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 . 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., ). 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.
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
2.1 Reconstruction Problem
Let us denote the unobservable part of the boundary as
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
is the differentiation operator with respect to
In the case of odd dimension
It should be noted that the formula
In the following, we will work with functions
As we already mentioned, this is, for example, the case for circular and elliptical domains. In such a situation, it can be shown that
2.3 Limited View Problem
In practice, the wave data u is frequently given on a subset
Let us mathematically specify the stable recovery of f. Let
It is sufficient to show the two-side estimate
for some constants
To show the left-hand estimate, we decompose
The right-hand side estimate can be found in [19, Theorem 3.4]. The required visibility condition
is satisfied due to the assumption that
It is worth to mention that despite the boundedness of
Recall that in order to give the exact reconstruction, the formula
At the same time, the use of formula
An extension of the limited view data u from the observable part of the boundary
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
With the introduced extension operator
3.2 Proposed Learned Extension Operator
In this paper, we propose to construct an operator
Our construction of the approximate operator
So, how to construct (or, using the terminology of the statistical learning, how to learn) an approximation
Further, note that for all
3.3 Computation of Learned Approximation
How to compute the learned approximation
where the coefficients
Denote the matrix corresponding to the above linear system as
Further, denote the vector of unknowns as
Note that the matrix
Finally, with the coefficients
4 Approximate Reconstructions and Their Error Analysis
For obtaining an approximate reconstruction of f using
the limited view data
Then we apply the formula
In the following theorem, we estimate the
Let a set of linearly independent training functions
If additionally, the domain Ω is such that (2.3) holds, then we have the following
We first prove (4.2). From the definition of the operator
From the properties of the projection operators, we also have
For an element
and therefore, there exists an element
As we see from Theorem 2, the estimates of the
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.
Comparing the error estimates (4.2) and (4.3) for the learned approximations with
correspondingly for learned approximations with
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
Using properties of the projection operators in Hilbert spaces, one can show that
then inequality (4.12) is strict, i.e.
5 Numerical Results
In this section, we present results of the numerical realization of the proposed operator learning approach.
We consider the spatial dimension
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
Left: the function f that we use in our numerical experiments and the chosen observation boundary
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
Let us note that we use the rectangular region K for illustration purpose.
If the region containing
The extended limited view data
Let us discuss the reconstructions in Figure 6. First of all, as expected, one observes strong artifacts
in the reconstruction
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
Calculation times in seconds for the parts involved in the proposed reconstruction approach.
The calculation of the learned data extension
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
It could be interesting to look at the behavior of the proposed learned data extension without knowledge of a rectangular
region K containing
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
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.
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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