[Paper Review] The Vertical Slice Transform in Spherical Tomography

Boris Rubin|arXiv (Cornell University)|Jul 20, 2018

Photoacoustic and Ultrasonic Imaging26 references4 citations

TL;DR

This paper presents new inversion formulas and singular value decomposition for the vertical slice transform in spherical tomography, applicable to thermoacoustic and photoacoustic imaging. By relating the transform to the hyperplane Radon transform and using analytic continuation, it achieves explicit reconstruction of functions on the sphere from integrals over vertical slices, solving the inverse problem for the Euler-Poisson-Darboux equation in spherical geometry.

ABSTRACT

The vertical slice transform takes a function on the n-dimensional unit sphere to integrals of that function over spherical slices parallel to the last coordinate axis. This transform arises in thermoacoustic tomography. We obtain new inversion formulas for the vertical slice transform and its singular value decomposition. The results can be applied to the inverse problem for the Euler-Poisson-Darboux equation associated to the corresponding spherical means.

Motivation & Objective

• To develop inversion formulas for the vertical slice transform in spherical tomography on the unit sphere $S^n$ for $n \geq 2$.
• To address the inverse problem of reconstructing a function from its integrals over vertical hyperplane sections parallel to the last coordinate axis.
• To connect the vertical slice transform to the hyperplane Radon transform via a change of variables and surface measure adjustment.
• To provide explicit inversion formulas using analytic continuation and singular value decomposition for the transform.
• To apply the results to the inverse problem of the Euler-Poisson-Darboux equation on the sphere, particularly for initial data reconstruction from equatorial measurements.

Proposed method

• The vertical slice transform $Vf$ is defined as integration of a function $f$ over spherical slices parallel to the last coordinate axis.
• By exploiting the evenness of $f$ in the last variable, the problem is reduced to the upper hemisphere, leading to the hemispherical transform $V_+f$.
• A change of variables maps the spherical integral to a Radon transform in $\mathbb{R}^n$, using the relation $\varphi(x') = f(x', \sqrt{1-|x'|^2}) / \sqrt{1-|x'|^2}$.
• The inverse is constructed via the inverse Radon transform $R^{-1}$, yielding $f(x', x_{n+1}) = x_{n+1} \cdot (R^{-1}\Phi)(x')$ with $\Phi(\theta,t) = (V_+f)(\theta,t)/\sqrt{1-t^2}$.
• For $n=2$, an alternative method using analytic continuation and logarithmic potentials is developed, leading to an integral formula involving $\Delta$ and $\log|t - \theta \cdot x'|$.
• The method explicitly inverts the transform via a double integral over $S^1 \times [-1,1]$ with a logarithmic kernel and normalization by $\sqrt{1-t^2}$.

Experimental results

Research questions

• RQ1 How can the vertical slice transform on $S^n$ be inverted for $n \geq 2$ using known transforms in Euclidean space?
• RQ2 What is the relationship between the vertical slice transform and the hyperplane Radon transform in $\mathbb{R}^n$?
• RQ3 Can the inverse of the vertical slice transform be constructed via analytic continuation and potential theory for $n=2$?
• RQ4 How does the singular value decomposition of the vertical slice transform relate to its inversion properties?
• RQ5 What is the connection between the vertical slice transform and the Euler-Poisson-Darboux equation on the sphere?

Key findings

• The vertical slice transform $Vf$ is shown to be equivalent to the hyperplane Radon transform $R\varphi$ via the relation $(V_+f)(\theta,t) = \sqrt{1-t^2} \cdot (R\varphi)(\theta,t)$, where $\varphi$ is derived from $f$ by a change of variables.
• For $n \geq 2$, the function $f$ can be reconstructed from $Vf$ via $f(x', x_{n+1}) = x_{n+1} \cdot (R^{-1}\Phi)(x')$ with $\Phi(\theta,t) = (V_+f)(\theta,t)/\sqrt{1-t^2}$, providing a general inversion formula.
• For $n=2$, an alternative inversion formula is derived: $f(x) = \frac{|x_3|}{8\pi^2} \Delta \int_{S^1} d\sigma(\theta) \int_{-1}^1 (Vf)(\theta,t) \frac{\log|t - \theta \cdot x'|}{\sqrt{1-t^2}} dt$, using logarithmic potentials and analytic continuation.
• The singular value decomposition of the vertical slice transform is explicitly computed, revealing its spectral structure and stability properties.
• The results are applied to the inverse problem of the Euler-Poisson-Darboux equation: initial data $f$ on $S^n$ can be reconstructed from the solution $u(x,\omega)$ restricted to the equator $S^{n-1}$, particularly for $\alpha = (1-n)/2$ corresponding to the wave equation.
• The method provides a constructive solution to the inverse problem for geodesic spheres of fixed radius $0 < \theta \leq \pi/2$, extending prior results to a broader class of problems.