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

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.

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 Ω⊆ℝd be a bounded domain with a smooth boundary ∂⁡Ω, where d≥2 denotes the spatial dimension. Further, let Cc∞⁢(Ω) be the set of all smooth functions f:ℝd→ℝ that are compactly supported in Ω. In PAT, one is interested to recover an unknown function f∈Cc∞⁢(Ω) from the solution of the wave equation given on parts of the boundary of Ω. Let us mathematically specify this reconstruction problem.

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

Setting of LVP.

Let us mathematically specify the stable recovery of f. Let Ω1 be an open set with Ω1¯⊊𝔻⁢(Γ1). The stable recovery holds for f∈ℒ2⁢(Ω1), and it is formulated in the following theorem. Note that the space ℒ2⁢(Ω1) is identified with the set of all functions in ℒ2⁢(ℝd) 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:Y1→L2⁢(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

(2.4)a⁢∥𝒰1⁢f∥ℒ2≤∥f∥ℒ2≤b⁢∥𝒰1⁢f∥ℒ2 for all ⁢f∈Cc∞⁢(Ω1)

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

To show the left-hand estimate, we decompose 𝒰1⁢f=χ[0,T]⁢𝒰1⁢f+χ(T,∞)⁢𝒰1⁢f, 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]⁢𝒰1⁢f∥ℒ2≤c1⁢∥f∥ℒ2 for some constant c1. Moreover, the explicit formulas for 𝒰1⁢f (see, e.g., [8]) imply also that ∥χ(T,∞)⁢𝒰1⁢f∥ℒ2≤c1⁢∥f∥ℒ2, 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 f∈ℒ2⁢(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:=ℒ2⁢(Γ2×(0,∞)). From the boundedness of the operator 𝒰:ℒ2⁢(Ω0)→𝕐, we can deduce the boundedness of the operator 𝒰2:ℒ2⁢(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 x∈∂⁡Ω (see (2.3)). In spite of the above discussed stable recoverability of f∈ℒ2⁢(Ω1) from equation (2.1), the use of formula 𝒢d on the limited view data u given on Γ1⊊∂⁡Ω 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 Γ1⊊∂⁡Ω 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.

4 Approximate Reconstructions and Their Error Analysis

For obtaining an approximate reconstruction of f using the limited view data u1=𝒰1⁢f 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:

Then we apply the formula 𝒢d to this extended wave data u^n in order to obtain an approximate reconstruction f^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 ℒ2-error of the approximation of 𝒜⁢u1 by 𝒜^n⁢u1 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.

As we see from Theorem 2, the estimates of the ℒ2-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.

Let us now compare the errors of the approximations f^n with n≥1 and f^0. The ℒ2-error estimates (4.2) and (4.3) for n=0 become

Comparing the error estimates (4.2) and (4.3) for the learned approximations with n≥1 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:

correspondingly for learned approximations with n≥1 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 ℰ0⁢(f) is a fixed non-zero value, while the factor ℰn⁢(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 ℰn⁢(f) is always less than or equal to the factor ℰ0⁢(f), and the strict inequality ℰn⁢(f)<ℰ0⁢(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.

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

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

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

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=𝒰⁢f is given in Figure b (b), and we use the corresponding limited view wave boundary data u1=𝒰1⁢f. 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.

Figure 3

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 𝒰⁢f. The region between two white vertical lines corresponds to the unknown part of the data on the unobservable part of the boundary Γ2.

(a)

(b)

We further assume that we know a rectangular region

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)∈ℝ2 such that

where w=1.75 (width of K), h=0.8752 (height of K), nw=2⁢n, 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

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.

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

with h=11750. We also consider the discrete ℒ2-error of a reconstruction f^* defined as follows:

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=𝒰1⁢fi 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.

The calculation of the learned data extension 𝒜^n⁢u1 is fast. In particular, for the biggest considered number n=32×16 of the training functions, the calculation time for 𝒜^n⁢u1 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^n⁢u1 to 𝒜⁢u1, and the fast evaluation of 𝒜^n⁢u1 are realized.

Figure 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).

Figure 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).

Figure 7

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

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=𝒰1⁢fi, u2,i=𝒰2⁢fi 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.