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

TL;DR

This work develops a framework for tensor tomography by introducing generalized Radon transforms on symmetric $m$-tensor fields in $\mathbb{R}^n$, including the longitudinal and transversal transforms and their weighted variants. The authors obtain kernel descriptions for both transforms and prove that a tensor field $f$ admits a unique decomposition $f = \sum_{i=0}^m d^i v_i$ with $v_i$ solenoidal, enabling full recovery from combined data: either $LRT$ with weighted $LRT$ of order $k$, or $TRT$ with weighted $TRT$ of order $k$. They provide explicit reconstruction procedures that use componentwise Radon transforms, Radon inversion, and elliptic solvability to recover each $v_i$ sequentially, first the solenoidal part $v_0$ (or $v_m$ in the transversal setting) and then the higher-order components. The results generalize prior vector-field inversions to symmetric $m$-tensor fields in arbitrary dimensions, with potential impact on applications in tensor tomography and imaging where tensor-valued data arise.

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.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3