Inversion formulas for quarter-spherical Radon transforms

1.   Introduction

A spherical Radon transform maps a function to its integral over a family of spheres. The problem of inverting a spherical Radon transform from a set of integrals along all spheres is overdetermined as the family of all spheres in $\mathbb{R}^{n}, n\ge2$, is determined using $(n + 1)$-dimensional variables. Hence, spherical Radon transforms can be inverted in numerous ways from a set of integrals along some spheres. The centers of the spheres are centered at any point in the whole space with a fixed radius ^[39], or all spheres of integration pass through the origin ^{[3,4,25,30,31]}. In case of a variable radius, the centers can also be restricted on a hyperplane ^{[2,7,27,32,38]}, a sphere ^[8,9,26,40], a cylinder ^[13,20], etc.^{[12,18,19,28,36]}.

The relevance of the spherical Radon transform to solving the wave equation has garnered significant interest in the context of Partial Differential Equations (PDEs) ^[5]. This interest has grown as the wave equation became intertwined with fields such as sonar, seismic waves and radar ^[7,29]. For example, a simplified model of Photoacoustic Tomography can be described as follows ^[1,17,24]: When a body of interest is irradiated with short pulses of electromagnetic waves, some of the radiation is absorbed by the body and heats the tissue. As a result, the body experiences thermoelastic expansion and ultrasound waves are produced. Transducers placed at the boundary of the body record this ultrasound. Assuming that the speed of the ultrasound is a constant $c$, at any instant $t$, one transducer records several signals generated by a location at a fixed distance $ct$ from the detector. In other words, the detector measurements can be represented as the integral of a function along a sphere centered at the detector location with varying radii (depending on time). We can measure enough data using an array of these transducers (or by moving the detector to the boundary surface) to recover an unexposed image function that essentially contains the biological information of the body. As similar mathematical problems arise in other imaging models, such as sonar and radar imaging, the inversion of spherical Radon transforms has become a subject of considerable interest.

Recent research has also explored incomplete spherical Radon transforms. For instance, restrictions have been imposed on the set of radii of integration spheres ^[1,11]. Another variant is the spherical-cap Radon transform, which assigns a given function $f$ to its integrals over a part of a sphere of integration, rather than the entire sphere ^[11,33], (where limited angular-aperture detectors are employed, similar to ^[15,35]). While the uniqueness of the spherical-cap Radon transform has been established, obtaining an exact inversion formula remains a challenge to the best of our knowledge.

In this study, we investigate two types of quarter-spherical Radon transforms whose integrals are a quarter of a sphere: 1) centers of spheres of integration lie on a plane, and 2) centers of spheres of integration lie on a line and a quarter sphere of integration revolves around the line. To our knowledge, we are the first to study these types of Radon transforms and find the exact inversion formulas. These quarter-spherical Radon transforms are developed from endoscopic photoacoustic imaging that can provide structural and functional information of biological luminal structures such as the coronary arteries and digestive tract ^{[14,16,22,34,37,41]}. In ^[34], the author built a forward imaging model and inverted the forward model through iterative optimization, and in ^[41], a Convolutional Neural Network (CNN) based on deep gradient descent was developed. Assuming our model is successful, our results help reconstruct higher quality images than ^[34] and provide a theory as to why CNNs ^[41] work.

The shape of the detector is frequently depicted in endoscopic photoacoustic images as a pentahedron with two right-triangle faces, as shown in Figure 1(a); for the design of an actual detector, see ^[37] Figure 1. The motion of the detector corresponds to one of the centers of the integral sphere, and the time may vary with radius. We study the quarter-spherical Radon transforms for reasons that may be somewhat related to endoscopic photoacoustic imaging. However, it is reasonable that these transforms could appear in photoacoustic imaging using pentahedron detectors, like a spherical-cap Radon transform arises in one with a specific detector ^[33]. Additionally, such a problem of determining a function from the integral value on a particular family of manifolds is known as the integral geometry problem ^[10,21]; it is a complex task that is worth studying.

The rest of this paper is organized as follows. Quarter-spherical Radon transforms are precisely formulated in Section 2. Two inversion formulas for the two types of quarter-spherical Radon transforms are derived in Section 3. The paper concludes with a short discussion in Section 4.

2.   Formulation of the quarter-spherical Radon transform

In this section, we define two quarter-spherical Radon transforms. First, let us define a quarter-spherical Radon transform with the centers of integration spheres on the hyperplane for the $n$ dimensions, as follows:

Definition 1. For $({\bf u}, r)\in \mathbb{R}^{n-1}\times[0, \infty)$, the quarter-spherical Radon transform of $f$ on the hyperplane is defined as

where ${\rm d}S({\bf x}, z)$ is the surface measure on the sphere and $S^{n-1}_{r, q} = \{({\bf x}, z)\in \mathbb{R}^n:|({\bf x}, z)| = r, x_1 > 0, z > 0\}$ if the integral exists (see Figure 1(b) for $n = 3$).

Next, let us define the quarter-spherical Radon transform with the centers of integration spheres on a line and the rotation of a quarter sphere, as follows:

Definition 2. For $(u, \theta, r)\in \mathbb{R}^{1}\times [0, 2\pi)\times[0, \infty)$, the quarter-spherical Radon transform of $f$ is defined as

where $S^{2}_{\theta, r, q} = \{({\bf x}, z)\in \mathbb{R}^3:|({\bf x}, z)| = r, z > 0, (\cos\theta, \sin\theta)\cdot {\bf x} > 0\}$ if the integral exists (see Figure 1(c)).

Actually, we are interested in the 3 dimension cases for $R_Hf$ and $R_Lf$, but the inversion of $R_Hf$ can be easily generalized, and the inversion of $R_Lf$ requires an inversion formula for 2 dimension $R_Hf$. Therefore, for $R_Lf$, only the 3 dimension case is discussed.

3.   Inversion formulas

In this section, we derive inversion formulas for two quarter-spherical Radon transforms. We first develop an inversion formula for the quarter-spherical Radon transform $R_H f$. For finding the inversion formula for $R_L f$, we need an inversion formula for the quarter-spherical Radon transform $R_H f$ when considering $n = 2$.

Theorem 3. For $f\in C(\mathbb{R}^n)$ with compact support in $\{({\bf x}, z)\in \mathbb{R}^n:z\geq0\}$, we have

where $\mathcal F_{n-1}$ is the $n-1$-dimensional Fourier transform and $\operatorname{erf}$ is the error function defined as $\operatorname{erf} z = \frac2{\sqrt\pi}\int^z_0 e^{-t^2}{\rm d}t$^*.

^*for details of the error function see [23, Chapter 2]

Proof. By definition, $R_Hf$ can be written as

Let us define the function $F$ on $\mathbb{R}^n$ by

Then, we have $f({\bf x}, z) = zF({\bf x}, |z|^2)$ for $z\geq0$. Substituting $F$ into (3.2) gives

Taking the $n-1$-dimensional Fourier transform of $R_Hf$ with respect to ${\bf u}$, we have

where $\mathcal F_{n-1}(R_Hf)$ and $\mathcal F_{n-1}F$ are the $n-1$-dimensional Fourier transforms of $R_Hf$ and $F$ with respect to ${\bf u}$ and ${\bf x}$, respectively.

By multiplying $r e^{r^2(-\mathrm i\sigma-\epsilon)}$ for $\epsilon > 0$ and integrating with respect to $r$, we have

where $\mathbb{R}^{n-1}_+ = \{ {\bf x}\in \mathbb{R}^{n-1}:x_1 > 0\}$. Changing the variables $r^2-| {\bf x}|^2\to t$ gives

The last equality follows from the compact support of $F$ on $z > 0$. Hence, it holds that

where $\mathcal F_{n}F$ is the $n$-dimensional Fourier transform of $F$. To compute the integral with respect to ${\bf x}$, we employ the following identities [6, (11) on page 15 and (18) on page 73]: for $\operatorname{Re} a > 0$,

where $\operatorname{Re} a$ is the real part of $a\in\mathbb C$. Hence, we have

which implies

with

Thus (3.4) is equal to

The compact support of $F$ on the right-hand side implies the boundedness of $\mathcal F_{n}F$, which belongs to a Schwarz space, so the right-hand side has a finite value for every $\epsilon$, which means it is bounded. Taking the limit on $\epsilon$ gives

or equivalently,

Now, we can recover $F$ by taking the inverse Fourier transform as

which implies the conclusion. □

To obtain the inversion of $R_Lf$, we find the relation with the 2-dimensional quarter-spherical Radon transform $R_Hf$ and then apply (3.1) with $n = 2$.

Theorem 4. For $f\in C(\mathbb{R}^3)$ with compact support and $f(- {\bf x}, z) = -f({\bf x}, z)$, we have

where $\theta_{ {\bf x}}$ is the polar angle of ${\bf x}$.

The assumption $f(- {\bf x}, z) = -f({\bf x}, z)$ in the above Theorem 4 is necessary to prove it mathematically.

Proof. Notice that

Differentiating with respect to $\theta$ gives

Let $F_\theta(\hat{x}, \hat{z}) = [f(\hat{z}\cos(\theta+\frac\pi2), \hat{z}\sin(\theta+\frac\pi2), \hat{x})-f(\hat{z}\cos(\theta-\frac\pi2), \hat{z}\sin(\theta-\frac\pi2), \hat{x})]\hat{z}, \hat z\geq 0$. Then, it follows that

where $R_HF_\theta$ is the 2-dimensional quarter-spherical Radon transform on the line. By exploiting (3.1) with $n = 2$, we have, for $\hat{z} > 0$,

which implies the conclusion. □

4.   Conclusion and additional remarks

In this paper, we studied two quarter-spherical Radon transforms which may arise in endoscopic PAT with a certain type of detectors. These spherical Radon transforms, one of the incomplete spherical Radon transforms, are first suggested and their exact inversion fomulas are provided. Considering applications of the spherical Radon transform, our results are expected to have a significant impact both mathematically and in terms of applicability in endoscopic PAT as well as sesimic waves, radar and PDEs.

Lastly, we would like to note the following:

$1)$ Theorems 3.1 and 3.2 imply that the mapping $f \longmapsto R_Hf$ is one-to-one on $C_c(\mathbb{R}^n) = \{f\in C(\mathbb{R}^n): f \mbox{ has compact support}\}$ and the mapping $f \longmapsto R_Lf$ is one-to-one on $\{f \in C_c(\mathbb{R}^3): f(- {\bf x}, z) = -f({\bf x}, z)\}.$

$2)$ The smoothness and decay conditions for $f$ in both inversion formulae (Theorems 3.1 and 3.2) are not optimized. This formula can hold under weaker requirements, such as piecewise continuity of $f$, or $f$ decaying rapidly on the $z$ axis.

$3)$ Accurately and efficiently numerically implementing the inversion formula derived in the paper is an interesting problem in its own right. The authors plan to address this problem in future work.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The work of G. Hwang was supported by the 2020 Yeungnam University research grant. The work of S. Moon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (NRF-2022R1C1C1003464). We are thankful to the referees for multiple suggestions that helped to improve this paper. S. M. also thanks J. Lee, S. Mah and S. Park for fruitful discussions.

Conflict of interest

The authors declare that they have no competing interests.