Operator Learning Approach for the Limited View Problem in Photoacoustic Tomography

  • 1 Department of Mathematics, University of Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria
  • 2 Department of Neuroradiology, Medical University of Innsbruck, Anichstraße 35, 6020, Innsbruck, Austria
Florian Dreier, Sergiy Pereverzyev JrORCID iD: https://orcid.org/0000-0002-8627-3995 and Markus HaltmeierORCID iD: https://orcid.org/0000-0001-5715-0331

Abstract

In photoacoustic tomography, one is interested to recover the initial pressure distribution inside a tissue from the corresponding measurements of the induced acoustic wave on the boundary of a region enclosing the tissue. In the limited view problem, the wave boundary measurements are given on the part of the boundary, whereas in the full view problem, the measurements are known on the whole boundary. For the full view problem, there exist various fast and robust reconstruction methods. These methods give severe reconstruction artifacts when they are applied directly to the limited view data. One approach for reducing such artefacts is trying to extend the limited view data to the whole region boundary, and then use existing reconstruction methods for the full view data. In this paper, we propose an operator learning approach for constructing an operator that gives an approximate extension of the limited view data. We consider the behavior of a reconstruction formula on the extended limited view data that is given by our proposed approach. Approximation errors of our approach are analyzed. We also present numerical results with the proposed extension approach supporting our theoretical analysis.

1 Introduction

Photoacoustic tomography (PAT) is an emerging non-invasive imaging technique. It is based on the photoacoustic effect, and it has a big potential for a successful use in biomedical studies, including preclinical research and clinical practice. Applications include tumor angiogenesis monitoring, blood oxygenation mapping, functional brain imaging, and skin melanoma detection [49, 31, 5, 47]

The principle of PAT is the following. When short pulses of non-ionising electromagnetic energy are delivered into a biological (semi-transparent) tissue, then parts of the electromagnetic energy become absorbed. The absorbed energy leads to a nonuniform thermoelastic expansion depending on the tissue structure. This gives rise to an initial acoustic pressure distribution, which further is the source of an acoustic pressure wave. These waves are detected by a measurement device on the boundary of the tissue. The mathematical task in PAT is to reconstruct the spatially varying initial pressure distribution using these measurements. The values of the initial pressure distribution inside the tissue allow to make a judgment about the directly unseen structure of the tissue. For example, whether there are some abnormal formations inside the investigated tissue, such as a tumor.

Consider the part of the boundary of a region enclosing the tissue where the wave measurements are available. This part is called observation boundary. If the tissue is fully enclosed by the observation boundary, then one speaks about the full view problem. Otherwise, if some part of the tissue boundary is not accessible, then one has the so-called limited view problem (LVP). The LVP frequently arises in practice, for example in breast imaging (see, e.g., [50, 27]).

The LVP can be approached using iterative reconstruction algorithms (see, e.g., [39, 37, 23, 52, 25, 19, 42]). Although these algorithms can provide accurate reconstruction, they are computationally expensive and time consuming. Approaches for the full view problem, such as time reversal [7, 24], Fourier domain algorithms [16, 29, 51], explicit reconstruction formulas [10, 9, 30, 28, 35], are faster than iterative reconstructions and additionally are robust and accurate. However, when they are directly applied on the limited view data, then one obtains severe reconstruction artifacts.

And so, an idea appears to try to extend the limited view data to the whole boundary, and then use efficient algorithms for the full view data on the extended data to obtain a reconstruction of the initial pressure. Knowing characterizations of the range of the forward operator, which maps the initial pressure distribution to the wave data on the whole boundary of the tissue, may be used for this purpose (see, e.g., [3, 11, 1, 27] and the references therein). This knowledge is expressed with so-called range conditions. In [40, 41], some of these conditions, the so-called moment conditions, were realized for the extension of the limited view data.

The data extension process based on the moment conditions is unstable, and therefore, mostly low frequencies of the limited view wave data can be extended. This instability is connected with the following issue. The observation boundary defines a so-called detection region, which, for typical measurement configurations, is the convex hull of the observation boundary [26]. It is known (see, e.g., [26, 44, 27]) that if the support of the initial pressure is contained in this detection region, then a stable recovery of the initial pressure from the limited view wave data is theoretically possible. However, the data extension process based on the moment conditions does not use information about the support of the initial pressure, and so, it does not employ advantages of the possible stable recovery.

In this paper, we propose a stable method for the extension of the limited view wave data that uses advantages of the mentioned possible stable recovery. Our method is based on the observation that in the case of the stable recovery, there exists a continuous data extension operator that maps the limited view wave data to the unknown wave data on the unobservable part of the boundary. We formally define this operator in Section 3.1. However, this operator is not explicitly known. In our method, we therefore propose to construct an approximate data extension operator using an operator learning approach that is inspired by the methods of the statistical learning theory (see, e.g., [22]). We suggest an operator learning procedure that uses the projection on the linear subspace defined by the training inputs.

Having an approximately extended limited view wave data, one can employ reconstruction methods for the full view wave data, such as time reversal or methods based on the explicit inversion formulas. As an example, we consider an explicit reconstruction formula for that purpose. We demonstrate that the resulting reconstruction algorithm corrects most limited view reconstruction artifacts, while the computational time remains to be low. The involved steps in the proposed reconstruction approach are illustrated in Figure 1.

Figure 1
Figure 1

Illustration of the proposed approach for limited view PAT. In the first step, we extend the limited view data to the whole boundary via operator learning. In the second step, we apply a standard direct PAT reconstruction algorithm to thecompleted data.

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

The rest of the paper is organized as follows. In Section 2, we present a mathematical background for PAT, give the used explicit reconstruction formula, and discuss the LVP. Our operator learning approach to the extension of the limited view wave data is given in Section 3. In Section 4, we analyze the approximation errors of our approach. We look at the approximation errors for the unknown wave data and for the corresponding reconstructions obtained by explicit reconstruction formulas. We present the numerical results in Section 5. Finally, we finish the paper with conclusion and outlook in Section 6.

2 Mathematics of PAT

Let Ωd be a bounded domain with a smooth boundary Ω, where d2 denotes the spatial dimension. Further, let Cc(Ω) be the set of all smooth functions f:d that are compactly supported in Ω. In PAT, one is interested to recover an unknown function fCc(Ω) from the solution of the wave equation given on parts of the boundary of Ω. Let us mathematically specify this reconstruction problem.

2.1 Reconstruction Problem

Let 𝔘f:d×(0,) denote the solution of the following initial value problem for the wave equation:

{(t2-Δx)u(x,t)=0for (x,t)d×(0,),u(x,0)=f(x)for xd,(tu)(x,0)=0for xd.

Here t denotes differentiation with respect to the second variable t, and Δx is the Laplacian with respect to x. Then the reconstruction problem in PAT consists in recovering the unknown function fCc(Ω) from the corresponding wave boundary data

u(x,t)=(𝔘f)(x,t)for (x,t)Γ1×(0,),

where Γ1Ω. If Γ1=Ω, then (2.1) is called full view problem; otherwise, if Γ1Ω, we have the limited view problem (LVP). In this paper, we are particularly interested in the limited view case, which we consider in some detail in Section 2.3.

Let us denote the unobservable part of the boundary as Γ2:=ΩΓ1. We define also the following restrictions of 𝔘f:

𝒰f:=𝔘f|Ω×(0,),𝒰1f:=𝔘f|Γ1×(0,),𝒰2f:=𝔘f|Γ2×(0,).

Let us note that in practice, the reconstruction problem (2.1) arises in PAT in spatial dimensions two and three. The three-dimensional problem appears when the so-called point-like detectors are used (see, for example, [49, 26, 12]). When one uses linear or circular integrating detectors, then the reconstruction problem (2.1) is considered in two spatial dimensions (see [6, 15, 38, 53]).

2.2 Explicit Inversion Formula

The reconstruction problem (2.1) can be approached by various solution techniques. Among these techniques, the derivation of the explicit inversion formulas of the so-called back-projection type is particularly appealing. A numerical realization of these formulas typically gives reconstruction algorithms that are accurate and robust, and at the same time are faster than iterative approaches.

An inversion formula consists of an explicitly given operator 𝒢d that recovers the function f from the data u. Such formulas are currently known only for special domains and only for the full view data, i.e. u must be given for all xΩ. In this paper, we consider the formula that first has been derived in [48, 30, 6]. In addition to the data u, the formula 𝒢d also depends on the boundary Ω of the domain Ωd and on the reconstruction point x0Ω. The structure of the formula further depends on whether the spatial dimension d is even or odd.

If d2 is an even integer, then

𝒢d(Ω,u,x0):=κdΩνx,x0-x|x0-x|(t𝒟t(d-2)/2t-1u)(x,t)t2-|x0-x|2dtds(x).

Here κd:=(-1)(d-2)/2/πd/2 is a constant, νx denotes the outward pointing unit normal to Ω, and

𝒟t:=(2t)-1t

is the differentiation operator with respect to t2. Further, , and || denote the standard inner product and the corresponding Euclidian norm on d, respectively.

In the case of odd dimension d3, the formula 𝒢d is defined as follows:

𝒢d(Ω,u,x0):=κdΩνx,x0-x|x0-x|(t𝒟t(d-3)/2t-1u)(x,|x0-x|)ds(x)

with constant κd:=(-1)(d-3)/2/(2π(d-1)/2).

The formula 𝒢d has been introduced in [48] for dimension d=3, and in [6] for dimension d=2. In [30], it has been studied for the case when Ω is a ball in arbitrary dimension. Further, in [34, 17, 18], it has been shown that for any elliptical domain Ω, the formula 𝒢d exactly recovers any smooth function f with support in Ω from data u=𝒰f. In [20], it was shown that the same result also holds for parabolic domains Ω with d=2. The formula 𝒢d in arbitrary spatial dimension d2 on certain quadric hypersurfaces, including the parabolic ones, has been analyzed in [21].

It should be noted that the formula 𝒢d can be in fact used for any convex bounded domain Ω. Then, however, the formula does not recover the function f exactly, and it introduces an approximation error. The form of this error has been analyzed in [34, 17, 18]. Numerical experiments indicate that this error is rather low for domains that can be well approximated by elliptic domains. This is also suggested by the microlocal analysis in [35].

The operator 𝒰 can be defined for functions f2(Ω0), where Ω0 is an open set with Ω0¯Ω. Define the image of 2(Ω0) under the operator 𝒰 as 𝕐:=𝒰(2(Ω0)). Then it is known (see, e.g., [26, 44, 27]) that 𝕐 is a closed subspace of 2(Ω×(0,)), and therefore, we will treat 𝕐 as a Hilbert space with the scalar product of 2(Ω×(0,)). Moreover, the operator 𝒰:2(Ω0)𝕐 is bounded, and it has the bounded inverse 𝒰-1:𝕐2(Ω0).

In the following, we will work with functions f2(Ω0), and we will assume that the domain Ω is such that the formula 𝒢d gives exact recovery of the function f from its wave data u=𝒰f, i.e. it holds that

f=𝒢d𝒰f.

As we already mentioned, this is, for example, the case for circular and elliptical domains. In such a situation, it can be shown that 𝒢d is a continuous extension of 𝒰-1 to 2(Ω×(0,)).

2.3 Limited View Problem

In practice, the wave data u is frequently given on a subset Γ1 of the boundary Ω (Figure 2). This subset Γ1, called observation boundary, defines the so-called detection region 𝔻(Γ1) (see, for example, [37, 26]). If supp(f)¯𝔻(Γ1), then the function f in (2.1) can be stably recovered from data on Γ1. The detection region 𝔻(Γ1) contains points x such that any line going through x intersects Γ1. For example, if Γ1 is a spherical or elliptical cap, then 𝔻(Γ1)=conv(Γ1).

Figure 2
Figure 2

Setting of LVP.

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

Let us mathematically specify the stable recovery of f. Let Ω1 be an open set with Ω1¯𝔻(Γ1). The stable recovery holds for f2(Ω1), and it is formulated in the following theorem. Note that the space 2(Ω1) is identified with the set of all functions in 2(d) that vanish outside of Ω1¯.

Theorem 1.

The operator U1:L2(Ω1)L2(Γ1×(0,)) is well defined and bounded. Moreover, it has bounded inverse U1-1:Y1L2(Ω1), where Y1:=U1(L2(Ω1))L2(Γ1×(0,)) denotes the range of U1. In particular, Y1 is closed.

Proof.

It is sufficient to show the two-side estimate

a𝒰1f2f2b𝒰1f2for all fCc(Ω1)

for some constants a,b(0,). The claims then follow by continuous extension.

To show the left-hand estimate, we decompose 𝒰1f=χ[0,T]𝒰1f+χ(T,)𝒰1f, where T is larger than the diameter of Ω. Since the operator 𝒰1 is the sum of two Fourier integral operators of order zero (see [19]), we have χ[0,T]𝒰1f2c1f2 for some constant c1. Moreover, the explicit formulas for 𝒰1f (see, e.g., [8]) imply also that χ(T,)𝒰1f2c1f2, which gives the left-hand side estimate in (2.4).

The right-hand side estimate can be found in [19, Theorem 3.4]. The required visibility condition is satisfied due to the assumption that f2(Ω1). ∎

It is worth to mention that despite the boundedness of 𝒰1-1, no theoretically exact direct solution methods are available. Let us note that if the condition Ω1¯𝔻(Γ1) is not satisfied, the visibility condition in [19, Theorem 3.4] is also not valid, and the inverse of the operator 𝒰1 is severely ill-posed (see, e.g., [19, 44, 27]).

Denote 𝕐2:=2(Γ2×(0,)). From the boundedness of the operator 𝒰:2(Ω0)𝕐, we can deduce the boundedness of the operator 𝒰2:2(Ω1)𝕐2. We will use this for the data extension operator below.

Recall that in order to give the exact reconstruction, the formula 𝒢d requires the full view wave data u, which is given for all xΩ (see (2.3)). In spite of the above discussed stable recoverability of f2(Ω1) from equation (2.1), the use of formula 𝒢d on the limited view data u given on Γ1Ω leads to serious artifacts in the reconstruction; see, e.g., [20], where the numerical results of the application of 𝒢2 on finite parabolas are presented. The reconstruction artefacts in the case of the limited view data are also discussed in [50, 13, 45, 4, 14, 36].

At the same time, the use of formula 𝒢d for reconstructing function f can be attractive from various points of view. For example, as we already pointed out, the reconstruction using a numerical realization of 𝒢d is faster than iterative reconstruction algorithms. Another point may be connected with the nature of the software development. Namely, having already a tested and trusted computer code of the numerical realization of formula 𝒢d, it could be tempting to develop its extensions for the LVP.

An extension of the limited view data u from the observable part of the boundary Γ1Ω to the whole boundary Ω may give a possibility to improve the reconstruction quality of the formula 𝒢d. In this paper, we propose to realize this extension using the operator learning approach, which we consider in the next section.

3 Data Extension Using Operator Learning Approach

The extension of the limited view data to the whole boundary can be in principle done by the extension operator that we define in the next subsection. This operator is however not explicitly known, and we propose an operator learning approach to construct its approximation in Section 3.2. In Section 3.3, we discuss computational aspects of the proposed learned approximation of the extension operator.

3.1 Extension Operator

Let us recall that Γ1Ω is the observation boundary, 𝔻(Γ1) is the corresponding detection region defined in Section 2.3, Γ2=ΩΓ1 is the unobservable part of the boundary, and Ω1 is an open set with Ω1¯𝔻(Γ1). Further, let us remind that the operators 𝒰1 and 𝒰2 are defined in (2.2).

The operator 𝒜:𝕐1𝕐2 that maps functions 𝒰1f to functions 𝒰2f for f2(Ω1) realizes the extension of the limited view data u1=𝒰1f to the unobservable part of the boundary Γ2. This operator 𝒜 can be written as 𝒜=𝒰2𝒰1-1. Because of this representation and the assumptions on Γ1 and Ω1, the operator 𝒜 is a linear continuous operator as a superposition of linear continuous operators. Recall that the continuity (or boundedness) of the operators 𝒰1-1 and 𝒰2 is discussed in Section 2.3.

With the introduced extension operator 𝒜, one could extend the limited view data u1 to the whole boundary Ω, and then use the formula 𝒢d on this extended data. In this way, the disadvantages of the use of the formula 𝒢d on the limited view data can be eliminated. However, the form of the operator 𝒜 is not explicitly known.

3.2 Proposed Learned Extension Operator

In this paper, we propose to construct an operator 𝒜^n that approximates the operator 𝒜. The role of the parameter n{0} is described below. The approximate operator 𝒜^n must satisfy the following two requirements. The first requirement concerns the approximation quality: 𝒜^nu1 must be close to 𝒜u1. The second requirement is related to the computational effort of the numerical evaluation of 𝒜^nu1. This evaluation must be fast such that the evaluation of the formula 𝒢d on the extended limited view data with the help of 𝒜^n remains to be computationally efficient.

Our construction of the approximate operator 𝒜^n is inspired by the statistical learning approach (see, e.g., [22]). For i=1,,n, consider training functions fi:Ω1. For each training function fi, we can determine the corresponding wave data u1,i:=𝒰1fi, u2,i:=𝒰2fi. By the definition of the extension operator 𝒜 we have that u2,i=𝒜u1,i. In the context of statistical learning, the set 𝒵:={(u1,i,𝒜u1,i):i=1,,n} is called a training set. Define for future reference the set 𝐔1,n:={u1,i:i=1,,n}.

So, how to construct (or, using the terminology of the statistical learning, how to learn) an approximation 𝒜^nu1 of 𝒜u1 using the training set 𝒵? It should be noted that many statistical learning algorithms are designed for learning a small number of scalar-valued functions. These algorithms are not applicable in our case because the function that we need to learn is an operator. Recently, the development of the statistical learning methods for learning vector-valued functions and also functions with values in function spaces, i.e. operators, has been started (see, e.g., [33, 2]). For obtaining good results, these methods require an a priori knowledge of the dependence between different components of the output vector that is given by the function to be learned. This knowledge is not readily available in our case. However, as we observe below, the linear structure of the extension operator 𝒜 that we want to learn allows to employ a projection operator for the learning.

For any n{0}, define the linear subspace

Vn:={j=1ncju1,j:cj},V0:={0}𝕐1,

and let 𝒫n:2(Γ1×(0,))Vn be the orthogonal projection on Vn in 2(Γ1×(0,)). Then we define the learned approximation 𝒜^nu1 as follows:

𝒜^nu1:=𝒜𝒫nu1.

Note that Vn𝕐1, and therefore, the operator composition 𝒜𝒫n is well-defined, and

𝒜^n:2(Γ1×(0,))𝕐2

is bounded. Further, note that for all u12(Γ1×(0,)), 𝒜^0u1=0𝕐2.

3.3 Computation of Learned Approximation

How to compute the learned approximation 𝒜^nu1 using the training set 𝒵 for n1? First of all, observe that since 𝒫nu1Vn, the projection 𝒫nu1 has the following representation:

𝒫nu1=j=1ncju1,j,

where the coefficients cj can be determined from the conditions 𝒫nu1-u1,u1,i=0 for i=1,,n. These conditions can be written in the form of the system of linear equations for the coefficients cj

j=1ncju1,i,u1,j=u1,u1,i,i=1,,n.

Denote the matrix corresponding to the above linear system as 𝐏n, i.e. the elements of 𝐏n are

(𝐏n)ij=u1,i,u1,jfor i,j=1,,n.

Further, denote the vector of unknowns as 𝐜n, and the right-hand side as 𝐮n, i.e.

(𝐜n)i=ciand(𝐮n)i=u1,u1,ifor i=1,,n.

The matrix 𝐏n is the Gram matrix of the functions in 𝐔1,n, and it is invertible if the set 𝐔1,n is linearly independent. Since the operator 𝒰1 is invertible, the set 𝐔1,n is linearly independent if the set {fi:i=1,,n} is linearly independent, and for the following, we assume that this is the case.

Note that the matrix 𝐏n does not depend on the limited view wave data u1 that we want to extend. Therefore, the inverse matrix 𝐏n-1 can be precomputed once the set of the learning inputs 𝐔1,n is given. This will make the determination of the coefficients cj very fast.

Finally, with the coefficients cj in (3.3), i.e. 𝐜n=𝐏n-1𝐮n, the approximation 𝒜^nu1 is calculated as follows:

𝒜^nu1=𝒜𝒫nu1=𝒜(j=1ncju1,j)=j=1ncju2,j=j=1ncj𝒰2fj.

4 Approximate Reconstructions and Their Error Analysis

For obtaining an approximate reconstruction of f using the limited view data u1=𝒰1f and the formula 𝒢d, we can now proceed as follows. First, we extend the limited view data u1 to the whole boundary Ω using the learned extension operator 𝒜^n in the following way:

u^n(x,t)={u1(x,t)if xΓ1,(𝒜^nu1)(x,t)if xΓ2.

Then we apply the formula 𝒢d to this extended wave data u^n in order to obtain an approximate reconstruction f^n:

f^n=𝒢du^n.

Note that u^0 is obtained by extending the limited view data u1 to the whole boundary Ω with zero values on Γ2. As we already discussed, the corresponding approximate reconstruction f^0 contains significant errors, and it is desirable to have better reconstructions of f using u1. Additionally, one may desire that the reconstruction f^n improves as n increases.

In the following theorem, we estimate the 2-error of the approximation of 𝒜u1 by 𝒜^nu1 and of the approximation of f by f^n. From the derived estimates, we see that the above aims can be realized if the training functions fi, i=1,,n, are chosen appropriately.

Theorem 2.

Let a set of linearly independent training functions {fi:i=1,,n}L2(Ω1) be given, and denote Wn:={i=1ncifi:ciR}, W0:={0}L2(Ω1). Define the training limited view wave data u1,i:=U1fi, the corresponding linear subspace Vn in (3.1), and the learned extension operator A^n in (3.2). Consider a function fL2(Ω1), its limited view wave data u1:=U1f, and its approximation f^n defined in (4.1). Then the following L2-error estimate for the unobservable data holds:

𝒜u1-𝒜^nu1𝒜𝒰1mingWnf-g.

If additionally, the domain Ω is such that (2.3) holds, then we have the following L2-error estimate for the reconstruction:

f-f^n𝒢d𝒜𝒰1mingWnf-g.

Proof.

We first prove (4.2). From the definition of the operator 𝒜^n, we have

𝒜u1-𝒜^nu1=𝒜𝒰1f-𝒜𝒫n𝒰1f𝒜𝒰1f-𝒫n𝒰1f.

From the properties of the projection operators, we also have

𝒰1f-𝒫n𝒰1f=minhVn𝒰1f-h.

For an element hVn, there are unique constants ci, i=1,,n, such that

h=i=1nciu1,i=i=1nci𝒰1fi=𝒰1(i=1ncifi),

and therefore, there exists an element gWn such that h=𝒰1g. Using this fact, we can estimate

minhVn𝒰1f-h=mingWn𝒰1f-𝒰1g𝒰1mingWnf-g.

Then combining (4.4), (4.5), (4.6), we obtain estimate (4.2) for the 2-error 𝒜u1-𝒜^nu1.

Now, consider (4.3). Using (2.3) and (4.1), we have

f-f^n=𝒢d𝒰f-𝒢du^n𝒢d𝒰f-u^n.

Since (𝒰f)(x,t)=u^n(x,t)=u1(x,t) for xΓ1, it follows that

𝒰f-u^n=𝒰2f-𝒜^nu1=𝒜u1-𝒜^nu1.

Thus, the error estimate (4.3) is obtained from (4.7), (4.8), and the error estimate (4.2). ∎

Remark 1.

Let 𝒬n:2(Ω1)Wn be the orthogonal projection on Wn in the space 2(Ω1). Then, since we have mingWnf-g=f-𝒬nf, we can write f-𝒬nf instead of mingWnf-g in (4.2) and (4.3).

As we see from Theorem 2, the estimates of the 2-errors given by our learning procedure depend on the minimal distance from the unknown function f to the linear subspace Wn defined by the training functions fi. This gives us an indication for the choice of the training functions. Namely, one should choose the training functions fi such that the unknown function f can be well approximated by their linear combination.

Estimates (4.2) and (4.3) also allow us to state the condition for the exact approximation given by our learning procedure and for the convergence of the learned approximation when the number of the training functions n goes to infinity. We present these conditions in the following two corollaries.

Corollary 1.

If fWn, then the learned approximation A^nu1 and the reconstruction f^n are exact, i.e.

𝒜u1-𝒜^nu1=f-f^n=0.

Corollary 2.

If n1Wn¯=L2(Ω1), then the learned approximation A^nu1 and the reconstruction f^n converge respectively to Au1 and f as n, i.e.

limn𝒜u1-𝒜^nu1=limnf-f^n=0.

Let us now compare the errors of the approximations f^n with n1 and f^0. The 2-error estimates (4.2) and (4.3) for n=0 become

𝒜u1-0𝒜𝒰1f,
f-f^0𝒢d𝒜𝒰1f.

Comparing the error estimates (4.2) and (4.3) for the learned approximations with n1 and the error estimates (4.9) and (4.10) for the approximations using zero extension of the limited view wave data, one sees that these error estimates differ regarding the following factors:

n(f):=mingWnf-g,0(f):=f,

correspondingly for learned approximations with n1 and approximation using zero extension.

The factors (4.11) can be seen as indicators for the expected approximation quality of the considered algorithms. For a fixed non-zero function f, the factor 0(f) is a fixed non-zero value, while the factor n(f) can be zero, or can be made arbitrary small, see Corollaries 1 and 2. Therefore, the approximation quality of the learned approximations is expected to be better than of the approximations using zero extension of the data. This expectation will be confirmed by the numerical results in the next section. In fact, one can show (see Remark 2 below) that the factor n(f) is always less than or equal to the factor 0(f), and the strict inequality n(f)<0(f) holds under rather mild conditions on the function f and the training functions fi. Generally, this condition can be expected to hold in practice.

Remark 2.

Using properties of the projection operators in Hilbert spaces, one can show that the sequence n(f) is non-increasing, i.e.

n(f)m(f)for n>m0.

If additionally

f,fi0for some i{m+1,,n},

then inequality (4.12) is strict, i.e.

n(f)<m(f)for n>m0.

Condition (4.13) is also necessary for (4.14), i.e. if (4.14) holds, then we have (4.13).

5 Numerical Results

In this section, we present results of the numerical realization of the proposed operator learning approach.

We consider the spatial dimension d=2, and we take the elliptical domain

Ω={(x1,x2)2:(x1a1)2+(x2a2)2<1},

with a1=2, a2=1. We use the following parametrization of the boundary:

Ω={(a1cosθ,a2sinθ):θ[-π,π)},

and we assume that the unobservable part of the boundary is (see Figure b (a))

Γ2={(a1cosθ,a2sinθ):θ[0.97,2.17)}.

Thus, approximately 19 % of the angular values are missing.

We work with the function f presented in Figure b (a). Its numerical full view wave boundary data u=𝒰f is given in Figure b (b), and we use the corresponding limited view wave boundary data u1=𝒰1f. The observation boundary Γ1 is discretized such that the distance between two consecutive points is in the interval [0.0099,0.0101]. We take the time step size as 0.01.

Left: the function f that we use in our numerical experiments and the chosen observation boundary Γ1. Right: the corresponding numerical full view wave boundary data 𝒰f. The region between two white vertical lines corresponds to the unknown part of the data on the unobservable part of the boundary Γ2.

(a)
(a)

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

(b)
(b)

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

We further assume that we know a rectangular region

K={(x1,x2)2:-1.25x1<0.5,-0.7x2<0.1752}

that contains the support of f (Figure 4 (top and bottom)). We use this region K for defining training functions fi. Namely, we consider partitions of the region K into squares Ki, i{1,,n}. The square Ki contains points (x1,x2)2 such that

-1.25+(inh-1)wnwx1<-1.25+inhwnw,
-0.7+(imodnh-1)hnhx2<-0.7+(imodnh)hnh,

where w=1.75 (width of K), h=0.8752 (height of K), nw=2n, nh=nw2 (see Figure 4 (middle)). Then we define the training function fi as the indicator function of the square Ki. We take the number of the training functions in the form n=n1×n2, where n1 and n2 are the numbers of the partitioning intervals along the coordinate x1 and x2 correspondingly. We present the numerical results for n=4×2,8×4,16×8,32×16.

Let us note that we use the rectangular region K for illustration purpose. If the region containing supp(f) is not known, then one may consider squares filling the whole subset Ω1 of the detection region 𝔻(Γ1). Further, note that other type of basis functions can be used in a similar manner. Kaiser–Bessel functions, which are frequently used in computed tomography (see, e.g., [32, 46, 43]), would be another reasonable choice.

Figure 4
Figure 4

Top: the rectangular region K containing supp(f). Middle: the example of the partition of K into 8×4 squares. The training functions fi are numbered starting from the bottom-left square from bottom to top and from left to right. Bottom: the position of supp(f) in K with the partition of K into 8×4 squares.

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

The extended limited view data u^n using the learned extension operator 𝒜^n for the considered values of n are presented in Figure 5. We observe that as n increases, the extended data u^n approaches the full view data u in Figure b (a). Note that the chosen training functions fi satisfy the condition of Corollary 2. Therefore, the approach of u^n to the full view data u is in agreement with our theoretical analysis.

The reconstructions f^n using the extended data u^n are presented in Figure 6 (second and third rows). For comparison purpose, we also present the reconstruction f^ using the full view wave boundary data u, and the reconstruction f^0 using the zero extended data u^0 (Figure 6 (first row)). We evaluate the reconstructions at the points from the discrete set

Ωh:={(-2.2+n1h,-2.2+n2h)2:n1,n2{0,1,,300}}Ω,

with h=11750. We also consider the discrete 2-error of a reconstruction f^* defined as follows:

E2(f^*):=(xΩh|f(x)-f^*(x)|2h2)1/2.

Let us discuss the reconstructions in Figure 6. First of all, as expected, one observes strong artifacts in the reconstruction f^0, especially outside of supp(f). These artifacts are considerably corrected in the reconstruction f^4×2, and as the number of the training functions n increases, the artifacts become weaker such that the reconstruction f^32×16 is very similar to the reconstruction f^. This observation is also reflected in E2-errors that are presented in Figure 7. Note that f^ differs from f due to the discretization error of the numerical realization of the formula 𝒢2. Thus, as in the case of the data u^n, the approach of f^n to f is in agreement with Corollary 2.

Finally, in Table 1, we present the calculation times for the parts involved in the proposed reconstruction approach. Our numerical results are performed with MATLAB version R2015b on the PC lenovo e31 with four processors Intel(R) Xeon(R) CPU 3.20 GHz. We see that the most time consuming part is the calculation of the matrix 𝐏n-1, which is used for solving the system of linear equations (3.4). Here, the calculation of u1,i=𝒰1fi is the most computationally expensive. But for a given set of the training functions fi, u1,i and the matrix 𝐏n-1 have to be calculated only once and prior to the actual image reconstruction process.

Table 1

Calculation times in seconds for the parts involved in the proposed reconstruction approach.

n𝐏n-1𝒜^nu1𝒢2
4×21179.730.534.20
8×44707.310.683.55
16×819036.231.413.90
32×1675874.876.074.33

The calculation of the learned data extension 𝒜^nu1 is fast. In particular, for the biggest considered number n=32×16 of the training functions, the calculation time for 𝒜^nu1 is near the calculation time for the formula 𝒢2. Thus, our proposed operator learning approach fulfills the requirements that we stated at the beginning of Section 3.2. Namely, the closeness of the approximation 𝒜^nu1 to 𝒜u1, and the fast evaluation of 𝒜^nu1 are realized.

Figure 5
Figure 5Figure 5Figure 5Figure 5

The extended limited view data u^n using the learned extension operator 𝒜^n for n=4×2,8×4,16×8,32×16 (from left to right and from top to bottom). The gray scaling is as in Figure b (a).

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

Figure 6
Figure 6Figure 6Figure 6Figure 6Figure 6Figure 6

From left to right and from top to bottom: the reconstructions f^, f^0, and f^n, for n=4×2,8×4,16×8,32×16.The gray scaling is as in Figure b (a).

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

Figure 7
Figure 7

E2-errors of the considered reconstructions f^, f^0, and f^n, for n=4×2,8×4,16×8,32×16.

Citation: Computational Methods in Applied Mathematics 19, 4; 10.1515/cmam-2018-0008

6 Conclusion and Outlook

In this paper, we demonstrated that an approximate extension of the limited view data in PAT can be realized using an operator learning approach. Our numerical results show that the learned extension of the limited view data with a good approximation quality and a low computational cost is possible. A good approximation quality is especially achieved for the biggest number n=32×16 of considered training functions. This makes the proposed learned data extension attractive for the algorithms that are designed for the full view data. As an example, we demonstrated a satisfactory performance of a reconstruction formula with the proposed learned data extension.

It could be interesting to look at the behavior of the proposed learned data extension without knowledge of a rectangular region K containing supp(f). As we already noted, in this case, one could consider partitions of the whole detection region Ω1. Also other training functions, such as generalized Kaiser–Bessel functions (see, e.g., [32, 46, 43]), can be tried.

It is appealing to consider a comparison of the reconstruction quality and computation time of the proposed reconstruction approach and iterative reconstruction algorithms. Implementation of the proposed learned extension of the limited view data to three spatial dimensions is an interesting aspect of future research. In this case, the choice of the generalized Kaiser–Bessel functions as the training functions fi is particularly convenient because for them the wave data u1,i=𝒰1fi, u2,i=𝒰2fi are known analytically (see, e.g., [46]). This makes the determination of the entries of the matrix 𝐏n fast. Also, the solution of the system of linear equations (3.4) can be done either using iterative methods, such as conjugate gradient method, or an approximate inverse matrix to 𝐏n can be determined.

Finally, it seems to be worth to examine applications of the presented operator learning approach to the limited data problems in other tomographic modalities, such as sparse angle or region of interest computed tomography.

Acknowledgements

Sergiy Pereverzyev Jr. would like to thank Alessandro Verri, Vera Kurkova, Linh Nguyen, Jürgen Frikel, Xin Guo, Ding-Xuan Zhou, and members of Ding-Xuan Zhou’s group at the City University of Hong Kong for discussions concerning this work.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    M. Agranovsky, D. Finch and P. Kuchment, Range conditions for a spherical mean transform, Inverse Probl. Imaging 3 (2009), no. 3, 373–382.

    • Crossref
    • Export Citation
  • [2]

    M. A. Alvarez, L. Rosasco and N. D. Lawrence, Kernels for vector-valued functions: A review, Found. Trends Mach. Learn. 4 (2012), no. 3, 195–266.

    • Crossref
    • Export Citation
  • [3]

    G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2006), no. 2, 681–692.

    • Crossref
    • Export Citation
  • [4]

    L. L. Barannyk, J. Frikel and L. V. Nguyen, On artifacts in limited data spherical Radon transform: Curved observation surface, Inverse Problems 32 (2016), no. 1, Article ID 015012.

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    P. Beard, Biomedical photoacoustic imaging, Interf. Focus 1 (2011), no. 4, 602–631.

    • Crossref
    • Export Citation
  • [6]

    P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors, Inverse Problems 23 (2007), no. 6, S65–S80.

    • Crossref
    • Export Citation
  • [7]

    P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Rev. E 75 (2007), no. 4, Article ID 046706.

  • [8]

    R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. II, Interscience, New York 1962.

  • [9]

    D. Finch, M. Haltmeier and R. Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math. 68 (2007), no. 2, 392–412.

    • Crossref
    • Export Citation
  • [10]

    D. Finch, S. K. Patch and R. Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal. 35 (2004), no. 5, 1213–1240.

    • Crossref
    • Export Citation
  • [11]

    D. Finch and R. Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems 22 (2006), no. 3, 923–938.

    • Crossref
    • Export Citation
  • [12]

    D. Finch and R. Rakesh, Recovering a function from its spherical mean values in two and three dimensions, Photoacoustic Imaging and Spectroscopy, CRC Press, Boca Raton (2009), 77–88.

  • [13]

    J. Frikel and E. T. Quinto, Characterization and reduction of artifacts in limited angle tomography, Inverse Problems 29 (2013), no. 12, Article ID 125007.

  • [14]

    J. Frikel and E. T. Quinto, Artifacts in incomplete data tomography with applications to photoacoustic tomography and sonar, SIAM J. Appl. Math. 75 (2015), no. 2, 703–725.

    • Crossref
    • Export Citation
  • [15]

    H. Grün, T. Berer, P. Burgholzer, R. Nuster and G. Paltauf, Three-dimensional photoacoustic imaging using fiber-based line detectors, J. Biomed. Optics 15 (2010), no. 2, Article ID 021306.

  • [16]

    M. Haltmeier, Frequency domain reconstruction for photo- and thermoacoustic tomography with line detectors, Math. Models Methods Appl. Sci. 19 (2009), no. 2, 283–306.

    • Crossref
    • Export Citation
  • [17]

    M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, Comput. Math. Appl. 65 (2013), no. 7, 1025–1036.

    • Crossref
    • Export Citation
  • [18]

    M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, SIAM J. Math. Anal. 46 (2014), no. 1, 214–232.

    • Crossref
    • Export Citation
  • [19]

    M. Haltmeier and L. V. Nguyen, Analysis of iterative methods in photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci. 10 (2017), no. 2, 751–781.

    • Crossref
    • Export Citation
  • [20]

    M. Haltmeier and S. Pereverzyev Jr., Recovering a function from circular means or wave data on the boundary of parabolic domains, SIAM J. Imaging Sci. 8 (2015), no. 1, 592–610.

    • Crossref
    • Export Citation
  • [21]

    M. Haltmeier and S. Pereverzyev Jr., The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces, J. Math. Anal. Appl. 429 (2015), no. 1, 366–382.

    • Crossref
    • Export Citation
  • [22]

    T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer Ser. Statist., Springer, New York, 2009.

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    G. T. Herman, Fundamentals of Computerized Tomography. Image Reconstruction from Projections, 2nd ed., Adv. Pattern Recognit., Springer, Dordrecht, 2009.

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    Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems 24 (2008), no. 5, Article ID 055006.

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    C. Huang, K. Wang, L. Nie, L. V. Wang and M. A. Anastasio, Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media, IEEE Trans. Med. Imag. 32 (2013), no. 6, 1097–1110.

    • Crossref
    • Export Citation
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    P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), no. 2, 191–224.

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    P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging. Vol. 1, 2, 3, Springer, New York (2015), 1117–1167.

  • [28]

    L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems 27 (2011), no. 2, Article ID 025012.

  • [29]

    L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems 23 (2007), no. 6, S11–S20.

    • Crossref
    • Export Citation
  • [30]

    L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems 23 (2007), no. 1, 373–383.

    • Crossref
    • Export Citation
  • [31]

    C. Li and L. V. Wang, Photoacoustic tomography and sensing in biomedicine, Phys. Med. Biol. 54 (2009), no. 19, R59–R97.

    • Crossref
    • PubMed
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CMAM considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. The journal is interdisciplinary while retaining the common thread of numerical analysis, readily readable and meant for a wide circle of researchers in applied mathematics.

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