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

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.