On solenoidal-injective and injective ray transforms of tensor fields on surfaces

TL;DR

This paper addresses tensor tomography on simple surfaces, introducing new injective integral transforms for tensor fields through weighted and transverse approaches, advancing the understanding of tensor field reconstruction.

Contribution

It provides a constructive solution to the attenuated tensor tomography problem and proposes two novel fully injective vector-valued integral transforms for tensor fields.

Findings

Constructive solution to attenuated tensor tomography on simple surfaces

Development of weighted geodesic flow transforms generalizing moment transforms

Introduction of transverse ray transforms for tensor field injectivity

Abstract

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields. The first approach is by construction of appropriate weights which vary along the geodesic flow, generalizing the moment transforms. The second one is by changing the pairing with the tensor field to generate a collection of transverse ray transforms.