Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

Abstract

Let Im denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f , and I * m be its formal L 2 adjoint. We study a unique continuation result for the normal operator Nm = I * m Im. More precisely, we show that if Nm vanishes to infinite order at a point x 0 and if the Saint-Venant operator W acting on f vanishes on an open set containing x 0 , then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.

Inversion of a restricted transverse ray transform with sources on a curve

“…To the best of authors' knowledge, there are only two works available with restricted TRT; the first one is [8], where authors have developed a slice-by-slice filtered back projection reconstruction algorithm for symmetric 2-tensor fields in R 3 while the second one [23] deals with microlocal inversion for symmetric m-tensor fields in R 3 . The articles [1,2,15,21] address the injectivity questions for TRT in different settings. For more literature related to transverse ray transform, please refer to [12,16,25,26,31,36], and the references therein.…”