Exact reconstruction formulas for a Radon transform over cones

Technical Summary: Exact Reconstruction Formulas for a Radon Transform over Cones

Problem Statement
This paper addresses the mathematical inversion of a specific variant of the Radon transform known as the conical Radon transform, denoted as $R^{(p)}$. The transform integrates a smooth function $f$ supported in the upper half-space $\mathbb{R}^{d-1} \times (0, \infty)$ over one-sided conical surfaces. These cones are characterized by vertices located on the hyperplane $\mathbb{R}^{d-1} \times \{0\}$, a symmetry axis orthogonal to this plane (aligned with the $y$-axis), and a half-opening angle $\theta \in (0, \pi/2)$. The integration includes a radial weight $1/s^p$, where $s$ represents the distance along the cone's generator.

The primary motivation for studying this transform lies in computed tomography (CT) that accounts for Compton scattering. In such imaging modalities, the detection of scattered photons allows one to trace the source location back to the surface of a cone rather than a line or plane. While Fourier reconstruction formulas exist for specific cases in three dimensions ($p \in \{0, 2\}$) and back-projection formulas exist for two dimensions, the general case for arbitrary spatial dimension $d \geq 2$ and arbitrary weight parameter $p \in \mathbb{R}$ had not been previously studied.

Methodology
The author, Markus Haltmeier, employs harmonic analysis techniques, specifically leveraging the Fourier transform and the Hankel transform, to derive the inversion formulas. The core of the methodology involves establishing a "Fourier slice identity" for the conical Radon transform, analogous to the well-known Fourier slice theorem for the classical Radon transform.

The derivation proceeds in three logical steps:

1. Reduction to the Unweighted Case: A lemma is established showing that the transform $R^{(p)}$ for an arbitrary weight $p$ can be related to the unweighted transform $R^{(0)}$ via simple multiplicative operators involving $\cos(\theta)$ and $y^{-p}$. This allows the authors to focus their derivation on the case $p=0$ and generalize the results later.
2. Derivation of the Fourier Slice Identity: By applying the Fourier transform with respect to the spatial variable $u \in \mathbb{R}^{d-1}$ and utilizing the integral representation of Bessel functions over the sphere $S^{d-2}$, the author derives an identity linking the Fourier-Hankel transform of the original function $f$ to the Fourier transform of the conical Radon transform data.
3. Inversion via Back-Projection: Using the derived Fourier slice identity, the author inverts the transform by applying the inverse Fourier and Hankel transforms. This process leads to explicit reconstruction formulas of the filtered back-projection type. The "filtering" step is identified as the application of the Riesz potential $I^{(1-d)}$ with respect to the first argument.

Key Contributions and Results
The paper provides two primary explicit reconstruction formulas for recovering $f(x, y)$ from the data $R^{(p)}f$:

1. Angular Parametrization (Theorem 1.1, Eq. 1.3):
   $$f(x, y) = \frac{y^p}{(2\pi)^{d-1}} \int_0^{\pi/2} \frac{1}{\cos(\theta)^{1+p}} \int_{S^{d-2}} \left( I^{(1-d)} R^{(p)} f \right)(x + y \tan(\theta) n, \theta) \, dn \, d\theta$$
   This formula integrates over all cones passing through the reconstruction point $(x, y)$, parametrized by the opening angle $\theta$ and the direction $n$ on the unit sphere.

2. Spatial Parametrization (Theorem 1.1, Eq. 1.4):
   $$f(x, y) = \frac{1}{(2\pi)^{d-1}} \int_{\mathbb{R}^{d-1}} \frac{(|u-x|^2 + y^2)^{\frac{p-1}{2}}}{|u-x|^{d-2}} \left( I^{(1-d)} R^{(p)} f \right)\left(u, \arctan\frac{|u-x|}{y}\right) \, du$$
   This form integrates over the base plane $\mathbb{R}^{d-1}$, summing contributions from cones whose vertices lie at $u$.

Specific Dimensions:

• For $d=2$: The Riesz potential $I^{-1}$ corresponds to the composition of the derivative and the Hilbert transform ($\partial_u H_u$).
• For $d=3$: The Riesz potential $I^{-2}$ corresponds to the negative Laplacian ($-\Delta_u$). Consequently, the 3D reconstruction formula involves applying the Laplacian to the data followed by a conical back-projection.

Significance and Claims
The paper claims that these are the first explicit reconstruction formulas of the filtered back-projection type for the conical Radon transform in arbitrary dimensions $d \geq 2$ and for any real weight $p$.

• Locality: The author notes a distinction based on the parity of the dimension $d$. For odd dimensions, the reconstruction is local, meaning the value of $f$ at a point $(x, y)$ depends only on integrals over cones passing through an arbitrarily small neighborhood of that point. For even dimensions, the reconstruction is non-local, requiring knowledge of integrals over all conical surfaces. This behavior mirrors that of the classical Radon transform.
• Novelty: While equivalent formulas for $d=2$ were previously derived in other works, the formulas for $d=3$ and higher dimensions are presented as new contributions.
• Practicality: The paper asserts that the 3D reconstruction formulas are particularly simple, consisting of a Laplacian application followed by conical back-projection. The author suggests that numerical implementation should be straightforward, following established methods for classical and spherical Radon transforms, though numerical studies are left for future research.

The work is framed as a theoretical foundation for emission tomography using Compton cameras, providing the necessary mathematical tools to invert the data generated by such devices.