## 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.

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

### 2.1 Reconstruction Problem

Let

Here *t*,
and *x*.
Then the reconstruction problem in PAT consists in recovering the unknown function

where

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
*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 *u*, the formula *d* is even or odd.

If

Here

is the differentiation operator with respect to

In the case of odd dimension

with constant

The formula *f* with support in Ω from data

It should be noted that the formula *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

In the following, we will work with functions *f* from its wave data

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 *f* in (2.1) can be stably recovered from data on *x* such that any line going through *x* intersects

Let us mathematically specify the stable recovery of *f*. Let

*The operator *

It is sufficient to show the two-side estimate

for some constants

To show the left-hand estimate, we decompose
*T* is larger than the diameter of Ω.
Since the operator

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

Denote

Recall that in order to give the exact reconstruction, the formula *u*,
which is given for all *u* given on

At the same time, the use of formula *f* can be attractive from various points of view. For example, as we already pointed out, the reconstruction using a numerical realization of

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

The operator

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

For any

and let

Note that

is bounded.
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

The matrix

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

Note that *f* using *n* increases.

In the following theorem, we estimate the *f* by

*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

Then combining (4.4), (4.5), (4.6), we obtain estimate (4.2) for the

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

Since

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

Let

As we see from Theorem 2, the estimates of the *f* to the linear subspace *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.

*If *

*If *

*f*as

Let us now compare the errors of the approximations

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 *f* and the training functions

Using properties of the projection operators in Hilbert spaces, one can show that
the sequence

If additionally

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

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

with

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 *K* into squares

where *K*), *K*),

Let us note that we use the rectangular region *K* for illustration purpose.
If the region containing

The extended limited view data *n*
are presented in Figure 5. We observe that as *n* increases, the extended data *u* in Figure b (a). Note that the chosen training functions *u* is in agreement with our theoretical analysis.

The reconstructions *u*, and the
reconstruction

with

Let us discuss the reconstructions in Figure 6. First of all, as expected, one observes strong artifacts
in the reconstruction *n* increases, the artifacts become weaker such that the reconstruction *f* due to the discretization error of the numerical realization of the formula *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

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

1179.73 | 0.53 | 4.20 | |

4707.31 | 0.68 | 3.55 | |

19036.23 | 1.41 | 3.90 | |

75874.87 | 6.07 | 4.33 |

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.

## References

- [1]↑
M. Agranovsky, D. Finch and P. Kuchment, Range conditions for a spherical mean transform, Inverse Probl. Imaging 3 (2009), no. 3, 373–382.

- [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.

- [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.

- [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.

- [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.

- [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.

- [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.

- [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.

- [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.

- [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.

- [17]↑
M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, Comput. Math. Appl. 65 (2013), no. 7, 1025–1036.

- [18]↑
M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, SIAM J. Math. Anal. 46 (2014), no. 1, 214–232.

- [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.

- [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.

- [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.

- [22]↑
T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, 2nd ed., Springer Ser. Statist., Springer, New York, 2009.

- [23]↑
G. T. Herman, Fundamentals of Computerized Tomography. Image Reconstruction from Projections, 2nd ed., Adv. Pattern Recognit., Springer, Dordrecht, 2009.

- [24]↑
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.

- [25]↑
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.

- [26]↑
P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), no. 2, 191–224.

- [27]↑
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.

- [30]↑
L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems 23 (2007), no. 1, 373–383.

- [31]↑
C. Li and L. V. Wang, Photoacoustic tomography and sensing in biomedicine, Phys. Med. Biol. 54 (2009), no. 19, R59–R97.

- [32]↑
S. Matej and R. M. Lewitt, Practical considerations for 3-D image reconstruction using spherically symmetric volume elements, IEEE Trans. Med. Imag. 15 (1996), no. 1, 68–78.

- [33]↑
C. A. Micchelli and M. Pontil, On learning vector-valued functions, Neural Comput. 17 (2005), no. 1, 177–204.

- [34]↑
F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging 6 (2012), no. 2, 315–320.

- [35]↑
L. V. Nguyen, On a reconstruction formula for spherical Radon transform: A microlocal analytic point of view, Anal. Math. Phys. 4 (2014), no. 3, 199–220.

- [36]↑
L. V. Nguyen, On artifacts in limited data spherical Radon transform: Flat observation surfaces, SIAM J. Math. Anal. 47 (2015), no. 4, 2984–3004.

- [37]↑
G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors, Inverse Problems 23 (2007), no. 6, S81–S94.

- [38]↑
G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Photoacoustic tomography using a Mach–Zehnder interferometer as an acoustic line detector, App. Opt. 46 (2007), no. 16, 3352–3358.

- [39]↑
G. Paltauf, J. A. Viator, S. A. Prahl and S. L. Jacques, Iterative reconstruction algorithm for optoacoustic imaging, J. Acoust. Soc. Am. 112 (2002), no. 4, 1536–1544.

- [40]↑
S. K. Patch, Thermoacoustic tomography – Consistency conditions and the partial scan problem, Phys. Med. Biol. 49 (2004), 2305–2315.

- [41]↑
S. K. Patch, Photoacoustic and thermoacoustic tomography: Consistency conditions and the partial scan problem, Photoacoustic Imaging and Spectroscopy, CRC Press, Boca Raton (2009), 103–116.

- [42]↑
A. Rosenthal, V. Ntziachristos and D. Razansky, Acoustic inversion in optoacoustic tomography: A review, Curr. Med. Imag. Rev. 9 (2013), no. 4, 318–336.

- [43]↑
J. Schwab, S. Pereverzyev, Jr. and M. Haltmeier, A Galerkin least squares approach for photoacoustic tomography, SIAM J. Numer. Anal. 56 (2018), no. 1, 160–184.

- [44]↑
P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), no. 7, Article ID 075011.

- [45]↑
P. Stefanov and G. Uhlmann, Is a curved flight path in SAR better than a straight one?, SIAM J. Appl. Math. 73 (2013), no. 4, 1596–1612.

- [46]↑
K. Wang, R. W. Schoonover, R. Su, A. Oraevsky and M. A. Anastasio, Discrete imaging models for three-dimensional optoacoustic tomography using radially symmetric expansion functions, IEEE Trans. Med. Imag. 33 (2014), no. 5, 1180–1193.

- [47]↑
J. Xia, J. Yao and L. V. Wang, Photoacoustic tomography: Principles and advances, Prog. Electromagn. Res. 147 (2014), 1–22.

- [48]↑
M. Xu and L. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E 71 (2005), no. 1, Article ID 0167067.

- [49]↑
M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Rev. Sci. Instruments 77 (2006), no. 4, Article ID 041101.

- [50]↑
Y. Xu, L. V. Wang, G. Ambartsoumian and P. Kuchment, Reconstructions in limited-view thermoacoustic tomography, Med. Phys. 31 (2004), no. 4, 724–733.

- [51]↑
Y. Xu, M. Xu and L. V. Wang, Exact frequency-domain reconstruction for thermoacoustic tomography–II: Cylindrical geometry, IEEE Trans. Med. Imag. 21 (2002), 829–833.

- [52]↑
L. Yao and H. Jiang, Photoacoustic image reconstruction from few-detector and limited-angle data, Biomed. Opt. Express 2 (2011), no. 9, 2649–2654.

- [53]↑
G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci. 7 (2009), no. 3, 665–678.