The V-line transform with some generalizations and cone differentiation

TL;DR

This paper introduces new inversion formulas for the V-line and conical Radon transforms, providing theoretical insights and numerical demonstrations relevant to optical tomography and imaging applications.

Contribution

It presents novel explicit inversion formulas for VLT and CRT, along with a generalized Fundamental Theorem of Calculus called Cone Differentiation Theorem.

Findings

Derived new explicit inversion formulas for VLT and CRT

Described the range of VLT for broad function classes

Demonstrated efficiency through numerical examples

Abstract

The paper studies various properties of the V-line transform (VLT) in the plane and conical Radon transform (CRT) in $\mathbb{R}^n$. VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for VLT and CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the Fundamental Theorem of Calculus, which we call Cone Differentiation Theorem.