Inversion of generalized Radon transform over symmetric $m$-tensor fields in $\mathbb{R}^n$

TL;DR

This paper introduces and analyzes generalized Radon transforms for symmetric m-tensor fields in Euclidean space, providing kernel descriptions and demonstrating unique recoverability of the tensor fields from these transforms.

Contribution

It extends previous work on vector fields to symmetric m-tensor fields, establishing invertibility and kernel properties of the generalized Radon transforms.

Findings

Kernel descriptions for Radon transforms of tensor fields

Unique recovery of tensor fields from Radon transform data

Generalization of vector field results to m-tensor fields

Abstract

In this work, we study a set of generalized Radon transforms over symmetric $m$-tensor fields in $\mathbb{R}^n$. The longitudinal/transversal Radon transform and corresponding weighted integral transforms for symmetric $m$-tensor field are introduced. We give the kernel descriptions for the longitudinal and transversal Radon transform. Further, we also prove that a symmetric $m$-tensor field can be recovered uniquely from certain combinations of these integral transforms of the unknown tensor field. This generalizes a recent study done for the recovery of vector fields from its weighted Radon transform data to recovery of a symmetric $m$-tensor field from analogously defined weighted Radon transforms.