Exact inversion of an integral transform arising in Compton camera imaging

Abstract.

Purpose: The paper addresses exact inversion of the integral transform, called the Compton (or cone) transform, that maps a three-dimensional (3-D) function to its integrals over conical surfaces in ${\mathbb{R}}^3$. Compton transform arises in passive detection of gamma-ray sources with a Compton camera, which has promising applications in medical and industrial imaging as well as in homeland security imaging and astronomy.

Approach: A generalized identity relating the Compton and the Radon transforms was formulated. The proposed relation can be used to devise a method for converting the Compton transform data of a function into its Radon projections. The function can then be recovered using well-known inversion techniques for the Radon transform.

Results: We derived a two-step method that uses the full set of available projections to invert the Compton transform: first, the recovery of the Radon transform from the Compton transform, and then the Radon transform inversion. The proposed technique is independent of the geometry of detectors as long as a generous admissibility condition is met.

Conclusions: We proposed an exact inversion formula for the 3-D Compton transform. The stability of the inversion algorithm was demonstrated via numerical simulations.

1. Introduction

This paper addresses the exact inversion of the integral transform that maps a function on ${\mathbb{R}}^3$ to its integrals over conical surfaces. Our main motivation of studying this transform is its appearance in modeling the data acquired by a gamma-ray detection system called Compton camera. Therefore, its inversion is useful for image reconstruction in Compton camera imaging.^1,2

Owing to their use of the Compton scattering principle, Compton cameras can efficiently visualize gamma rays having energies as high as 10 MeV.^3,4 Unlike collimated cameras, they can collect measurements from all incoming gamma photons without any elimination. Other benefits include a wide field of view, the capacity to fully reconstruct three-dimensional (3-D) images, and the feasibility of a compact portable design for the camera.⁵ These features make Compton camera a promising tool in nuclear imaging applications, such as single-photon emission computed tomography (SPECT) and positron emission tomography. Compton cameras are also proven to be effective in online proton therapy monitoring as well as in the detection of radioactive contamination.^4,6,7

A Compton camera consists of two-layer (not necessarily planar) position and energy-sensitive detectors (Fig. 1). When an incoming gamma photon reaches the camera, it undergoes Compton scattering in the front detector (scatterer) and photoelectric absorption in the second detector (absorber). The recorded energies and interaction positions on both detectors provide a cone of possible directions to locate the gamma-ray source. The data provided by Compton camera can thus be modeled simply as integrals of the distribution of the sources of radiation over conical surfaces with a vertex on the scattering detector. The Compton (or cone) transform is the operator that maps the distribution of source intensity to its integrals over these cones. The objective of Compton camera imaging is to recover source distribution from this information.^1,2

Fig. 1.

Schematic representation of a Compton camera.

As the space of cones in three dimensions with vertices on a detector surface is five-dimensional, inverting the Compton transform is an overdetermined problem, that is, there exist multiple left inversions, which all agree on exact data but perform differently on noisy data. Choosing a 3-D subset of cones leads to a unique inversion formula.^1,8–10 However, we avoid such a restriction because it can lead to the elimination of essential information in the case of Compton imaging. Instead, we consider all cones with vertices on the scattering detector and use the resulting overdetermined data to stabilize the reconstruction. It is important to note that the reconstruction algorithms we develop are independent of the geometry of detectors, as long as a generous admissibility condition is met.^11–13

The paper is organized as follows. In Sec. 2, we define the Compton transform and briefly recall some other relevant transforms and their properties. In Sec. 3, we first formulate an integral relationship between the Compton and Radon transforms stating that the weighted average of the Compton transform of a function with respect to opening angles is the spherical convolution (with a kernel depending on the weight) of the Radon transform of that function. This result is a generalization of a previously obtained identity (see Ref. 14, Theorem 5), which was instrumental in deriving an inversion formula for the Compton transform,^11,14 in the sense that it allows for arbitrary kernels in the spherical convolution of the Radon transform. The gain is that the freedom of choice of the convolution kernel enables one to derive variety of inversion formulas for the Compton transform. We then choose a convolution kernel and derive the corresponding inversion formula, which is more stable than the previously developed one [see Ref. 11, Eqs. (12) and (14)]. In Sec. 4, we discuss the requirement on the geometry of detectors and provide examples of detector geometries satisfying this condition. Section 5 contains the results of the numerical implementation of the proposed inversion formula. The proofs of the presented theorems can be found in the Appendix section.

2. Definitions

The surface of a 3-D circular cone having vertex $u\in {\mathbb{R}}^3$, central axis direction $\beta \in {\mathbb{S}}^2$, and (half)-opening angle $\psi \in (0,\pi )$ (Fig. 1) can be defined as

$${\mathfrak{S}}_{u,\beta ,\psi }=\{x\in {\mathbb{R}}^3:(x-u)·\beta =|x-u|\cos \psi \}.$$

Let $f$ be a function on ${\mathbb{R}}^3$ that is supported away from the detector surface. The Compton transform of $f$, denoted by $Cf$, are integrals of $f$ over all circular cones in ${\mathbb{R}}^3:$

$$Cf(u,\beta ,\psi )={\int }_{{\mathfrak{S}}_{u,\beta ,\psi }}f(x)\mathrm{d}S(x)=\kappa (\psi )\sin \psi {\int }_{{\mathbb{R}}^3}f(x)\delta [(x-u)·\beta -|x-u|\cos \psi ]\mathrm{d}x,$$ (1)

where $dS(x)$ is the surface measure on the cone and $\delta$ is the Dirac-delta distribution. Here, the function $\kappa (\psi )$ accounts for the distribution of scattering angles. This system-dependent function can be determined in advance^15,16 and can easily be incorporated into the weight when integrating over $\psi$ in the inversion process (see Theorem 2), and so we consider $\kappa (\psi )=1$.

In the following, we assume for simplicity that $f$ is smooth and rapidly decaying, that is, $f$ belongs to the Schwartz space $\mathcal{S}({\mathbb{R}}^3)$. The Radon transform integrates a function $f\in \mathcal{S}({\mathbb{R}}^3)$ over the planes in ${\mathbb{R}}^3$. Namely, if $\omega \in {\mathbb{S}}^2$ and $s\in \mathbb{R}$,

$$Rf(\omega ,s)={\int }_{x·\omega =s}f(x)\mathrm{d}x={\int }_{{\mathbb{R}}^3}f(x)\delta (x·\omega -s)\mathrm{d}x.$$ (2)

The inversion of the Radon transform is well studied, and in three-dimensions, it reads as

$$f=\frac{-1}{8\pi }{\int }_{{\mathbb{S}}^2}\frac{{\partial }^2}{\partial s^2}{Rf(\omega ,s)|}_{s=x·\omega }\mathrm{d}\omega ,$$ (3)

which is known as the filtered backprojection formula.^17,18

3. Inversion of the Compton Transform

In the following, we present a two-step method for inverting the Compton transform. We start with an identity that relates the Compton and Radon transforms. It states that the weighted integral of the Compton transform data of a function with respect to opening angles is the spherical convolution (with a kernel depending on the weight) of the Radon transform data of that function.

Theorem 1.

Let $h$ be an integrable function on $[-\mathrm{1,1}]$ and $w:[-\mathrm{1,1}]\to \mathbb{R}$ be defined as

$$w(s)=2{\int }_0^{\pi }h\left(\sqrt{1-s^2}\cos \theta \right)\mathrm{d}\theta .$$

Then, for any $f\in \mathcal{S}({\mathbb{R}}^3)$, $u\in {\mathbb{R}}^3$ and $\beta \in {\mathbb{S}}^2$,

$${\int }_0^{\pi }Cf(u,\beta ,\psi )w(\cos \psi )\mathrm{d}\psi ={\int }_{{\mathbb{S}}^2}Rf(\omega ,u·\omega )h(\omega ·\beta )\mathrm{d}\omega .$$ (4)

(See Appendix for the proof.)

We remark that previously in Smith (Ref. 19, Theorems 1 and 3), the Compton and Radon transforms were linked via a different integral identity. A generalization of this integral relation to arbitrary weight functions for weighted cone transforms was presented in Proposition 4.1 of Ref. 12.

Applying spherical deconvolution on the right-hand side of Eq. (4), one can recover the Radon data of $f$ from its Compton data. Different choices of the convolution kernel $h$ lead to various methods for the recovery of the Radon data, and hence various inversion formulas for the Compton transform. We note that a special case of the above result, namely for $h(t)=|t|$, was proven in Terzioglu¹⁴ and was used in deriving an inversion formula for the Compton transform.^11,14

As the spherical convolution on the right-hand side of Eq. (4) is a smoothing operator, one expects its inversion to involve differentiation. The degree of smoothness of the resulting function, and thus the order of differentiation that is needed for the inversion, depends on the convolution kernel used. For example, when $n=3$, the absolute value function $h(t)=|t|$ leads to a spherical convolution whose inversion requires the application of Laplace–Beltrami operator twice.^11,14,20 On the other hand, the spherical convolution

$${\mathcal{S}}_{0}g(\beta )=\frac{-1}{\pi }{\int }_{{\mathbb{S}}^2}g(\omega )\log (1-{|\omega ·\beta |}^2)\mathrm{d}\omega ,$$ (5)

is known as the zero-order sine transform [see Ref. 20, p. 531, Eq. (A.13.41)] and can be inverted as

$$\frac{-{\mathrm{\Delta }}_{{\mathbb{S}}^2}}{8}{\mathcal{S}}_{0}g=g,$$ (6)

provided ${\int }_{{\mathbb{S}}^2}g(\omega )\mathrm{d}\omega =0$ (see Ref. 20, p. 300, Theorem 5.31 and the remark afterward).

The following inversion formula is obtained by taking $h(t)=\log (1-t^2)$. It is more stable than the one obtained by using the absolute value function as it requires the application of Laplace–Beltrami operator only once. This is confirmed by our numerical tests in Sec. 5.

Theorem 2.

Let $f\in \mathcal{S}({\mathbb{R}}^3)$. For any $u\in {\mathbb{R}}^3$ and $\beta \in {\mathbb{S}}^2$,

$$Rf(\beta ,u·\beta )=\frac{{\mathrm{\Delta }}_{{\mathbb{S}}^2}}{2}{\int }_0^{\pi }Cf(u,\beta ,\psi )\log (|\cos \psi |+1)\mathrm{d}\psi +\frac{1}{2}{\int }_0^{\pi }Cf(u,\beta ,\psi )\mathrm{d}\psi ,$$ (7)

where ${\mathrm{\Delta }}_{{\mathbb{S}}^2}$ is the Laplace–Beltrami operator on ${\mathbb{S}}^2$ acting on $\beta$. (See Appendix for the proof.)

Equation (7) is the first step of the Compton transform inversion. It remains to invert Radon transform to obtain the function $f$. Now if one has access to complete set of Radon projections, that is, if $Rf(\beta ,s)$ is available for all $s\in \mathbb{R}$ at each direction $\beta \in {\mathbb{S}}^2$, then the application of any Radon transform inversion leads to the reconstruction of $f$ from its Compton data. In the next section, we discuss the requirement on the geometry of Compton detectors for obtaining a complete set of Radon projections.

4. Admissibility of Compton Detector Geometries

To reconstruct $f$, one needs $Rf(\beta ,s)$ for all $s\in \mathbb{R}$ at each direction $\beta \in {\mathbb{S}}^2$. If we are required to have a Compton detector such that any plane intersecting the domain of reconstruction intersects a detection site of the scatterer, then for all $s\in \mathbb{R}$, there exists a $u$ such that $s=u·\beta$. Thus, this condition guarantees the availability of a complete set of Radon projections and is called the “Compton admissibility condition.”¹¹ We note that the same condition was needed in the inversion formulas developed by Smith¹⁹ and was referred as “the completeness condition.”

For example, when the imaged object lies strictly inside a sphere/cylinder/cube, detectors placed around this sphere/cylinder/cube satisfy the Compton admissibility condition. Figure 2 shows other examples of admissible Compton detectors.

Fig. 2.

Some admissible geometries for Compton detectors. Black dots represent the detection sites (aka. cone vertices) on the scattering detector of the Compton camera and are placed on (or a small planar Compton camera moving along) (a) two perpendicular great semicircles of the sphere with radius $\sqrt{2}$ and center at the origin (the imaged object is assumed to be strictly inside the unit sphere centered at the origin), (b) a curve composed of two circles and a helix on a cylinder enclosing the imaged object, (c) a closed curve composed of certain edges of a cube surrounding the imaged object.

On the other hand, a ring-/line-/planar-shaped detector is inadmissible. However, such detectors can be made admissible if they are moved around the imaging domain. For instance,

• •rotating a ring detector 180 deg around its diameter to form a sphere surrounding the imaged object;
• •moving a ring detector in orthogonal direction, or moving a line detector on a circular trajectory, to form a cylinder surrounding the imaged object;
• •moving a planar detector around the imaged object to form a polygonal prism surrounding the imaged object; or
• •moving a small planar detector on a helical trajectory surrounding the imaged object,

all lead to a complete set of Radon projections.

We now state the second step of our Compton inversion formula.

Theorem 3.

Suppose that the Compton detectors satisfy Compton admissibility condition. Then, one can recover the Radon data of $f$ for all $s\in \mathbb{R}$ and $\beta \in {\mathbb{S}}^2$ using Eq. (7), and hence the application of any Radon transform inversion leads to the reconstruction of $f$ from its Compton data.

5. Numerical Reconstructions

In this section, we present the results of numerical implementation of Theorems 2 and 3 using MATLAB for the phantom $f={\chi }_{B_1}-0.5{\chi }_{B_2}$. Here, ${\chi }_{B_i}$, $i=\mathrm{1,2}$, denotes the characteristic function of the 3-D ball with radius 0.4 units and center at $(\mathrm{0,0},-0.4)$, and with radius 0.5 units and center at the origin, respectively (see Fig. 3).

Fig. 3.

(a) Scanning geometry showing the position of the phantom $f$ in reference to the detector geometry described in Fig. 2(a). (b) The cross section of the phantom by the plane $x=0$. (c) Surface plot of the phantom’s cross section by the plane $x=0$. The vertical axis corresponds to the density of the phantom.

In Fig. 4, we provide the reconstructions of $f$ on a 3-D grid that was obtained by uniformly discretizing $[-\mathrm{1,1}{]}^3$ into a $90\times 90\times 90$ array of cubic voxels. The detector configuration is shown in Fig. 3(a). Figure 4(a) shows a cross section of the reconstructed image, Fig. 4(c) shows the surface plot of this cross section, and Fig. 4(e) shows the $z$ axis profile of the reconstruction in comparison to the phantom. The corresponding plots for the reconstruction from Compton data that are contaminated with 5% Gaussian white noise are shown in Figs. 4(b), 4(d), and 4(f).

Fig. 4.

The reconstruction (size $90\times 90\times 90$) of the phantom shown in Fig. 3 from its Compton data that is synthetically simulated using 360 counts for vertices $u$ (represented by black dots on the sphere in Fig. 3(a), 7446 counts for directions $\beta \in {\mathbb{S}}^2$, and 180 counts for opening angles $\psi$. (a) The cross section of the reconstruction by the plane $x=0$. (b) The cross section by the plane $x=0$ of the reconstruction from Compton data that is contaminated with 5% Gaussian white noise. (c) Surface plot of the reconstruction’s cross section by the plane $x=0$. (The vertical axis corresponds to the density of the reconstruction.) (d) Surface plot of the noisy reconstruction’s cross section by the plane $x=0$. (e) The comparison of $z$ axis profiles of the phantom and the reconstruction. (f) The comparison of $z$ axis profiles of the phantom and the reconstruction from noisy Compton data.

The forward data were simulated by numerically computing the integrals of $f$ over cones for 360 counts of vertices $u$ that are uniformly distributed over the two perpendicular great semicircles (corresponding to points with azimuthal and polar angles $({\theta }_k,\pi /2)$, and $(3\pi /2,{\theta }_k)$, ${\theta }_k=k\pi /180\mathrm{rad}$, $k=1,\dots ,180$) of the sphere of radius $\sqrt{2}$ units and centered at the origin, 7446 counts of central axis directions $\beta \in {\mathbb{S}}^2$, and 180 counts of uniformly distributed opening angles $\psi \in [0,\pi ]$.

For the central axis directions, we generated a triangular mesh with 7446 points on the unit sphere ${\mathbb{S}}^2$ using the algorithm given in Persson and Strang.²¹ The discretization of the Laplace–Beltrami operator on ${\mathbb{S}}^2$ was done by a linear approximation of gradient and divergence operators on each triangle followed by a weighted averaging over the first ring of each vertex in terms of the triangle area.²² We preferred this method over the algorithm described in Belkin et al.²³ and the cotangent formula²⁴ as it led to more accurate recovery of the regions near discontinuities of the phantom.

In obtaining the Radon data for 128 counts of uniformly sampled $s\in [-\mathrm{1,1}]$ from $Rf(\beta ,u·\beta )$ at each direction $\beta \in {\mathbb{S}}^2$ (with 7446 counts), we used MATLAB’s Curve Fitting Toolbox cftool with spline fitting. We then inverted the Radon transform using filtered backprojection [Eq. (3)], and obtained the reconstructions depicted in Fig. 4.

For a quantitative comparison, we approximately computed the $L^2$ error between the phantom $f$ and its reconstruction $f_{\mathrm{rec}}$, which is given as ${\Vert f_{\mathrm{rec}}-f\Vert }_{L^2({\mathbb{R}}^3)}={[{\int }_{{\mathbb{R}}^3}{|f_{\mathrm{rec}}(x)-f(x)|}^2\mathrm{d}x]}^{1/2}$. The $L^2$-error values for reconstruction from Compton data without additive noise and with 5% Gaussian noise were 0.18 and 0.22, respectively.

To demonstrate the degree of improvement in stability achieved by the inversion Eq. (7), we have reimplemented the inversion formula previously obtained by using the convolution kernel $h(t)=|t|$ [see Ref. 11, Eqs. (12) and (14)]. Although it had been successfully implemented¹¹ using a Compton data of size $1080\times \mathrm{30,054}\times 90$ in the case of a spherical detector geometry, it failed to produce a meaningful result when we used a Compton data of size $360\times 7446\times 180$ obtained from the detector geometry shown in Fig. 3(a). This can be attributed to the fact that this time we use a more challenging detector geometry and much coarser forward data.

6. Discussion and Conclusion

We developed a generalized integral relation between the Compton and the Radon transforms and proposed an exact inversion formula for the 3-D Compton transform. Numerical implementations showed that the presented inversion formula leads to stable reconstructions. We note that one can obtain better spatial resolution in the reconstruction by considering the following. First, our forward data were generated by computing the area of intersection of the phantom and the cone surfaces numerically rather than analytically, and hence involved errors. The inversion algorithm would certainly perform better with exact forward data. Second, the discrete Laplace–Beltrami operator would produce more accurate results if one considers a finer mesh on the unit sphere for the central axis directions $\beta$.¹¹ Likewise, the interpolation/approximation of Radon data would produce better results if more detection points $u$ are available. In addition, the reconstruction quality may be further enhanced by smoothing the data in the spherical variable before and after applying the Laplace–Beltrami operator as well as by denoising²⁵ the reconstructed image.

As the main objective of our numerical experiments was to demonstrate the stability of the proposed inversion formula, we used idealized forward data assuming the availability of a complete set of cone projections. To test the performance of the inversion algorithm for image reconstruction from Compton camera data, one needs more realistically simulated forward data, e.g., obtained by using Monte-Carlo techniques. Then, the count of Compton events scattering through each small range of cone angles may not be equal. Investigation of methods to imputing missing data and analyzing its effects on the quality of reconstruction are important issues we plan to address in a future study.

The model we used for Compton camera measurements is simplified and neglects weight factors that may arise in the surface integral [Eq. (1)] depending on the application and the design of detectors.^12,26 For example, it is known that nonuniform attenuation of the photons by the imaged object before detection in the Compton camera introduces an exponential weight factor, which is a phenomenon arising especially in SPECT. In the future, we plan to use the ideas and techniques of this work to derive inversion algorithms that account for these weight factors. Moreover, we assumed that the measurements acquired by a Compton camera are ideal, but in practice imperfections in detectors cause uncertainties in measurements. For example, Doppler broadening is an important factor limiting the angular resolution of a Compton camera. The result of numerical implementation testing the reaction of the inversion algorithm to noisy data is a partial relief. Still, in a future study, we plan to explore the convolution kernel that takes this effect into account and derive the corresponding inversion formula.

7. Appendix

Proof of Theorem 1.

We first prove that, for any $\sigma$, $\beta \in {\mathbb{S}}^2$,

$${\int }_{{\mathbb{S}}^2}\delta (\omega ·\sigma )h(\omega ·\beta )\mathrm{d}\omega =2{\int }_0^{\pi }h\left(\sqrt{1-{(\sigma ·\beta )}^2}\cos \theta \right)\mathrm{d}\theta =w(\sigma ·\beta ).$$ (8)

To this end, let $A$ be the rotation that transforms the north pole $e_3=(\mathrm{0,0},1)$ to $\sigma$, and $A^{-1}\beta =:\eta \in {\mathbb{S}}^2$. Since the Lebesgue measure on the sphere is rotation-invariant, we have

$${\int }_{{\mathbb{S}}^2}\delta (\omega ·\sigma )h(\omega ·\beta )\mathrm{d}\omega ={\int }_{{\mathbb{S}}^2}\delta (A\omega ·\sigma )h(A\omega ·\beta )\mathrm{d}\omega ={\int }_{{\mathbb{S}}^2}\delta (\omega ·e_3)h(\omega ·\eta )\mathrm{d}\omega .$$

Now since $\omega =((\sin \varphi )\zeta ,\cos \varphi )$ for some $\zeta \in {\mathbb{S}}^1$, $\varphi \in [0,\pi ]$, and $\eta =(\overline{\eta },{\eta }_3)\in {\mathbb{R}}^2\times [-\mathrm{1,1}]$, we obtain

$$\begin{gathered} {\int }_{{\mathbb{S}}^2}\delta (\omega ·e_3)h(\omega ·\eta )\mathrm{d}\omega ={\int }_{{\mathbb{S}}^1}{\int }_0^{\pi }\delta (\cos \varphi )h[(\sin \varphi )\zeta ·\overline{\eta }+(\cos \varphi ){\eta }_3]\sin \varphi \mathrm{d}\varphi \mathrm{d}\zeta \\ ={\int }_{{\mathbb{S}}^1}{\int }_{-1}^1\delta ({\omega }_3)h(\sqrt{1-{\omega }_3^2}\zeta ·\overline{\eta }+{\omega }_3{\eta }_3)\mathrm{d}{\omega }_3\mathrm{d}\zeta ={\int }_{{\mathbb{S}}^1}h(\zeta ·\overline{\eta })\mathrm{d}\zeta =2{\int }_0^{\pi }h(|\overline{\eta }|\cos \varphi )\mathrm{d}\varphi , \end{gathered}$$

where the last equality follows from the rotation invariance of the Lebesgue measure on the unit circle. Observing that ${\eta }_3=\eta ·e_3=A^{-1}\beta ·A^{-1}\sigma =\beta ·\sigma$ and $|\overline{\eta }|=\sqrt{{|\eta |}^2-{\eta }_3^2}=\sqrt{1-{(\beta ·\sigma )}^2}$, we obtain Eq. (8).

Now let $f\in \mathcal{S}({\mathbb{R}}^3)$. By definition of the Compton transform Eq. (1), for any $u\in {\mathbb{R}}^3$ and $\beta \in {\mathbb{S}}^2$:

$${\int }_0^{\pi }Cf(u,\beta ,\psi )w(\cos \psi )\mathrm{d}\psi ={\int }_0^{\pi }\sin \psi {\int }_{{\mathbb{R}}^3}f(u+y)\delta (y·\beta -|y|\cos \psi )\mathrm{d}yw(\cos \psi )\mathrm{d}\psi ={\int }_{{\mathbb{R}}^3}f(u+y){\int }_0^{\pi }\delta (y·\beta -|y|\cos \psi )w(\cos \psi )\sin \psi \mathrm{d}\psi \mathrm{d}y={\int }_{{\mathbb{R}}^3}f(u+y){\int }_{-1}^1\frac{1}{|y|}\delta (\frac{y}{|y|}·\beta -t)w(t)\mathrm{d}t\mathrm{d}y={\int }_{{\mathbb{R}}^3}f(u+y)w(\frac{y}{|y|}·\beta )\frac{\mathrm{d}y}{|y|},$$

where we changed the order of integrals and then used the homogeneity and integration property of the Dirac delta distribution. Now, we use Eq. (8) to write as follows:

$$\begin{gathered} {\int }_{{\mathbb{R}}^3}f(u+y)w(\frac{y}{|y|}·\beta )\frac{\mathrm{d}y}{|y|}={\int }_{{\mathbb{R}}^3}f(u+y){\int }_{{\mathbb{S}}^2}\delta (\frac{y}{|y|}·\omega )h(\omega ·\beta )\mathrm{d}\omega \frac{\mathrm{d}y}{|y|} \\ ={\int }_{{\mathbb{R}}^3}f(x){\int }_{{\mathbb{S}}^2}\delta [(x-u)·\omega ]h(\omega ·\beta )\mathrm{d}\omega \mathrm{d}x={\int }_{{\mathbb{S}}^2}{\int }_{{\mathbb{R}}^3}f(x)\delta [(x-u)·\omega ]\mathrm{d}xh(\omega ·\beta )\mathrm{d}\omega \\ ={\int }_{{\mathbb{S}}^2}Rf(\omega ,u·\omega )h(\omega ·\beta )\mathrm{d}\omega , \end{gathered}$$

which yields the result.

Proof of Theorem 2.

For $h(t)=\log (1-t^2)$, we have

$$w(s)=2{\int }_0^{\pi }h\left(\sqrt{1-s^2}\cos \theta \right)\mathrm{d}\theta =4\pi \log \left(\frac{|s|+1}{2}\right),$$

[see Ref. 27, p. 532, Eq. (4.226-2)]. Then, for all $f\in \mathcal{S}({\mathbb{R}}^3)$, $u\in {\mathbb{R}}^3$, and $\beta \in {\mathbb{S}}^2$, the identity Eq. (4) reads as

$$\begin{gathered} 4\pi {\int }_0^{\pi }Cf(u,\beta ,\psi )\log \left(\frac{|\cos \psi |+1}{2}\right)\mathrm{d}\psi ={\int }_{{\mathbb{S}}^2}Rf(\omega ,u·\omega )\log (1-{|\omega ·\beta |}^2)\mathrm{d}\omega \\ =-\pi {\mathcal{S}}_0[Rf(\omega ,u·\omega )](\beta ), \end{gathered}$$ (9)

where ${\mathcal{S}}_0$ is as in Eq. (5). We observe that for each $u\in {\mathbb{R}}^3$, the function

$$g(\omega )≔Rf(\omega ,u·\omega )-\frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}Rf(\sigma ,u·\sigma )\mathrm{d}\sigma ,$$

has zero spherical mean, and that ${\mathcal{S}}_{0}g$ and ${\mathcal{S}}_0(Rf)$ differ by a constant. Therefore,

$$Rf(\beta ,u·\beta )-\frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}Rf(\sigma ,u·\sigma )\mathrm{d}\sigma =g(\beta )=\frac{-{\mathrm{\Delta }}_{{\mathbb{S}}^2}}{8}{\mathcal{S}}_{0}g(\beta )=\frac{-{\mathrm{\Delta }}_{{\mathbb{S}}^2}}{8}{\mathcal{S}}_0[Rf(\omega ,u·\omega )](\beta ).$$

From Eq. (9), we know that

$$\begin{gathered} {\mathcal{S}}_0[Rf(\omega ,u·\omega )](\beta )=-4{\int }_0^{\pi }Cf(u,\beta ,\psi )\log \left(\frac{|\cos \psi |+1}{2}\right)\mathrm{d}\psi \\ =-4{\int }_0^{\pi }Cf(u,\beta ,\psi )\log (|\cos \psi |+1)\mathrm{d}\psi +4\log 2{\int }_0^{\pi }Cf(u,\beta ,\psi )\mathrm{d}\psi . \end{gathered}$$

Now letting $h(t)=1$ in Eq. (4) yields

$$2\pi {\int }_0^{\pi }Cf(u,\beta ,\psi )\mathrm{d}\psi ={\int }_{{\mathbb{S}}^2}Rf(\sigma ,u·\sigma )\mathrm{d}\sigma ,$$

which is independent of $\beta$. Combining the last two identities and applying $-1/8{\mathrm{\Delta }}_{{\mathbb{S}}^2}$ in the variable $\beta$, we obtain

$$Rf(\beta ,u·\beta )=\frac{{\mathrm{\Delta }}_{{\mathbb{S}}^2}}{2}{\int }_0^{\pi }Cf(u,\beta ,\psi )\log (|\cos \psi |+1)\mathrm{d}\psi +\frac{1}{2}{\int }_0^{\pi }Cf(u,\beta ,\psi )\mathrm{d}\psi .$$

Biography

Fatma Terzioglu is a William H. Kruskal instructor in the Statistics Department at the University of Chicago. She received her PhD degree in mathematics in 2018 from Texas A&M University. Her research interests primarily lie in the fields of inverse problems and imaging, and applied harmonic analysis.

Disclosures

This work was partially supported by the NSF DMS Grant 1211463.