mathbf{x}\right)v_{tt}(\mathbf{x},t)-\Delta v(\mathbf{x},t)\right]d\mathbf{x}dt$. By taking v as the solution of (1)–(3) this yields ${\int }_{{\mathbb{R}}^3}{c}^{-2}\left(\mathbf{x}\right)g\left(\mathbf{x}\right){\mathbf{W}}_T\left(f\right)\left(\mathbf{x}\right)d\mathbf{x}={\int }_{{\mathbb{R}}^3}{c}^{-2}\left(\mathbf{x}\right)u_t(\mathbf{x},0)f\left(\mathbf{x}\right)d\mathbf{x}$, which implies ${\mathbf{W}}_T^{*}g={\chi }_{B(0,R)}{u}_t(·,0)={\chi }_{B(0,R)}q(·,0)$ and completes the proof. □

We can reformulate the adjoint operator as follows.

Corollary 1.

For $g\in C_0^{\infty }\left({\mathbb{R}}^3\right)$, let q be the solution of

$$q_{tt}(\mathbf{x},t)-c^2\left(\mathbf{x}\right)\Delta q(\mathbf{x},t)=0\mathit{for}(\mathbf{x},t)\in {\mathbb{R}}^3\times (0,\infty )$$

(16)

with initial conditions $(q(·,0),q_t(·,0)=(g,0)$. Then, ${\mathbf{W}}_T^{*}g={\chi }_{B(0,R)}q(·,T)$.

Proof. 

Clearly q solves (16) with initial data $(q(·,0),q_t(·,0)=(g,0)$ if and only if $(x,t)\mapsto q(x,T-t)$ solves (15). Therefore, the claim follows from Theorem 2. □

3.4. Application of the Steepest 6hhod

We propose solving the finite time wave inversion problem (9) via gradient type methods applied to residual functional $\mathsf{\Phi }\left(f\right)\frac{1}{2}{∥{\mathbf{W}}_{T}f-h∥}^2$. Using standard PAT data, various iterative methods for PAT accounting for variable sound speed have been investigated in [36,37,38,39,40]. In particular, in [40] the steepest descent method has been demonstrated to be numerically efficient and robust. For the results shown in this paper we will therefore use the steepest-descent method. Our numerical experiments confirm that the steepest-descent method is also efficient for FFD-PAT and the final time wave inversion problem, where it reads as follows.

Because of the injectivity and boundedness of ${\mathbf{W}}_T$, the sequence of iterates generated by Algorithm 1 converges to the unique solution of (9). The stability result (11) even implies that the steepest descent method converges linearly for FFD-PAT. More precisely, the sequence ${\left(f_k\right)}_{k\in \mathbb{N}}$ generated by Algorithm 1 satisfies the estimate ${∥f_{k+1}-f∥}_2\le c^k{∥f_0-f∥}_2^2$ for some constant $c\in (0,1)$.

Algorithm 1: Steepest descent method for solving ${\mathbf{W}}_{T}f=g$.

(S1) Choose $a_{★}\in (0,1]$ and initialize $f_0=0$, $k\leftarrow 0$ (S2) While stopping criteria not satisfied, do ■ $s_k={\mathbf{W}}_T^{*}({\mathbf{W}}_T{f}_k-g)$ ■ $a_k=a_{★}{\parallel s_k\parallel }^2/{∥{\mathbf{W}}_T{s}_k∥}^2$ ■ $f_{k+1}=f_k-a_k{s}_k$ ■ $k\leftarrow k+1$

4. Numerical Simulations

In this section, we numerical present results using Algorithm 1 for FFD-PAT with variable sound speed. Recall that for constant speed of sound, reconstructions from experimental FFD-PAT data are shown in [30,31]. In the present study, we give numerical results for spatially varying speed of sound. Performing experimental measurements for samples with variable speed of sound and applying our algorithms to such data is subject of future research.

4.1. Discretization

To implement the steepest descent method (or other gradient type schemes), one has to discretize the final time wave operator ${\mathbf{W}}_T$ and its adjoint ${\mathbf{W}}_T^{*}$. For that purpose we solve the forward and adjoint wave Equations (9) and (16) on a cubical grid with side length $2L$ and nodes $\mathbf{x}\left[\mathbf{i}\right]-(L,L,L)+\mathbf{i}2L/N_x$ for $\mathbf{i}=(i_1,i_2,i_3)\in \left\{0,\dots ,N_x-1\right\}$ with the k-space method [52,53], which we briefly recall in Appendix A. Note that the implementation of the k-space method yields a $2L$-periodic solution. The parameter L is chosen such that $a+T<L$ which implies that inside $B(0,R)$, for $t\in [0,T]$, the solution of (16) coincides with its $2L$-periodic extension.

We denote by $\mathcal{F}\subseteq {\mathbb{R}}^{N_x\times N_x\times N_x}$ the set of all $\mathbf{f}\in {\mathbb{R}}^{N_x\times N_x\times N_x}$ with $\mathbf{f}\left[\mathbf{i}\right]=0$ for $\mathbf{x}\left[\mathbf{i}\right]\notin B(0,R)$. The discretized versions of ${\mathbf{W}}_T$ and its adjoint ${\mathbf{W}}_T^{*}$ are defined by $\mathcal{W}:\mathcal{F}\to {\mathbb{R}}^{N_x\times N_x\times N_x}:\mathbf{f}\mapsto \left(\mathcal{K}\mathbf{f}\right)(·,N_t)$ and ${\mathcal{W}}^{\mathsf{T}}:{\mathbb{R}}^{N_x\times N_x\times N_x}\to \mathcal{F}:\mathbf{h}\mapsto {\chi }_R\left(\mathcal{K}\mathbf{h}\right)(·,N_t)$, respectively, where $\mathcal{K}:{\mathbb{R}}^{N\times N\times N}\to {\mathbb{R}}^{N_x\times N_x\times N_x\times (N_t+1)}$ denotes the discretized wave propagation defined by the k-space method using the discrete time steps $jT/N_t$ for $0\le j\le N_t$, and ${\chi }_R$ is the discretized indicator function of the ball $B(0,R)$. The linear projection $\mathbf{X}$ and its left inverse ${\mathbf{X}}^{♯}$ are implemented using the Matlab build in functions for the Radon and inverse Radon transforms, respectively, with $N_{\theta }$ equally spaced projection angles covering $[0,\pi ]$. The resulting discretized transforms are denoted by $\mathcal{X}$ and ${\mathcal{X}}^{♯}$, respectively.

4.2. Data Simulation

For the presented numerical results, we use $R=0.4$, $T=1$, $L=1.5$, $N=300$, $M=300$ and collect FFD-data for $N_{\theta }=300$ projection directions. As initial pressure ${\mathbf{f}}_{★}$ we take the sum of three solid spheres, as show in the left picture in Figure 3. The used sound speed ${\mathbf{c}}_{★}$ is shown in the right picture in Figure 3, and consist of two Gaussian peaks with opposite signs, added to the constant sound speed ${\mathbf{c}}_0\left[\mathbf{i}\right]=1$. The top row in Figure 4 shows the simulated FFD-PAT data ${\mathbf{g}}_{★}=\mathcal{X}\mathcal{W}{\mathbf{f}}_{★}$ (left) and the noisy FFD-PAT data $\mathbf{g}{\mathbf{g}}_{★}+\mathrm{noise}$ (right) at projection angle $\theta =0$ for both cases. To generate the noisy FFD data we have added Gaussian white noise with a standard deviation of 10% of the maximal value of ${\mathbf{g}}_{★}$, resulting in a relative $L^2$-data error $\parallel \mathrm{noise}\parallel /\parallel {\mathbf{g}}_{★}\parallel \simeq 85.3$%. For the first reconstruction step we apply the inverse Radon transform to the FFD-PAT data, resulting in the approximate 3D final time pressure ${\mathcal{X}}^{♯}\mathbf{g}\simeq \mathcal{W}{\mathbf{f}}_{★}$. The reconstruction of final time pressure from noisy data is shown in the bottom right image in Figure 4. For comparison purpose, the simulated final time pressure ${\mathbf{h}}_{★}\mathcal{W}{\mathbf{f}}_{★}$ is shown in the bottom left image in Figure 4.

4.3. Reconstruction Results

To recover the initial pressure ${\mathbf{f}}_{★}$, we apply the steepest descent method (Algorithm 1) with $a_{★}=0.8$ to the point-wise approximation $\mathbf{h}\simeq \mathcal{W}{\mathbf{f}}_{★}$ that is recovered in step one. Reconstruction results after 5 iterations are shown in Figure 5. The top row shows reconstruction results from simulated data and the middle row shows reconstruction results from noisy data, both using the correct sound speed ${\mathbf{c}}_{★}$ for the reconstruction process. One observes, that the reconstruction results from simulated as well as from noisy data are quite accurate. This demonstrates that the proposed algorithm is efficient, accurate and stable with respect to noise. In both cases, the whole reconstruction procedure takes about 90 $\min$ on a standard desktop PC.

The bottom row in Figure 5 shows reconstruction results using the wrong constant sound speed ${\mathbf{c}}_0$ in Algorithm 1 (while data generation still uses the inhomogeneous sound speed ${\mathbf{c}}_{★}$). In this case, the reconstruction error is much larger, showing the relevance of integrating sound speed variations in image reconstruction. Finally, in Figure 6 we show reconstruction results with the previous Fourier method that has been derived in [31] under a constant sound speed assumption. The results with the Fourier method show a large reconstruction error and are very similar to the reconstruction result using the steepest descent method assuming constant sound speed. This demonstrates that the artifacts in both cases are due to the wrong wave propagation model, which further supports the importance of taking sound speed variations into account in FFD-PAT image reconstruction.

4.4. Quantitative Error Analysis

For all results shown in Figure 5, we stopped the steepest decent method after five iterations, since the reconstructions did not significantly improve by using more iterations. In fact, even after a single iteration, the ${\ell }^2$-reconstruction error and the ${\ell }^2$-residual error are quite close to their minimal values. This phenomenon is investigated in a more quantitative manner in Figure 7. The images on the left-hand side show the evaluation of the relative $L^2$-reconstruction error $\parallel {\mathbf{f}}_k-{\mathbf{f}}_{★}\parallel /\parallel {\mathbf{f}}_{★}\parallel$ in dependence of the iteration index k. The images on the right show the relative residual errors $\parallel \mathcal{W}{\mathbf{f}}_k-{\mathbf{h}}_{★}\parallel /\parallel {\mathbf{h}}_{★}\parallel$. The rapid convergence of both error metrics indicates that the finite time inversion problem is well conditioned, which is also suggested by our theoretical analysis presented in Section 3. The relative data errors, residuals and reconstruction errors for all reconstructions shown above are summarized in

Table 1. Relative error metrics for the FFD data and reconstructions. The first column shows the relative ${\ell }^2$-norm of the noise added to the data. The second column shows the relative ${\ell }^2$-reconstruction error after step one (not applicable to Fourier reconstruction). The third column shows the relative residual error which are minimized in step 2 of the iterative algorithms. The last column shows the relative ${\ell }^2$-error of final reconstruction.

	$\frac{\parallel \mathbf{g}-{\mathbf{g}}_{★}\parallel }{\parallel {\mathbf{g}}_{★}\parallel }$	$\frac{\parallel {\mathcal{X}}^{♯}\mathbf{g}-{\mathbf{h}}_{★}\parallel }{\parallel {\mathbf{h}}_{★}\parallel }$	$\frac{\parallel \mathcal{W}{\mathbf{f}}_{\mathit{rec}}-{\mathbf{h}}_{★}\parallel }{\parallel {\mathbf{h}}_{★}\parallel }$	$\frac{\parallel {\mathbf{f}}_{\mathit{rec}}-{\mathbf{f}}_{★}\parallel }{\parallel {\mathbf{f}}_{★}\parallel }$
	Noise	Error (after Step 1)	Residual (for Step 2)	Error (after Step 2)
Correct SS (simulated data)	0	$\boxed{0.208}$	$\boxed{0.052}$	$\boxed{0.170}$
Correct SS (noisy data)	0.853	1.760	0.874	0.227
Wrong SS (simulated data)	0	0.208	0.510	0.560
Fourier method [31]	0	-	-	0.564

.

5. Conclusions

In this paper, we investigated FFD-PAT, where projection data of acoustic pressure are measured. For the first time, the variable speed of sound has been taken into account for this kind of setup. We developed a two-step reconstruction procedure that recovers the 3D pressure $p(·,T)$ in a first step, which is then used as input for a finite time wave inversion problem in a second step. As the main theoretical contribution of this paper we prove uniqueness and stability results for the finite time wave inversion problem. For the actual solution, we propose the steepest descent iteration which we found to be numerically efficient and stable for FFD-PAT. Moreover, our numerical results demonstrate that ignoring sound speed variations significantly degrades the reconstruction quality. We point out that the novelties of the paper are the development of a reconstruction method together with a mathematical uniqueness and stability analysis of the arising final time wave inversion problem for FFD-PAT with variable sound speed. The experimental feasibility of FFD-PAT has been demonstrated previously in [30,31]. In future work we will perform experiments using FFD-PAT with spatially variable sound speed and experimentally verify our two-step algorithm. Moreover, in upcoming work we will study limited view problems for FFD-PAT which naturally arise in applications, for instance in the case of breast imaging.