# An Efficient Numerical Algorithm for the Inversion of an Integral Transform Arising in Ultrasound Imaging

## Abstract

We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.

### Keywords

Circular Radon transform Elliptical Radon transform Volterra integral equations Truncated singular value decomposition Theromoacoustic tomography Photoacoustic tomography Ultrasound reflectivity imaging Intravascular imaging Radar imaging## 1 Introduction

In several imaging modalities such as ultrasound reflectivity imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing and radar imaging, circular or elliptical Radon transforms arise naturally. These are transforms that associate to a function, its integrals along a family of circles or ellipses.

In ultrasound imaging, ultrasonic pulses emitted from a transducer moving along a curve (typically a circle), propagate inside the medium and reflect off inhomogeneities which are measured by the same or a different moving transducer. Assuming that the speed of sound propagation within the medium is constant and that the medium is weakly reflecting, the pulses registered at the receiver transducer is the superposition of all the pulses reflected from those inhomogeneities such that the total distance traveled by the reflected pulse is a constant. This leads to the consideration of an integral transform of a function on a plane (which models the image to be reconstructed), given its integrals along a family of circles (for the case of identical emitter/receiver) or ellipses (for spatially separated emitter/receiver). The goal is to recover an image of the medium given these integrals. In other words, one is interested in the inversion of a circular or elliptical Radon transform. For a detailed discussion of the mathematical model of ultrasound imaging, we refer the reader to [25, 26, 27]. Similarly, the mathematical formulation of problems in thermoacoustic and photoacoustic tomography, non-desctructive testing, intravascular imaging, radar and sonar imaging all lead to inversion of circular or elliptical Radon transforms. For details, we refer the reader to the following references [4, 8, 23].

The inversion of circular Radon transforms has been extensively studied by several authors [1, 2, 7, 9, 10, 12, 13, 14, 15, 16, 29, 30, 31, 32, 37], and to a lesser extent, that of elliptical Radon transforms [3, 17, 24, 28, 36, 39]. All these papers deal with full data in the radial direction. In some imaging problems, full data in the radial direction is not available, as is the case of imaging the region surrounding a bone. To this end, Ambartsoumian, Gouia-Zarrad and Lewis in [5] found explicit inversion formulas for the circular Radon transform with circular acquisition geometry (one of the most widely used ways of collecting data) when half of the data in the radial variable is available. These results were recently generalized by Ambartsoumian and Krishnan for a class of elliptical and circular Radon transforms in [6]. The inversion formulas in these papers are given for three cases: support of the function is inside, outside and on both sides of the acquisition circle. The case when the support is inside the circle of acquisition is of importance in ultrasound reflectivity imaging, thermoacoustic and photoacoustic tomography, and non-destructive testing. When the support is outside and on both sides of the acquisition circle, the inversion formulas are applicable in intravascular and radar imaging, respectively. Given the importance of these inversion formulas in several imaging modalities, efficient numerical inversion is of great interest. The main contribution of this paper is a novel implementation of the inversion formulas for a class of circular and elliptical Radon transforms with radially partial data obtained in these papers.

The inversion formulas given in [5, 6] were based on an inversion strategy due to Cormack [11] that involved Fourier series techniques. As shown in these papers, the \(n{\mathrm{th}}\) Fourier coefficient of the circular (elliptical) Radon transform data is related to the \(n{\mathrm{th}}\) Fourier coefficient of the unknown function by a Volterra-type integral equation of the first kind with a weakly singular kernel. This can be transformed to a Volterra-type integral equation of the second kind in which the singularity is removed [38]. It is well known that such an integral equation has a unique solution and this can be obtained by the Picard’s process of successive approximations, leading to an exact inversion formula given by an infinite series of iterated kernels; see [38].

In this paper, we numerically invert Volterra-type integral equation of the first kind adopting a numerical method given in [41] (see also [33]) and combine it with a truncated singular value decomposition to recover the Fourier coefficients of the unknown function from the circular or elliptical Radon transform data. The same method can also be implemented for numerical inversion of Volterra-type integral equation of the second kind proved in the papers [5, 6], but the numerical inversion is less accurate (see Remark 1). The numerical implementation of the exact inversion formula for the Volterra integral equation of the second kind involving an infinite series of iterated kernels is very unstable and implementing them is still an open problem. To the best of our knowledge, ours is the first successful numerical inversion of circular and elliptical Radon transforms for the circular geometry of acquisition with radially partial data, the theoretical results of which, as already mentioned, were presented in [5] and [6].

This paper is organized as follows. Section 2 gives the relevant theoretical background recalling the inversion formulas for the circular and elliptical Radon transforms based on which the numerical simulations in this paper are performed. Section 3 gives the numerical algorithm for inverting a Volterra-type integral equation of first kind and second kind. In Sect. 4, we present the numerical simulations, and Sect. 5 summarizes the results obtained.

## 2 Theoretical Background

As mentioned in the introduction, we consider two generalized Radon transforms in the plane: (a) Circular Radon transform and (b) Elliptical Radon transform.

### 2.1 Circular Radon Transform

### 2.2 Elliptical Radon Transform

The transforms \(R^{C}\) and \(R^{E}\) with radially partial data were considered in [5, 6] and explicit inversion formulas were given there. The inversion of these transforms leads to the inversion of a Volterra-type integral equation of the first kind with a weakly singular kernel. This in turn is transformed to an integral equation of the second kind with the singularity removed and the inversion of such an integral equation is given as an infinite series involving iterated kernels. Since we perform numerical inversion of Volterra-type integral equations of the first kind here, in each of the set up below, we only recall the corresponding integral equations of the first kind derived in [5, 6] instead of the explicit inversion formulas given as an infinite series.

*Int*,

*Ext*or

*Both*to denote the cases when the support of the function is an annular region in the interior, exterior or on both sides of the circle \(\partial B(0,R)\), respectively.

**Theorem 1**

- 1.
[5, Thm. 1] (Circular transform) Let \(0<\varepsilon <R\) and \(f(r,\theta )\) in polar coordinates be an unknown continuous function supported inside the annular region \(A(\epsilon ,R) = \lbrace {(r,\theta ):r\in (\epsilon ,R), \theta \in [0,2\pi ]\rbrace }\). If \(R^{C}f(\rho ,\phi )\) is known for \(\phi \in [0,2\pi ]\) and \(\rho \in [0,R-\epsilon ]\), then \(f(r,\theta )\) can be uniquely recovered in \(A(\epsilon ,R)\).

- 2.
[6, Thm.3.1] (Elliptical transform) Let \(f(r,\theta )\) be a continuous function supported inside the annulus \(A(\varepsilon ,b)\). Suppose \(R^{E}f(\rho ,\phi )\) is known for all \(\phi \in [0,2\pi ]\) and \(\rho \in (0,b-\varepsilon )\), then \(f(r,\theta )\) can be uniquely recovered.

- 1.Circular case, see [5]where$$\begin{aligned} g_{n}^{\mathrm {C,Int}}(\rho )= \int \limits _{0}^{\rho } \frac{K_{n}(\rho ,u)F_{n}(u)}{\sqrt{\rho -u}} \mathrm {d}u, \end{aligned}$$(1)$$\begin{aligned} F_{n}(u)&=f_{n}(R-u), \qquad T_{n}(x)=\cos (n\arccos (x))\nonumber \\ K_n(\rho ,u)&= \frac{4\rho (R-u)T_n\left[ \frac{(R-u)^2\!+\!R^{2}\!-\!\rho ^2}{2R(R-u)}\right] }{\sqrt{(u\!+\!\rho )(2R+\rho -u)(2R-\rho -u)}}. \end{aligned}$$(2)
- 2.Elliptical case, see [6]where$$\begin{aligned} g_{n}^{\mathrm {E,Int}}(\rho )= \int \limits _{0}^{\rho } \frac{K_{n}(\rho ,u)F_{n}(u)}{\sqrt{\rho -u}} \mathrm {d}u, \end{aligned}$$(3)$$\begin{aligned}&F_{n}(u)=f_{n}(b-u)\nonumber \\&K_{n}(\rho ,u)=\frac{\widetilde{K}_{n}(\rho ,b-u)\sqrt{\rho -u}}{\sqrt{a^{2}+b\rho -\sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-(b-u)^{2})}}} \end{aligned}$$(4)$$\begin{aligned}&\widetilde{K}_{n}(\rho ,r)=\frac{2ar T_{n}\left( \frac{b(\rho ^{2}+a^{2})-\rho \sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}{a^{2}r}\right) }{\sqrt{a^{2}+(\sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}-b\rho )}}\nonumber \\&\quad \times \frac{\sqrt{2R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})-2b\rho \sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}}{\sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}. \end{aligned}$$(5)

**Theorem 2**

(Functions supported in an annulus exterior to \(\partial B(0,R)\)) [5, Thm. 6]

(Circular transform) Let \(f(r,\theta )\) be a continuous function supported inside the annulus centered at \(0\): \(A(R,3R) = \lbrace {(r,\theta ):r\in (R,3R), \theta \in [0,2\pi ]\rbrace }\). If \(R^{C}f(\rho ,\phi )\) is known for \(\phi \in [0,2\pi ]\) and \(\rho \in [0,R_1]\), where \(0 <R_1<2R\) then \(f(r,\theta )\) can be uniquely recovered in \(A(R,R_1)\).

**Theorem 3**

- 1.
[6, Thm. 3.3](Circular transform) Let \(f(r,\theta )\) be a continuous function supported inside the disc \(D(0,R_{2})\) with \(R_{2}>2R\). Suppose \(R^{C}(\rho ,\phi )\) is known for all \(\phi \in [0,2\pi ]\) and \(\rho \in [R_{2}-R,R_{2}+R]\), then \(f(r,\theta )\) can be uniquely recovered in the annulus \(A(R_{1},R_{2})\) where \(R_{1}=R_{2}-2R\).

- 2.
[6, Thm. 3.4] (Elliptical Radon transform) Let \(f(r,\theta )\) be a continuous function supported inside the disc \(D(0,R_{2})\) with \(R_{2}>2b\). Suppose \(R^{E}(\rho ,\phi )\) is known for all \(\phi \in [0,2\pi ]\) and \(\rho \in [R_{2}-b,R_{2}+b]\), then \(f(r,\theta )\) can be uniquely recovered in \(A(R_{1},R_{2})\) where \(R_{1}=R_{2}-2b\).

- 1.Circular case, see [6]where$$\begin{aligned} g_{n}^{\mathrm {C,Both}}(R_{2}+R-\rho )= \int \limits _{0}^{\rho } \frac{K_{n}(\rho ,u)F_{n}(u)}{\sqrt{\rho -u}} \mathrm {d}u, \end{aligned}$$(8)$$\begin{aligned}&F_{n}(u)=f_{n}(R_{2}-u)\nonumber \\&K_{n}(\rho ,u)\nonumber \\&\quad =\frac{4(R_{2}+R-\rho )(R_{2}-u)\,T_{n}\left( \frac{(R_{2}-u)^{2}+R^{2}-(R_{2}+R-\rho )^{2}}{2(R_{2}-u)R}\right) }{\sqrt{(\rho -u)(2R_{2}-\rho -u)(2R+u-\rho )(2R +2R_{2}-\rho -u)}}.\nonumber \\ \end{aligned}$$(9)
- 2.Elliptical case, see [6]where$$\begin{aligned} g_{n}^{\mathrm {E,Both}}(R_{2}+b-\rho )= \int \limits _{0}^{\rho } \frac{K_{n}(\rho ,u)F_{n}(u)}{\sqrt{\rho -u}} \mathrm {d}u, \end{aligned}$$(10)$$\begin{aligned}&F_{n}(u)= f_{n}(R_{2}-u)\nonumber \\&K_{n}(\rho ,u) \nonumber \\&\quad =\frac{\widetilde{K}_{n}(R_{2}+b-\rho ,R_{2}-u)\sqrt{\rho -u}}{\sqrt{a^{2}\!+\!b(R_{2}\!+\!b\!-\!\rho )\!-\!\sqrt{R^{2}(R_{2}\!+\!b\!-\!\rho )^{2}\!+\!a^{2}(R^{2}\!-\!(R_{2}\!-\!u)^{2})}}}\nonumber \\ \end{aligned}$$(11)$$\begin{aligned}&\widetilde{K}_{n}(\rho ,r)=\frac{2ar T_{n}\left( \frac{b(\rho ^{2}+a^{2})-\rho \sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}{a^{2}r}\right) }{\sqrt{a^{2}+(\sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}-b\rho )}}\nonumber \\&\quad \times \frac{{\sqrt{2R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})-2b\rho \sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}}}{ {\sqrt{R^{2}\rho ^{2}+a^{2}(R^{2}-r^{2})}}.}\nonumber \\ \end{aligned}$$(12)

## 3 Numerical Algorithm

In this section, we describe the numerical scheme used to invert the integral equations listed in the previous section.

### 3.1 Fourier Coefficients of the Circular and Elliptical Radon Data in the Angular Variable

- 1.
Let \(N\) be even and \(\{\phi _1, \phi _2,\ldots ,\phi _N\}\) be a discretization of \(\phi \). We break the array \(g^{C}(\rho ,\phi _{k})\) for \(1\le k\le N\) into two equal length arrays, \(A\) for the odd numbered \(k\) and \(B\) for the even numbered \(k\). In other words, we let \(A = \{g^{C}(\rho ,\phi _{2j-1})\}\) and \(B = \{ g^{C}(\rho , \phi _{2j})\}\) for \(j = 1,2,\ldots ,N/2\).

- 2.
We then create a complex array \(h^{c}_{\rho }(j) = A(j) + \mathrm {i}B(j),\quad j = 1,2,\ldots ,N/2\).

- 3.
Next we perform a discrete FFT on \(h^c_\rho \) to get \(\widehat{h^{c}_\rho }(n), \quad n = 1, 2,\ldots , N/2\).

- 4.The Fourier series of \(g^{C}\) in the \(\phi \) variable is then given by$$\begin{aligned} g^{C}_n(\rho ) = {\left\{ \begin{array}{ll} \frac{1}{2}\left\{ \left( \widehat{h^c_\rho }(n)+ \overline{\widehat{h^c_\rho }(\frac{N}{2}-n+2)}\right) \right. \\ \left. -\mathrm {i}\left( \widehat{h^c_\rho }(n)- \overline{\widehat{h^c_\rho }(\frac{N}{2}-n+2)}\right) \cdot e^{\frac{2\pi {i}(n-1)}{N}}\right\} ,\\ \qquad \text {for } n = 1,\ldots ,\frac{N}{2}+1 \\ \widehat{h^c_\rho }(N -n+2), n = \frac{N}{2}+2,\ldots ,N. \end{array}\right. } \end{aligned}$$

### 3.2 Trapezoidal Product Integration Method [41]

**Theorem 4**

- 1.The functionsare continuous \(0\le u\le \rho \le R\),$$\begin{aligned} K_n(\rho ,u) \text { and } \frac{\partial }{\partial \rho } K_n({\rho },u) \end{aligned}$$
- 2.
\(K_n(\rho ,\rho )\ne 0\) for all \(\rho \in [0,R]\),

- 3.The functionis continuous for \(\rho \in [0,R].\)$$\begin{aligned} G_n(\rho )=\frac{\partial }{\partial \rho }\int _0^{\rho } \frac{g_n(s)}{(\rho -s)^{1/2}} \mathrm {d}s \end{aligned}$$

Under the assumptions of the theorem and using the method of kernel transformation [40, §50], one can transform Volterra equation of the first kind to Volterra equation of the second kind which has a unique solution (see [40, §3]). This derivation was used in the results of [5, 6] to provide analytical inversion formulas for a class of circular and elliptical Radon transforms with radially partial data. Such an exact inversion formula, as it turns out, is numerically unstable. Therefore, we approach the numerical inversion problem by solving (13) directly. We use the so-called trapezoidal product integration method proposed in [41]; see also [33]. For the sake of completeness, we briefly sketch this method below.

**Theorem 5**

### 3.3 Truncated Singular Value Decomposition (TSVD)

Furthermore \(\Vert A_n-A_{n,r}\Vert _{2}=\sigma _{r+1}\). Therefore \(A_{n,r}\) for \(r\) large, would be a good approximation to \(A_{n}\), but with high condition number, whereas if \(r\) is small, the condition number would be small but the error in the approximation \(||A - A_{n,r}||_{2}\) would be large.

Figure 3 shows the relation between the condition number of the truncated matrix \(A_{n,r}\) and the error \(||A_{n} - A_{n,r}||_{2}\), where the norm is defined by (21), for the Fourier coefficients \(n=10, 80, 120\) and \(180\). For simplicity we considered the matrix arising out of the integral equation (1). The other cases are similar.

### 3.4 Numerical Solution of Volterra-Type Integral Equation of the Second Kind

The method of the previous section can also be applied to Volterra-type integral equations of the second kind. It again leads to a non-singular matrix \(A_{n}\) with high condition number.

*Remark 1*

One could apply the numerical algorithm given in this paper to the matrix equation (24). However, the evaluation of \(G_n\) and \(L_n\) involves calculating derivative of an integral which leads to numerical instabilities and hence a high percentage of error. Furthermore, numerical computation of \(G_n\) and \(L_n\) is time consuming.

## 4 Numerical Results

### 4.1 Functions Supported in an Interior Annulus

This corresponds to the case when the object we are interested in reconstructing is supported in an annulus centered at 0 of the circle \(\partial B(0,R)\) and the circular and elliptical Radon transforms are along circles (ellipses) with center (foci) on \(\partial B(0,R)\), see Theorem 1. For the circular Radon transform case, the matrix \(A_{n}\) consists of entries coming from the kernel equation (2), whereas for the elliptical Radon transform case, the matrix entries come from (4).

In both the circular and elliptical transform cases discussed below, we notice a good recovery of the image near the origin which is a point of singularity. There is reduction in the number of artifacts as we increase the number of discretization points and hence the relative \(L^2\) error decreases with increasing refinement.

#### 4.1.1 Circular Radon Transform Data

#### 4.1.2 Elliptical Radon Transform Data

For the computations we assumed that an object is placed inside the annulus \(A(\epsilon ,b)\) where \(b=R\cos \alpha \) with \(\alpha ={30^{\circ }}\) is the length of the semi-minor axis. The resulting integral equation to be solved is given by (3) with the kernel \(K_{n}(\rho ,u)\) given by (4). We see that all the objects in the image have been reconstructed even with the coarser discretization of 400 points. The relative \(L^{2}\) errors between the Fig. 7a and b, and between the Fig. 7a and c are 14.2 and 10.6 %, respectively.

### 4.2 Functions Supported Inside \(A(R,3R)\)

### 4.3 Functions Supported on Both Sides of \(\partial B(0,R)\)

#### 4.3.1 Circular Radon Transform Data for Functions Supported on Both Sides of \(\partial B(0,R)\)

#### 4.3.2 Elliptical Radon Transform Data for Functions Supported on Both Sides of \(\partial B(0,R)\)

## 5 Summary

We have developed a numerical technique to solve the inversion formulas for circular and elliptical Radon transforms arising in some imaging applications. The inversion formulas and the proposed numerical scheme have been demonstrated to give good reconstructions on some standard test problems involving both discontinuous and smooth images. While the absolute errors in the reconstructed image are large, especially for discontinuous images, what is more important is that the objects in the image are properly distinguished by the current method. The numerical algorithm requires the solution of ill-conditioned matrix problems which is accomplished using a truncated SVD method. The matrices and the SVD can be constructed in a pre-processing step and re-used for the subsequent computations leading to an efficient and fast algorithm.

## Notes

### Acknowledgments

VPK would like to thank Rishu Saxena for her valuable input and discussions during the initial stages of this work. SR and VPK would like to express their gratitude to Gaik Ambartsoumian and Eric Todd Quinto for several fruitful discussions and important suggestions. VPK was partially supported by NSF grant DMS 1109417. All authors benefited from the support of the Airbus Group Corporate Foundation Chair “Mathematics of Complex Systems” established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India.

### References

- 1.Agranovsky, Mark, Berenstein, Carlos, Kuchment, Peter: Approximation by spherical waves in \(L^p\)-spaces. J. Geom. Anal.
**6**(3), 365–383 (1997). 1996MathSciNetCrossRefGoogle Scholar - 2.Agranovsky, Mark, Quinto, Eric Todd: Injectivity sets for Radon transform over circles and complete systems of radial functions. J. Funct. Anal.
**139**, 383–414 (1996)MATHMathSciNetCrossRefGoogle Scholar - 3.Ambartsoumian, Gaik, Boman, Jan, Krishnan, Venkateswaran P., Quinto, Eric Todd: Microlocal analysis of an ultrasound transform with circular source and receiver trajectories. American Mathematical Society, Series Contemporary Mathematics
**598**, pp. 45–58 (2013)Google Scholar - 4.Ambartsoumian, Gaik, Felea, Raluca, Krishnan, Venkateswaran P., Nolan, Clifford, Quinto, Eric Todd: A class of singular Fourier integral operators in synthetic aperture radar imaging. J. Funct. Anal.
**264**(1), 246–269 (2013)MATHMathSciNetCrossRefGoogle Scholar - 5.Ambartsoumian, Gaik, Gouia-Zarrad, Rim, Lewis, Matthew A.: Inversion of the circular Radon transform on an annulus. Inverse Probl.
**26**(10), 105015, 11 (2010)MathSciNetCrossRefGoogle Scholar - 6.Ambartsoumian, Gaik, Krishnan, Venkateswaran P.: Inversion of a class of circular and elliptical Radon transforms. 2014. To appear in Contemporary Mathematics, Proceedings of the International Conference on Complex Analysis and Dynamical Systems VI, (2013)Google Scholar
- 7.Ambartsoumian, Gaik, Kuchment, Peter: On the injectivity of the circular Radon transform. Inverse Probl.
**21**(2), 473–485 (2005)MATHMathSciNetCrossRefGoogle Scholar - 8.Ambartsoumian, Gaik, Kunyansky, Leonid: Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Probl. Imaging
**8**(2), 339–359 (2014)MATHMathSciNetCrossRefGoogle Scholar - 9.Andersson, Lars-Erik: On the determination of a function from spherical averages. SIAM J. Math. Anal.
**19**(1), 214–232 (1988)MATHMathSciNetCrossRefGoogle Scholar - 10.Antipov, Yuri A., Estrada, Ricardo, Rubin, Boris: Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces. J. Anal. Math.
**118**(2), 623–656 (2012)MATHMathSciNetCrossRefGoogle Scholar - 11.Cormack, Allen M.: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys.
**34**(9), 2722–2727 (1963)MATHCrossRefGoogle Scholar - 12.Denisjuk, Alexander: Integral geometry on the family of semi-spheres. Fract. Calc. Appl. Anal.
**2**(1), 31–46 (1999)MATHMathSciNetGoogle Scholar - 13.Fawcett, John A.: Inversion of \(n\)-dimensional spherical averages. SIAM J. Appl. Math.
**45**(2), 336–341 (1985)MATHMathSciNetCrossRefGoogle Scholar - 14.Finch, David, Haltmeier, Markus, Rakesh: Inversion of spherical means and the wave equation in even dimensions. SIAM J. Appl. Math.
**68**(2), 392–412 (2007)Google Scholar - 15.Finch, David, Patch, Sarah K., Rakesh: Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal.
**35**(5), 1213–1240 (2004). (electronic)Google Scholar - 16.Finch, David, Rakesh: The spherical mean value operator with centers on a sphere. Inverse Probl.
**23**(6), S37–S49 (2007)Google Scholar - 17.Frikel, Jürgen, Quinto, Eric Todd: Artifacts in incomplete data tomography - with applications to photoacoustic tomography and sonar (2014). http://www.arxiv.org/abs/1407.3453
- 18.Greenleaf, Allan, Uhlmann, Gunther: Non-local inversion formulas for the X-ray transform. Duke Math. J.
**58**, 205–240 (1989)MATHMathSciNetCrossRefGoogle Scholar - 19.Guillemin, Victor: Some remarks on integral geometry. Technical Report, MIT (1975)Google Scholar
- 20.Guillemin, Victor, Sternberg, Shlomo: Geometric asymptotics. American Mathematical Society, Providence (1977). Mathematical Surveys, No. 14MATHCrossRefGoogle Scholar
- 21.Hansen, Per Christian: The truncated SVD as a method for regularization. BIT
**27**(4), 534–553 (1987)MATHMathSciNetCrossRefGoogle Scholar - 22.Horn, Roger A., Johnson, Charles R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)MATHGoogle Scholar
- 23.Kuchment, Peter, Kunyansky, Leonid: Mathematics of thermoacoustic tomography. Eur. J. Appl. Math.
**19**(2), 191–224 (2008)MATHMathSciNetCrossRefGoogle Scholar - 24.Lavrentiev, M.M., Romanov, V.G., Vasiliev, V.G.: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics, vol. 167. Springer, Berlin (1970)Google Scholar
- 25.Mensah, Serge, Franceschini, Émilie: Near-field ultrasound tomography. J. Acoust. Soc. Am.
**121**(3), 1423–1433 (2007)CrossRefGoogle Scholar - 26.Mensah, Serge, Franceschini, Émilie, Lefevre, Jean-Pierre: Mammographie ultrasonore en champ proche. Trait. Signal
**23**(3–4), 259–276 (2006)MATHGoogle Scholar - 27.Mensah, Serge, Franceschini, Émilie, Pauzin, Marie-Christine: Ultrasound mammography. Nucl. Instrum. Methods Phys. Res.
**571**(3), 52–55 (2007)CrossRefGoogle Scholar - 28.Moon, Sunghwan: On the determination of a function from an elliptical Radon transform. J. Math. Anal. Appl.
**416**(2), 724–734 (2014)MATHMathSciNetCrossRefGoogle Scholar - 29.Nguyen, Linh V.: A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging
**3**(4), 649–675 (2009)MATHMathSciNetCrossRefGoogle Scholar - 30.Nguyen, Linh V.: Spherical mean transform: a PDE approach. Inverse Probl. Imaging
**7**(1), 243–252 (2013)MATHMathSciNetCrossRefGoogle Scholar - 31.Norton, Stephen J.: Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution. J. Acoust. Soc. Am.
**67**(4), 1266–1273 (1980)MATHMathSciNetCrossRefGoogle Scholar - 32.Norton, Stephen J., Linzer, Melvin: Reconstructing spatially incoherent random sources in the nearfield: exact inversion formulas for circular and spherical arrays. J. Acoust. Soc. Am.
**76**(6), 1731–1736 (1984)MathSciNetCrossRefGoogle Scholar - 33.Plato, Robert: The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math.
**36**(2), 331–351 (2012)MATHMathSciNetCrossRefGoogle Scholar - 34.Press, William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992). The art of scientific computingMATHGoogle Scholar
- 35.Quinto, Eric Todd: Singularities of the X-ray transform and limited data tomography in \({\mathbb{R}}^2\) and \({\mathbb{R}}^3\). SIAM J. Math. Anal.
**24**, 1215–1225 (1993)MATHMathSciNetCrossRefGoogle Scholar - 36.Romanov, V.G.: An inversion formula in a problem of integral geometry on ellipsoids. Mat. Zametki
**46**(4), 124–126 (1989)MATHMathSciNetGoogle Scholar - 37.Rubin, Boris: Inversion formulae for the spherical mean in odd dimensions and the Euler–Poisson–Darboux equation. Inverse Probl.
**24**(2), 025021, 10 (2008)CrossRefGoogle Scholar - 38.Tricomi, F.G.: Integral Equations. Dover Publications Inc, New York (1985). Reprint of the 1957 originalGoogle Scholar
- 39.Volchkov, V.V.: Integral Geometry and Convolution Equations. Kluwer Academic Publishers, Dordrecht (2003)MATHCrossRefGoogle Scholar
- 40.Volterra, Vito: Theory of functionals and of integral and integro-differential equations. With a preface by G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by E. Whittaker. Dover Publications Inc, New York (1959)Google Scholar
- 41.Weiss, Richard: Product integration for the generalized Abel equation. Math. Comput.
**26**, 177–190 (1972)MATHCrossRefGoogle Scholar