Analysis for Full-Field Photoacoustic Tomography with Variable Sound Speed

Abstract

Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT measurement setup, the used data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full field PAT problem. In this paper, we build mathematical foundations of the latter problem for variable sound speed and settle its uniqueness and stability. Moreover, we introduce an exact inversion method using time-reversal and study its convergence. Our results demonstrate the suitability of both the full field approach and the proposed time-reversal technique for high resolution photoacoustic imaging.

1. Introduction.

Consider the following initial value problem for wave equation for an inhomogeneous isotropic medium

$$\{\begin{array}{ll}{\partial }_t^{2}p(x,t)-c^2(x)\Delta p(x,t)=0\hfill & \text{for }(x,t)\in {ℝ}^n\times (0,\infty )\hfill \\ p(x,0)=f(x)\hfill & \text{for }x\in {ℝ}^n\hfill \\ {\partial }_{t}p(x,0)=0\hfill & \text{for }x\in {ℝ}^n.\hfill \end{array}$$ (1.1)

Here $c\in C^{\infty }\left({ℝ}^n\right)$ denotes the sound speed and $f\in H_0^1\left({ℝ}^n\right)$ the initial data that is supported inside a bounded domain $\Omega \subseteq {ℝ}^n$ with Lipschitz boundary. We assume that the sound speed is positive everywhere and constant on the complement ${\Omega }^c≔{ℝ}^n\setminus \Omega$ of Ω. After rescaling we assume ${c|}_{{\Omega }^c}=1$. We refer to the solution $p:{ℝ}^n\times [0,\infty )\to ℝ$ of (1.1) as acoustic pressure field and f as the initial pressure.

Recall that $f:\Omega \to ℝ$ is an element of the Sobolev space H¹(Ω) if it is Lebesgue measurable and $\parallel f{\parallel }_{H^1(\Omega )}^2≔{\int }_{\Omega }|\nabla f(x){|}^2\text{ d}x+{\int }_{\Omega }|f(x){|}^2\text{ d}x$ is finite. Also, $H_0^1(\Omega )$ consists of all elements in H¹(Ω) that vanish on the boundary ∂Ω. The space $H_0^1(\Omega )$ is equipped with the norm $\parallel f{\parallel }_{H_0^1(\Omega )}≔\sqrt{{\int }_{\Omega }|\nabla f(x){|}^2\text{ d}x}$, which is equivalent to $\parallel \cdot {\parallel }_{H^1(\Omega )}$ when restricted to $H_0^1(\Omega )$. We note that each $f\in H_0^1(\Omega )$ can be extended to a function of $H^1\left({ℝ}^n\right)$ using the value zero on Ω^c, which is tacitly done in this paper.

Full field photoacoustic tomography.

The aim of photoacoustic tomography (PAT) is to recover the initial pressure from certain observations of the acoustic pressure field made outside of Ω. In standard PAT, the data is given by the restricted pressure p|_{S×[0, T]}, where $S\subseteq {ℝ}^n$ is an (n − 1)-dimensional observation surface [28, 38, 9, 20, 4, 19, 11, 30]. Opposed to that, in full field PAT introduced in [26, 27], the data provide the acoustic pressure only for a single and fixed time T but on an n-dimensional measurement domain.

To be more specific, for given T > 0, we define the following two operators

$$W_T:H_0^1(\Omega )\to H^1\left({ℝ}^n\right):f\mapsto p(\cdot ,T),$$ (1.2)

$$W_{T,\Omega }:H_0^1(\Omega )\to H^1\left({\overline{\Omega }}^c\right):{f\mapsto p(\cdot ,T)|}_{{\overline{\Omega }}^c},$$ (1.3)

where p is the solution of (1.1). We refer to W_T as the complete single time wave transform and to W_T,Ω as the exterior single time wave transform. Full field PAT provides approximations of W_T,Ωf, from which one aims to recover approximations to the initial pressure f. In [40] it is outlined how actual full field PAT data can be reduced to W_T,Ωf.

In this paper we prove uniqueness and stability of inverting W_T,Ω. We also propose a time-reversal technique to derive a Neumann series solution for the inversion.

Related work.

For the standard PAT problem there is a vast literature on various practical and theoretical aspects (see, for example, [38, 20, 4, 19, 11, 30]). In that context, the time-reversal method has been studied intensively [9, 15, 32, 33]. However, to the best of our knowledge, the time-reversal method has not developed for PAT with full field data.

Only few works exist [27, 26, 40, 13] on the full field inversion problem. The work [27] considers constant speed of sound and the problem is reduced to the inversion of the Radon transform. The work [40] deals with non-constant speed and uses the standard Landweber iterative method. However, the article uses the data in the whole space, not the exterior data as we consider here. In the proceeding [13], variational regularization is used with exterior data. Neither uniqueness nor stability has been proven there. In the present article, for the first time, we prove uniqueness and stability for inverting W_T,Ω. Moreover, we propose and analyze an iterative time-reversal procedure for its inversion.

2. Uniqueness and stability.

Let ${ℝ}^n$ be equipped with the metric g = c⁻²(x)dx². We denote by diam(Ω) the diameter of Ω, defined as the longest distance between any two points inside $\overline{\Omega }$ with respect to the metric g. We recall that T > 0 is a fixed observation time and $\Omega \subseteq {ℝ}^n$ a bounded domain with Lipschitz boundary.

2.1. Uniqueness of reconstruction.

Our first aim is to prove the injectivity of W_T,Ω, which implies that the full field PAT problem is uniquely solvable. For that purpose we start by recalling a uniqueness result for the wave equation, obtained by Stefanov and Uhlmann [32].

Lemma 2.1. Let $f\in H_0^1\left({ℝ}^n\right)$ and suppose $T>\frac{1}{2}\text{ diam}(\Omega )$. If the solution p of (1.1) satisfies ${p(\cdot ,T)|}_{{\Omega }^c}=0$ and ${\left({\partial }_{t}p\right)(\cdot ,T)|}_{{\Omega }^c}=0$, then f = 0.

Denote by $B_R\subseteq {ℝ}^n$ the ball of radius R > 0 in the Euclidean metric of ${ℝ}^n$. We have the following result:

Lemma 2.2. For ϵ > 0 and $h\in H_0^1\left({ℝ}^n\right)$, let u ∈ C([0, T], H¹(B_T+ϵ)) satisfy

$$\{\begin{array}{ll}{\partial }_t^{2}u(x,t)-\Delta u(x,t)=0\hfill & \text{for }(x,t)\in B_{T+ϵ}\times [0,T]\hfill \\ u(x,0)=h(x)\hfill & \text{for }x\in B_{T+ϵ}\hfill \\ {\partial }_{t}u(x,0)=0\hfill & \text{for }x\in B_{T+ϵ}.\hfill \end{array}$$ (2.1)

If h(x) = 0 for x ∈ B_T and u(x, T) = 0 for x ∈ B_ϵ, then h(x) = 0 for x ∈ B_T+ϵ.

Proof. For u satisfying the Euler-Poisson-Darboux equation with initial data (f, 0) instead of the wave equation (2.1), the result was proven in [2, 24]. The proof of the current situation is similar to [24, Theorem 2.1] and is therefore omitted. ■

In the following, for any a > 0, we write

$${\Omega }_a^{(1)}≔\left\{x\in {ℝ}^n\mid \text{dist}(x,\Omega )\le a\right\}$$

$${\Omega }_a^{(2)}≔\left\{x\in {ℝ}^n\mid \text{dist}(x,\Omega )\ge a\right\}.$$

Clearly, for $f\in H_0^1(\Omega )$ we have $W_{T}f\in H_0^1\left({\Omega }_T^{(1)}\right)$, due to finite speed of propagation.

Lemma 2.3. Let Ω be convex, $h\in H_0^1\left({ℝ}^n\right)$ and suppose u satisfies

$$\{\begin{array}{ll}{\partial }_t^{2}u(x,t)-\Delta u(x,t)=0\hfill & \text{for }(x,t)\in {\Omega }^c\times [0,T]\hfill \\ u(x,0)=h(x)\hfill & \text{for }x\in {\Omega }^c\hfill \\ {\partial }_{t}u(x,0)=0\hfill & \text{for }x\in {\Omega }^c.\hfill \end{array}$$

Then u(x, T) = 0 for all $x\in {\Omega }_T^{(2)}$ implies h(x) = 0 for all x ∈ Ω^c.

Proof. Using Lemma 2.2, the proof follows the lines of [2, Proof of Theorem 3] and for the sake of brevity is omitted. ■

Here is our main uniqueness result.

Theorem 2.4 (Main injectivity result). If T > diam(Ω)/2, then the exterior single time wave transform $W_{T,\Omega }:H_0^1(\Omega )\to H^1\left({\overline{\Omega }}^c\right)$ is injective. In particular, the equation W_T,Ωf = g has at most one solution in $H_0^1(\Omega )$ for $g\in H^1\left({\overline{\Omega }}^c\right)$.

Proof. Suppose $f\in H_0^1(\Omega )$ satisfies W_T,Ωf = 0 and denote by p the solution of (1.1). By definition we have $W_{T,\Omega }f={p(\cdot ,T)|}_{{\overline{\Omega }}^c}$ and thus p(x, T) = 0 and Δp(x, T) = 0 for all x ∈ Ω^c. Define u(x, t) ≔ ∂_tp(x, T − t). Then ${\partial }_{t}u(x,t)=-{\partial }_t^{2}p(x,T-t)=-\Delta p(x,T-t)$ in Ω^c. Consequently,

$$\{\begin{array}{ll}{\partial }_t^{2}u(x,t)-\Delta u(x,t)=0\hfill & \text{for }(x,t)\in {\Omega }^c\times [0,T]\hfill \\ u(x,0)={\partial }_{t}p(x,T)\hfill & \text{for }x\in {\Omega }^c\hfill \\ {\partial }_{t}u(x,0)=0\hfill & \text{for }x\in {\Omega }^c.\hfill \end{array}$$

Because u(x, T) = ∂_tp(x, 0) = 0 in Ω^c, Lemma 2.3 shows ∂_tp(x, T) = 0 for all x ∈ Ω^c. Now application of Lemma 2.1 gives f = 0. ■

2.2. Stability of inversion.

Let us first recall some microlocal analysis for the solution of the wave equation; see for example [32, 37] for more details. Let $\widehat{f}(\xi )={\int }_{{ℝ}^n}f(x)e^{-ix\cdot \xi }dx$ denote the Fourier transform of f. Let us for the moment assume that there are no two conjugate points within the distance T in (${ℝ}^n$, g). Then, up to infinitely smooth error, the solution p of (1.1) can be written as

$$p(x,t)=p^{+}(x,t)+p^{-}(x,t)≔\frac{1}{{(2\pi )}^n}\sum _{\sigma =\pm }{\int }_{{ℝ}^n}{e}^{i{\varphi }_{\sigma }(x,\xi ,t)}{a}_{\sigma }(x,\xi ,t)\widehat{f}(\xi )d\xi ,(x,t)\in {ℝ}^n\times [0,T].$$ (2.2)

Here, the phase functions ϕ_±(x, ξ, t) are positively homogenous of order 1 in ξ and solve the eikonal equations

$$\{\begin{array}{l}\mp {\partial }_t{\varphi }_{\pm }(x,\xi ,t)=c(x)|{\nabla }_x{\varphi }_{\pm }(x,\xi ,t)|\hfill \\ {\varphi }_{\pm }(x,\xi ,0)=x\cdot \xi .\hfill \end{array}$$

The functions a_± are classical amplitudes of order 0 satisfying a_±(x, ξ, 0) = 1/2. The principal terms $a_{\pm }^{(0)}(x,\xi ,t)$ satisfy $a_{\pm }^{(0)}(x,\xi ,t)=1/2$ and the homogenous equations

$$\left[\left({\partial }_t{\varphi }_{\pm }\right){\partial }_t-c^2\nabla {\varphi }_{\pm }\cdot {\nabla }_x+C_{\pm }\right]a_{\pm }^{(0)}=0,$$ (2.3)

where $C_{\pm }≔\left({\partial }_t^2-c^2\Delta \right){\varphi }_{\pm }/2$. Geometrically, each singularity (x, ξ) ∈ WF(f) is propagated by p₊ in the phase space along the positive bi-characteristic (γ_x,ξ(t), ζ_x,ξ(t)), while propagated by p₋ along the negative bicharacteristic given by (γ_x,−ξ(t), ζ_x,−ξ(t)) = (γ_x,ξ(−t), −ζ_x,ξ(−t)). Here, the bicharacteristic $\left({\gamma }_{x,\xi }(t),{\zeta }_{x,\xi }(t)\right)=(x(t),\xi (t))\in T^{*}{ℝ}^n$ is defined as the solution of

$$\{\begin{array}{l}\dot{x}(t)=\frac{\xi (t)}{|\xi (t){|}_{\text{g}}},x(0)=x,\hfill \\ \dot{\xi }(t)=-\frac{1}{2}\nabla \left(c^2(x(t))\right)|\xi (t){|}_g,\eta (0)=\xi ,\hfill \end{array}$$

where |ξ|_g is the length of ξ in the metric g. Let us note that the projection γ_x,ξ(t) of the bicharacteristic is a unit speed geodesic in (${ℝ}^n$, g). Its initial unit tangent vector is ξ/|ξ|_g.

We consider the following so-called non-trapping condition.

Condition 2.5 (Non-trapping condition). We assume that there exists T₀ > 0 such that there is no geodesic curve that intersects Ω with the length, in metric g, bigger than T₀.

It is worth noting that if Condition 2.5 holds then diam(Ω) ≤ T₀.

For $h\in H^1\left({ℝ}^n\right)$ consider the following time-reversed wave equation

$$\{\begin{array}{ll}{\partial }_t^{2}q(x,t)-c^2(x)\Delta q(x,t)=0\hfill & \text{for }(x,t)\in {ℝ}^n\times (0,T)\hfill \\ q(x,T)=h(x)\hfill & \text{for }x\in {ℝ}^n\hfill \\ {\partial }_{t}q(x,T)=0\hfill & \text{for }x\in {ℝ}^n.\hfill \end{array}$$ (2.4)

We define the time-reversal operator

$$W_T^{♯}:H^1\left({ℝ}^n\right)\to H^1(\Omega ):{h\to q(\cdot ,0)|}_{\Omega },$$ (2.5)

where q is the solution of (2.4). For a function $\Psi \in C_0^{\infty }\left({ℝ}^n\right)$ denote by Ψ the pointwise multiplication operator f ↦ Ψf.

Proposition 2.6. Let T > T₀/2, suppose $\Psi \in C_0^{\infty }\left({ℝ}^n\right)$ and set x_±(x, ξ) ≔ γ_x,ξ(±T). Then $W_T^{♯}\Psi W_T:H_0^1(\Omega )\to H^1(\Omega )$ is a pseudo-differential operator of order zero with principal symbol

$$\sigma (x,\xi )=\frac{1}{4}\left[\Psi \left(x_{+}(x,\xi )\right)+\Psi \left(x_{-}(x,\xi )\right)\right].$$

Proof. Our key idea is the construction of the parametrix of time-reversed wave equation (2.4), in the same spirit as [32, Proof of Theorem 3] but adapted to our context. From (2.2), up to smooth terms, we have $W_T^{♯}\Psi W_T=W_T^{♯}\Psi W_T^{(+)}+W_T^{♯}\Psi W_T^{(-)}$ with

$$W_T^{(\pm )}f(x)=p_{\pm }(x,T)=\frac{1}{{(2\pi )}^n}{\int }_{{ℝ}^n}{e}^{i{\varphi }_{\pm }(x,\xi ,T)}{a}_{\pm }(x,\xi ,T)\widehat{f}(\xi )d\xi .$$

It suffices to prove that $W_T^{♯}\Psi W_T^{(+)}$ is a pseudo-differential operator with principal symbol σ₊(x₀, ξ₀) = Ψ(x₊(x₀, ξ₀))/4, for all (x₀, ξ₀) ∈ T*Ω.

Consider the parametrix of the time-reversed wave equation (2.4) with initial data $h=\Psi W_T^{(+)}f$, which can be written in the form

$$q_{+}(x,t)=\frac{1}{2{(2\pi )}^n}{\int }_{{ℝ}^n}{e}^{i{\varphi }_{+}(x,\xi ,t)}b(x,\xi ,t)\widehat{f}(\xi )d\xi +\frac{1}{2{(2\pi )}^n}{\int }_{{ℝ}^n}{e}^{i{\varphi }_{+}(x,\xi ,2T-t)}b(x,\xi ,2T-t)\widehat{f}(\xi )d\xi ,$$

with b(x, ξ, T) = Ψ(x)a₊(x, ξ, T). Let us note that the first summand in q₊ is a modification of the (positive) forward solution p₊ which in the case that Ψ = 1 exactly equals p₊/2. The second summand is the time-reflection of the first part through the value t = T. This construction imposes zero velocity at t = T. Indeed, it is easy to check that q₊ satisfies the initial conditions $q_{+}(x,T)=\Psi W_T^{(+)}f$ and (q₊)_t(x, T) = 0. Therefore, from the definition of $W_T^{♯}$,

$$W_T^{♯}\Psi W_T^{(+)}f=q_{+}(x,0)=\frac{1}{2{(2\pi )}^n}{\int }_{{ℝ}^n}{e}^{i{\varphi }_{+}(x,\xi ,0)}b(x,\xi ,0)\widehat{f}(\xi )d\xi +\frac{1}{2{(2\pi )}^n}{\int }_{{ℝ}^n}{e}^{i{\varphi }_{+}(x,\xi ,2T)}b(x,\xi ,2T)\widehat{f}(\xi )d\xi ,$$ (2.6)

up to infinitely smoothing terms.

Note that both the principal term $a_{+}^{(0)}(x,\xi ,t)$ and b⁽⁰⁾(x, ξ, t) of a₊(x, ξ, t) and b(x, ξ, t), respectively, satisfy the homogeneous transport equation (2.3). Hence, their ratio on each bi-characteristic is constant. In particular,

$$\frac{b^{(0)}\left(x_0,{\xi }_0,0\right)}{a_{+}^{(0)}\left(x_0,{\xi }_0,0\right)}=\frac{b^{(0)}\left(x_{+}\left(x_0,{\xi }_0\right),{\xi }_{+}\left(x_0,{\xi }_0\right),T\right)}{a_{+}^{(0)}\left(x_{+}\left(x_0,{\xi }_0\right),{\xi }_{+}\left(x_0,{\xi }_0\right),T\right)}=\Psi \left(x_{+}\left(x_0,{\xi }_0\right)\right).$$

Let us consider (2.6). Since ϕ₊(0, x, ξ) = x · ξ, the first part on the right hand side is a pseudo-differential operator with principal symbol at (x₀, ξ₀) equals

$$\frac{1}{2}{b}^{(0)}\left(x_0,{\xi }_0,0\right)=\frac{1}{2}{a}_{+}^{(0)}\left(x_0,{\xi }_0,0\right)\Psi \left(x_{+}\left(x_0,{\xi }_0\right)\right)=\frac{1}{4}\Psi \left(x_{+}\left(x_0,{\xi }_0\right)\right).$$

The second summand of (2.6) is a Fourier integral operator that translates the singularity of f at (γ_x0,ξ₀(−2T), ζ_x0,ξ₀(−2T)) to (x₀, ξ₀). From the condition T > T₀/2, we have γ_x0, ξ₀(−2T) ∈ Ω^c. Therefore, f = 0 near γ_x0,ξ0(−2T), which implies the second part on the right hand side of (2.6) is infinitely smoothing. This concludes our proof. ■

Remark 2.7. Proposition 2.6 has been proven under the assumption that there are no two conjugate points within the distance T in (${ℝ}^n$, g). However, the proposition still holds when this condition fails. In that case, we split the time interval [0, T] into subintervals such that on each subinterval the geodesic γ_x,ξ(t) does not contain conjugate points. Applying above presented proof iteratively on each subinterval we derive Proposition 2.6 in the general case.

The following theorem provides the stability of solving the final time wave inversion problem.

Theorem 2.8 (Main stability result). Assume that T > T₀/2, with T₀ as in Condition 2.5. Then, there exists a constant C = C(Ω, T, c) > 0 such that

$$\forall f\in H_0^1(\Omega ):\parallel f{\parallel }_{H_0^1(\Omega )}\le C{\Vert W_{T,\Omega }f\Vert }_{H^1\left({\overline{\Omega }}^c\right)}.$$ (2.7)

Proof. Since T > T₀/2, there exists a > 0 such that for all (x, ξ) ∈ T*Ω either $x_{+}(x,\xi )\in {\Omega }_a^{(2)}$ or $x_{-}(x,\xi )\in {\Omega }_a^{(2)}$. Let $0\le \Psi \in C_0^{\infty }\left({ℝ}^n\right)$ be such that Ψ ≡ 0 on Ω and Ψ ≡ 1 on ${\Omega }_a^{(2)}$. Then ΨW_T,Ω = ΨW_T and thus Proposition 2.6 implies that $W_T^{♯}\Psi W_{T,\Omega }$ is a pseudo-differential operator with principal symbol (Ψ(x₊(x, ξ)) + Ψ(x₋(x, ξ)))/4 ≥ 1/4. Therefore,

$$\forall f\in H_0^1(\Omega ):\parallel f{\parallel }_{H^1(\Omega )}\le C_1\left({\Vert W_T^{♯}\Psi W_{T,\Omega }f\Vert }_{H^1(\Omega )}+\parallel f{\parallel }_{L^2(\Omega )}\right).$$

Because $W_T^{♯}\Psi W_{T,\Omega }f$ is supported inside ${\Omega }_{2T}^{(1)}$, we have

$$\begin{gathered} {\Vert W_T^{♯}\Psi W_{T,\Omega }f\Vert }_{H^1(\Omega )}\le {\Vert W_T^{♯}\Psi W_{T,\Omega }f\Vert }_{H^1\left({\Omega }_{2T}^{(1)}\right)} \\ \le C_2{\Vert W_T^{♯}\Psi W_{T,\Omega }f\Vert }_{H_0^1\left({\Omega }_{2T}^{(1)}\right)} \\ \le C_3{\Vert \Psi W_{T,\Omega }f\Vert }_{H_0^1\left({\Omega }_T^{(1)}\right)}. \end{gathered}$$

Above, the last inequality comes from the conservation of the energy ${\int }_{{ℝ}^n}\left[c^{-2}{|{\partial }_{t}q(\cdot ,t)|}^2+|\nabla q(\cdot ,t){|}^2\right]\text{ d}x$ for (2.4) and ∂_tq(·,T) ≡ 0. From the last two displayed equations we conclude

$$\forall f\in H_0^1(\Omega ):\parallel f{\parallel }_{H_0^1(\Omega )}\le C_4\left({\Vert W_{T,\Omega }f\Vert }_{H^1\left({\overline{\Omega }}^c\right)}+\parallel f{\parallel }_{L^2(\Omega )}\right).$$ (2.8)

Since W_T,Ω is injective, and the embedding $H_0^1(\Omega )\to L_2(\Omega ):f\mapsto f$ is compact, applying [35, Proposition 5.3.1] to (2.8) concludes the proof. ■

Let us briefly discuss the condition that T > T₀/2 posed in Theorem 2.8. It implies for any (x, ξ) ∈ T*Ω, at least either x₊(x, ξ) ≔ γ_x,ξ(T) or x₋(x,ξ) ≔ γ_x,ξ(−T) belongs to Ω^c. That is, if (x, ξ) ∈ WF(f) then either (x₊(x, ξ), ξ₊(x, ξ) ≔ ζ_x,ξ(t)) ∈ WF(W_T,Ωf) or (x₋(x, ξ), ξ₋(x, ξ) ≔ ζ_x,−ξ(t)) ∈ WF(W_T,Ωf). We, hence, say that all the singularities of f are observed by W_T,Ωf. Therefore, T > T₀/2 is called the visibility condition. We will always assume it in our subsequent presentation.

3. Iterative time-reversal.

Consider the extension operator $E_{\Omega }:H^1\left({\overline{\Omega }}^c\right)\to H^1\left({ℝ}^n\right)$ defined as follows. For any $g\in H^1\left({\overline{\Omega }}^c\right)$, we take ${E_{\Omega }(g)|}_{{\Omega }^c}=g$ on Ω^c and define E_Ω(g) on Ω as the harmonic extension of g to Ω. That is, h ≔ E_Ω(g)|_Ω satisfies the Dirichlet problem:

$$\{\begin{array}{ll}\Delta h=0\hfill & \text{in }\Omega \hfill \\ h={g|}_{\partial \Omega }\hfill & \text{on }\partial \Omega .\hfill \end{array}$$ (3.1)

Here, g|_∂Ω ∈ H^1/2(∂Ω) denotes the trace of $g\in H^1\left({\overline{\Omega }}^c\right)$ on ∂Ω. Note that the Dirichlet interior problem (3.1) has a unique solution h ∈ H¹(Ω) (see, for example, [22]); therefore, noting that E_Ω(g) = g on Ω^c, $E_{\Omega }(g)\in H^1\left({ℝ}^n\right)$. For notational conveniences, we sometimes use the short-hand notation $\overline{g}$ for E_Ω(g). We further define the orthogonal projection $P_{\Omega }:H^1(\Omega )\to H_0^1(\Omega ):g\mapsto g-h$, where h ∈ H¹(Ω) is the solution of (3.1).

Recall that our aim is the inversion of the restricted single time wave inversion operator $W_{T,\Omega }:H_0^1(\Omega )\to H^1\left({\overline{\Omega }}^c\right)$ defined by (1.2). Our proposed inversion approach is based on the modified time-reversal operator defined by

$$W_{T,\Omega }^{♯}≔P_{\Omega }{W}_T^{♯}{E}_{\Omega }:H^1\left({\overline{\Omega }}^c\right)\to H_0^1(\Omega ).$$

The modified time-reversal operator is itself the composition of harmonic extension E_Ω to ${ℝ}^n$, time-reversal $W_T^{♯}$ defined by (2.5) and projection P_Ω onto $H_0^1(\Omega )$.

3.1. Contraction property.

Consider the case n = 1, the sound speed is constant, and T > T₀. Then, from the D’Alembert formula, $2W_{T,\Omega }^{♯}$ is the exact inverse of W_T,Ω. In the general case this is not true. Nevertheless, as the basis our approach, we will show that the error operator $\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }$ is non-expansive for λ = 2 and a contraction for λ < 2. This will serve as the basis of the proposed iterative time-reversal procedure.

Throughout the following we denote by $E_u(t)≔{\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}u(x,t)|}^2+|\nabla u(x,t){|}^2\right]\text{ d}x$ the energy associated to a function u satisfying the wave equation ${\partial }_t^{2}u=c^2\Delta u$.

Theorem 3.1 (Contraction property of the error operator). Suppose T > T₀/2 and consider for any λ ∈ (0, 2] the error operator

$$K_{T,\Omega ,\lambda }≔\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }:H_0^1(\Omega )\to H_0^1(\Omega ).$$

Then the following hold:

1. K_T,Ω,2 satisfies $\forall f\in H_0^1(\Omega )\setminus \{0\}:{\Vert K_{T,\Omega ,2}f\Vert }_{H_0^1(\Omega )}<\parallel f{\parallel }_{H_0^1(\Omega )}$.

2. If λ ∈ (0, 2), then ∥K_T,Ω,λ∥ < 1.

Proof. (a): For $f\in H_0^1(\Omega )$, set g ≔ W_T,Ωf and $\tilde{g}≔W_{T}f$. That is, $\tilde{g}=p(\cdot ,T)$ and $g={\tilde{g}|}_{{\overline{\Omega }}^c}$, where p solves the forward problem (1.1). Let $\overline{g}≔E_{\Omega }(g)$ and q solve the time-reversal problem (2.4) with $h=2\overline{g}$. Then w ≔ p − q satisfies the wave equation ${\partial }_t^{2}w=c^2\Delta w$ and the corresponding energies at times 0 and T respectively satisfy

$$\begin{gathered} E_w(0)={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}q(x,0)|}^2+|\nabla [q(x,0)-f(x)]{|}^2\right]\text{d}x, \\ {E}_w(T)={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}p(x,T)|}^2+|\nabla [2\overline{g}(x)-\tilde{g}(x)]{|}^2\right]\text{d}x. \end{gathered}$$ (3.2)

Since ${\overline{g}|}_{\partial \Omega }={\tilde{g}|}_{\partial \Omega }$ and ${(\Delta \overline{g})|}_{\Omega }=0$,

$$\begin{gathered} {\int }_{\Omega }|\nabla [2\overline{g}(x)-\tilde{g}(x)]{|}^2-|\nabla \tilde{g}(x){|}^2\text{ d}x=4{\int }_{\Omega }\nabla \overline{g}(x)\cdot \nabla [\overline{g}(x)-\tilde{g}(x)]\text{d}x \\ =4{\int }_{\Omega }\Delta \overline{g}(x)[\overline{g}(x)-\tilde{g}(x)]\text{d}x=0. \end{gathered}$$

That is, ${\int }_{\Omega }|\nabla [2\overline{g}(x)-\tilde{g}(x)]{|}^2\text{d}x={\int }_{\Omega }|\nabla \tilde{g}(x){|}^2\text{ d}x$. Noting that $\tilde{g}=\overline{g}$ on Ω^c, we obtain ${\int }_{{ℝ}^n}|\nabla [2\overline{g}(x)-\tilde{g}(x)]{|}^2\text{ d}x={\int }_{{ℝ}^n}|\nabla \tilde{g}(x){|}^2\text{ d}x$. From (3.2) we deduce E_w(T) = E_p(T). With the conservation of energy we have E_p(0) = E_w(0) and therefore

$${\int }_{{ℝ}^n}|\nabla f(x){|}^2\text{ d}x={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}q(x,0)|}^2+|\nabla [q(x,0)-f(x)]{|}^2\right]\text{ d}x,$$ (3.3)

where we have used the explicit expressions for E_p(0) and E_w(0) respectively.

With $f^{*}≔2W_{T,\Omega }^{♯}{W}_{T,\Omega }f$ the error operator satisfies K_T,Ω,2f = f − f*. Moreover, writing q₀ ≔ q(·,0)|_Ω we have f* = P_Ω(q₀) and thus Δ[q₀ − f*] = 0 in Ω. From this we infer ${\int }_{\Omega }\nabla \left[q_0(x)-f^{*}(x)\right]\cdot \nabla \left[f^{*}(x)-f(x)\right]\text{ d}x=-{\int }_{\Omega }\Delta \left[q_0(x)-f^{*}(x)\right]\left[f^{*}(x)-f(x)\right]\text{ d}x=0$ and therefore

$$\begin{gathered} {\int }_{\Omega }{|\nabla \left[q_0(x)-f(x)\right]|}^2\text{ d}x={\int }_{\Omega }{|\nabla \left[q_0(x)-f^{*}(x)\right]|}^2\text{ d}x+{\int }_{\Omega }{|\nabla \left[f^{*}(x)-f(x)\right]|}^2\text{ d}x \\ \ge {\int }_{\Omega }{|\nabla \left[f^{*}(x)-f(x)\right]|}^2\text{ d}x. \end{gathered}$$ (3.4)

Together with (3.3) this implies $\parallel f{\parallel }_{H_0^1(\Omega )}^2\ge {\Vert f-f^{*}\Vert }_{H_0^1(\Omega )}^2$. That is, ${\Vert K_{T,\Omega ,2}f\Vert }_{H_0^1(\Omega )}\le \parallel f{\parallel }_{H_0^1(\Omega )}$.

In remains to show the strict inequality. To that end assume ${\Vert f-f^{*}\Vert }_{H_0^1(\Omega )}=\parallel f{\parallel }_{H_0^1(\Omega )}$. From (3.3) and (3.4) we obtain

$${\int }_{\Omega }{|\nabla \left[q_0(x)-f(x)\right]|}^2\text{d}x\ge {\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}q(x,0)|}^2+|\nabla [q(x,0)-f(x)]{|}^2\right]\text{ d}x,$$

and therefore

$${\int }_{{ℝ}^n}{c}^{-2}(x){|{\partial }_{t}q(x,0)|}^2\text{ d}x+{\int }_{{\Omega }^c}|\nabla q(x,0){|}^2\text{ d}x=0.$$

In particular, ∂_tq(·,0) vanishes on ${ℝ}^n$ and ∇q(·,0) vanishes on Ω^c. Because q(x, 0) vanishes for $x\in {\Omega }_{2T}^{(2)}$, it follows that q(·,0) vanishes on Ω^c. Applying Lemma 2.1 for u(·,t) ≔ q(·,T −t) yields $2\overline{g}=q(\cdot ,T)=u(\cdot ,0)=0$ on ${ℝ}^n$. In particular, W_T,Ωf = 0 on Ω^c. From Theorem 2.4, we infer f = 0 on ${ℝ}^n$, which concludes the proof.

(b): Let us first consider the case λ = 1. We have to show that there exists a constant L < 1 such that $\parallel \text{Id}-W_{T,\Omega }^{♯}{W}_{T,\Omega })\parallel \le L$. To that end, let $f\in H_0^1(\Omega )$, p solve the forward model (1.1) with initial data f, q solve the time-reversal problem (2.4) with initial data h = E_ΩW_T,Ωf and define the error term w ≔ q − p. The error term satisfies the wave equation ${\partial }_t^{2}w-c^2(x)\Delta w=0$ in ${ℝ}^n\times (0,T)$ and its energy at time T is given by

$$\begin{gathered} E_w(T)={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}w(x,T)|}^2+|\nabla w(x,T){|}^2\right]\text{ d}x \\ ={\int }_{{ℝ}^n}{c}^{-2}(x){|{\partial }_{t}p(x,T)|}^2\text{ d}x+{\int }_{{ℝ}^n}|\nabla [\overline{g}(x)-\tilde{g}(x)]{|}^2\text{ d}x, \end{gathered}$$

where for the second equality we used the conditions $q(\cdot ,T)=\overline{g}$ and ∂_tg(·,T) = 0. The functions $\tilde{g}$ and $\overline{g}$ are defined as above, in the proof of (a).

The second term in the above equation displayed satisfies

$$\begin{gathered} {\int }_{{ℝ}^n}|\nabla [\overline{g}(x)-\tilde{g}(x)]{|}^2\text{ d}x= \\ ={\int }_{{ℝ}^n}\nabla [\tilde{g}(x)-\overline{g}(x)]\cdot \nabla [\tilde{g}(x)-\overline{g}(x)]\text{ d}x \\ ={\int }_{{ℝ}^n}\nabla [\tilde{g}(x)-\overline{g}(x)]\cdot \nabla [\tilde{g}(x)+\overline{g}(x)]\text{ d}x-2{\int }_{{ℝ}^n}\nabla [\tilde{g}(x)-\overline{g}(x)]\cdot \nabla \overline{g}(x)\text{ d}x \\ ={\int }_{\Omega }|\nabla \tilde{g}(x){|}^2\text{ d}x-{\int }_{\Omega }|\nabla \overline{g}(x){|}^2\text{ d}x-2{\int }_{{ℝ}^n}(\tilde{g}(x)-\overline{g}(x))\Delta \overline{g}(x)\text{ d}x \\ ={\int }_{\Omega }|\nabla \tilde{g}(x){|}^2\text{ d}x-{\int }_{\Omega }|\nabla \overline{g}(x){|}^2\text{ d}x \\ \le {\int }_{\Omega }|\nabla \tilde{g}(x){|}^2\text{ d}x. \end{gathered}$$

As a consequence, recalling $g={\tilde{g}|}_{{\overline{\Omega }}^c}=W_{T,\Omega }f$, we obtain

$$\begin{gathered} E_w(T)\le {\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}p(x,T)|}^2+|\nabla \tilde{g}(x){|}^2\right]\text{ d}x-{\int }_{{\Omega }^c}|\nabla g(x){|}^2\text{ d}x \\ ={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}p(x,T)|}^2+|\nabla p(x,T){|}^2\right]\text{ d}x-{\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2 \\ =E_p(T)-{\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2. \end{gathered}$$

The conservation of energy for E and p then gives

$$\begin{gathered} E_w(0)+{\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2=E_w(T)+{\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2 \\ \le E_p(T)=E_p(0). \end{gathered}$$

Using the initial conditions p(x, 0) = f(x) and ∂_tp(x, 0) = 0 this shows

$$E_w(0)+{\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2\le \parallel f{\parallel }_{H_0^1(\Omega )}^2.$$

Noting that W_T,Ωf vanishes outside of ${\Omega }_T^{(2)}$, we have ${\Vert \nabla W_{T,\Omega }f\Vert }_{L^2\left({\overline{\Omega }}^c\right)}^2\ge \mu {\Vert W_{T,\Omega }f\Vert }_{H^1\left({\overline{\Omega }}^c\right)}^2$, for some μ > 0. Applying Theorem 2.8 we obtain

$$E_w(0)\le L^2\parallel f{\parallel }_{H_0^1(\Omega )}^2,$$

for some 0 < L < 1. Noting that $E_w(0)={\int }_{{ℝ}^n}\left[c^{-2}(x){|{\partial }_{t}q(x,0)|}^2+|\nabla [q(x,0)-f(x)]{|}^2\right]\text{ d}x$, we obtain

$${\int }_{\Omega }|\nabla [q(x,0)-f(x)]{|}^2\text{ d}x\le L^2\parallel f{\parallel }_{H_0^1(\Omega )}^2.$$ (3.5)

Let $f^{*}≔W_{T,\Omega }^{♯}{W}_{T,\Omega }f$ and q₀ = q(·,0)|_Ω, then f* = P_Ω(q₀). The left hand side in the above equation can be estimated as

$$\begin{gathered} {\int }_{\Omega }{|\nabla \left[q_0(x)-f(x)\right]|}^2\text{ d}x={\int }_{\Omega }{|\nabla \left[q_0(x)-f^{*}(x)\right]+\nabla \left[f^{*}(x)-f(x)\right]|}^2\text{ d}x \\ ={\int }_{\Omega }{|\nabla \left[q_0(x)-f^{*}(x)\right]|}^2+{|\nabla \left[f^{*}(x)-f(x)\right]|}^2\text{ d}x \\ \ge f^{*}-f_{H_0^1(\Omega )}^2, \end{gathered}$$

where we have used the fact that ${\int }_{\Omega }\nabla \left[q_0(x)-f^{*}(x)\right]\cdot \nabla \left[f^{*}(x)-f(x)\right]\text{ d}x={\int }_{\Omega }\Delta [q_0(x)-f^{*}(x)]\left[f^{*}(x)-f(x)\right]\text{d}x=0$. From 3.5, we arrive at

$${\Vert K_{1}f\Vert }_{H_0^1(\Omega )}^2={\Vert f-f^{*}\Vert }_{H_0^1(\Omega )}^2\le L^2\parallel f{\parallel }_{H_0^1(\Omega )}^2.$$

This finishes the proof for the case λ = 1.

For the general case note the identities

$$K_{T,\Omega ,\lambda }=\{\begin{array}{ll}(1-\lambda )\text{ Id}+\lambda K_{T,\Omega ,1}\hfill & \text{for }\lambda \in (0,1)\hfill \\ (\lambda -1)K_{T,\Omega ,2}+(2-\lambda )K_{T,\Omega ,1}\hfill & \text{for }\lambda \in (1,2).\hfill \end{array}$$

Using the already verified estimates ∥K_T,Ω,1∥ < 1 and ∥K_T,Ω,2∥ ≤ 1, these equalities together with the triangle inequality for the operator norm show ∥K_T,Ω,λ∥ < 1 for all λ ∈ (0, 2). ■

3.2. Neumann series solution.

According to Theorem 3.1 the error operator satisfies $\Vert \text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\Vert <1$ for any λ ∈ (0, 2). The Neumann series ${\sum }_{j=0}^{\infty }{\left(\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\right)}^j$, therefore, converges to ${\left(\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\right)}^{-1}$ with respect to the operator norm ∥·∥ in $H_0^1(\Omega )$. This results in the inversion formula

$$f=\sum _{j=0}^{\infty }{\left(\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\right)}^j\left(\lambda W_{T,\Omega }^{♯}g\right)\text{ with }g=W_{T,\Omega }f$$ (3.6)

valid for every initial data $f\in H_0^1(\Omega )$. Here $W_{T,\Omega }^{♯}=P_{\Omega }{W}_T^{♯}{E}_{\Omega }$ is the modified time-reversal operator formed by harmonic extension E_Ω of the missing data, time-reversal $W_T^{♯}$ defined by (2.4) and projection P_Ω onto $H_0^1$. Inversion formula (3.6) is the Neumann series solution for the inverse problem of full-field PAT.

Remark 3.2 (Iterative time-reversal algorithm). The Neumann series in (3.6) is the limit of its partial sums $f_k≔{\sum }_{k=0}^j{\left(\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\right)}^k\left(\lambda W_{T,\Omega }^{♯}g\right)$. These partial sums satisfy the recursion

$$\{\begin{array}{l}{f}_0=\lambda W_{T,\Omega }^{♯}g\hfill \\ f_j=f_{j-1}-\lambda W_{T,\Omega }^{♯}\left(W_{T,\Omega }{f}_{j-1}-g\right),\hfill \end{array}$$ (3.7)

with $W_{T,\Omega }^{♯}=P_{\Omega }{W}_T^{♯}{E}_{\Omega }$. This is an iterative algorithm producing a sequence ${\left(f_j\right)}_{j\in \mathbb{N}}$ converging to $f=W_{T,\Omega }^{-1}g$ in $H_0^1(\Omega )$. We call (3.7) iterative reversal algorithm for full field PAT. The form (3.7) will be used in the numerical solution. Because of the contraction property of the iteration $\Vert \text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }\Vert <1$ the iterative time-reversal reversal algorithm is linearly convergent.

We note that for standard PAT, the idea of using time-reversal was proposed in [9, 7] for the case of constant sound speed, and in [10, 15] for non-constant sound speed. The Neumann series solution was first proposed in [32] and further developed in [32, 33, 36, 14, 34, 25, 29, 18, 1]. Iterative reconstruction methods for variable sound speed based on an adjoint wave equation have been studied in [16, 5, 3, 11, 17]. Uniqueness and stability for standard PAT was studied in [39, 15, 32, 33, 23], just to name a few.

4. Numerical Simulations.

In this section, we present some of our numerical studies for the exterior single time wave transform. We consider the case of two spatial dimensions, and take $\Omega \subseteq {ℝ}^2$ as the unit disc. According to (3.6) any function $f\in H_0^1(\Omega )$ can be recovered from data g = W_T,Ωf via the iterative time-reversal algorithm (3.7). The numerical realization is described in the following subsection. The numerical simulations were performed for each of the three sound speed profiles shown in the top row of Figure 1 and additionally for the constant sound speed c_I = 1. The sound speed profiles c_III and c_IV are adopted from [31] where c_III is shown to be trapping and c_IV to be non-trapping. For the used radially symmetric sound speed profile c_III = c(∥x∥) the Herglotz condition $\frac{d}{dr}(r/(c(r)))>0$ is satisfied which implies that c_III is non-trapping. As phantom we used numerical approximations of one smooth and two piecewise constant functions which are visualized in the bottom row of Figure 1.

Figure 1. Sound speeds profiles and initial pressure distributions.

Top: Three non-trapping (c_II and c_III) and one trapping sound speed profile c_IV that we employed in our simulations besides the constant speed of sound c_I = 1. The white and black circles visualize the boundary of the imaging region which in our simulations is the unit disc. Bottom: A smooth phantom f^a (left) and the two piecewise constant phantoms f^b (middle) and f^c (right) are employed in our numerical simulations.

4.1. Numerical implementation.

In the numerical realization, any function $h:{ℝ}^2\to ℝ$ is represented by a discrete vector ${\left(h\left(x_i\right)\right)}_{i_1,i_2=0}^{N-1}\in {ℝ}^{N\times N}$, where

$$x_i=(-a,-a)+2ia/N\text{ for }i=\left(i_1,i_2\right)\in {\left\{0,\dots ,N-1\right\}}^2$$

are equidistant grid points in the square [−a, a]². The discrete domain I ⊆ {0, …, N − 1}² (where the discrete initial pressure is contained in) is defined as the set of all indices i with x_i ∈ Ω and we set J ≔ {0, …, N − 1}² \ I. Following [12], we define the discrete boundary of I as the set of all elements (i₁, i₂) ∈ J for which at least one of the discrete neighbors (i₁ + 1, i₂),(i₁ − 1, i₂),(i₁, i₂ + 1), (i₁, i₂ − 1) is contained in I. The discrete version of the initial data $f\in H_0^1(\Omega )$ is then an image $\mathtt{\text{f}}\in {ℝ}^I$ and the discrete version of the data $g\in L^2\left({\overline{\Omega }}^c\right)$ an image $g\in {ℝ}^J$.

In the iterative time-reversal algorithm the forward transform W_T,Ω as well as each factor in the modified time-reversal $W_{T,\Omega }^{♯}=P_{\Omega }{W}_T^{♯}{E}_{\Omega }$ are replaced by discrete approximations. The discrete forward operator and the discrete time-reversal operator are defined by

$${\mathtt{\text{W}}}_{T,I}:{ℝ}^I\to {ℝ}^J:\mathtt{\text{f}}\mapsto {\left({\mathtt{\text{W}}}_T\mathtt{\text{f}}\right)}_{I^c}$$ (4.1)

$${\mathtt{\text{W}}}_{T,I}^{♯}:{ℝ}^J\to {ℝ}^I:\mathtt{\text{g}}\mapsto \left({\mathtt{\text{P}}}_I{\mathtt{\text{W}}}_T^{♯}{\mathtt{\text{E}}}_I\right)\mathtt{\text{g}}.$$ (4.2)

Here ${\mathtt{\text{W}}}_T:{ℝ}^{N\times N}\to {ℝ}^{N\times N}$ and ${\mathtt{\text{W}}}_T^{♯}:{ℝ}^{N\times N}\to {ℝ}^{N\times N}$ are discrete analogs of the forward wave equation and its time reversed version, E_I a discretization of the harmonic extension operator and P_I a discretization of the projection of the projection onto $H_0^1(\Omega )$.

The numerical solution of the wave equation W_T f and likewise the numerical solution of the time reversed version ${\mathtt{\text{W}}}_T^{\#}$ are computed with the k-space method [6, 8, 21]. We use the k-space method with periodic boundary conditions on the rectangle [−a, a]² as described in [11]. We choose a ≥ T +1 such that W_T,I and ${\mathtt{\text{W}}}_{T,I}^{\#}$ are not affected by replacing the free space wave equation with its (2a)-periodic counterpart. The discrete harmonic extension E_I and the discrete projection P_I are constructed by numerically solving (3.1) with the MATLAB-routine solvepde.

4.2. Numerical results.

We first present results for data g = W_T f without added noise. Figure 2 shows results with constant speed with relaxation parameter λ = 2 and 80 iterations. For smaller values of λ slightly better reconstructions have been obtained but required a slightly larger number of iterations. Figure 3 visualizes the pointwise error map $f^a-f_{\text{rec}}^a$ for the non-constant sound speed profiles using λ = 1/2 and T = 2. We see that accurate results are obtained for all sound speed profiles. The best results were obtained for the sound speed profile c_II, and the error functions do not contain any visible information of the original phantom. Because all reconstructions look equally well and very similar to the original phantom f^a, we did not show them here. Additional simulations with other smooth and non-smooth phantoms indicate that smooth phantoms generally result in better reconstructions than non-smooth ones.

Figure 2. Results for constant sound speed.

Reconstructions (top) and corresponding point-wise errors (bottom) using constant sound speed c_I = 1.

Figure 3. Results for variable sound speed.

Point-wise errors (bottom) using different sound speed profiles and the smooth phantom f^a.

Next, in Figure 4, we present results for noisy data where ${\mathtt{\text{W}}}_T^{♯}\mathtt{\text{f}}$ has been contaminated with normally distributed noise with a standard deviation δ of two percent of the maximal pressure value. In order to avoid inverse crime, data are simulated using a three times finer discretization than used for the reconstruction. We observe that in all cases the iteration process is stable when λ is chosen sufficiently small, i.e. 0 < λ ≤ 1/2, and the use of a stopping rule is not necessary. Moreover, the reconstructions are more accurate for smooth phantoms than for piecewise constant phantoms. Finally in Figure 5 we show results for the trapping sound speed c_IV. Also in this case, the iterative time-reversal works well for all phantoms even though the theory developed in the previous Sections does not fully apply in this situation.

Figure 4. Exact versus noisy data for sound speed c_III.

From left to right: reconstruction, difference images to true phantom f^a, and logarithmic error plot in dependence of the number of iterations. The top row shows results for exact data, the bottom row shows results for noisy data with standard deviation δ = 0.02. Here λ = 1/2 and T = 4. In this example, the observation time is twice as large as for the example in Fig 3, i.e. we have more data. This leads to a much smaller pointwise error for zero noise.

Figure 5. Results for noisy data for trapping sound speed c_IV.

Top row shows the reconstructions of the phantom f^a, f^b and f^c. Bottom row: Corresponding difference images for to the true phantom. Here λ = 1/2 and T = 2

5. Conclusion.

In this work we studied an inverse source problem appearing in full field PAT. Image reconstruction amounts to the inversion of the exterior final time wave transform W_T,Ω that maps the initial data f supported in Ω to the solution of the wave equation at fixed time T restricted to the complement ${\overline{\Omega }}^c$. For non-constant sound speed, besides the work [13], to the best of our knowledge, inversion of W_T,Ω is studied for the first time. We, for the first time, derived uniqueness and stability results. Moreover we showed convergence of the proposed iterative time-reversal reconstruction algorithm. For that purpose we have proven that $\text{Id}-\lambda W_{T,\Omega }^{♯}{W}_{T,\Omega }$ is a contraction on $H_0^1(\Omega )$ for all λ ∈ (0, 2) where $W_{T,\Omega }^{♯}$ is a modified time-reversal operator. We also derived a numerical realization of the iterative time-reversal algorithm. Numerical results show accurate reconstruction for all sound speed profiles and all initial data.