Geodesic ray transform with matrix weights for piecewise constant functions

Abstract

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

“…The geodesic ray transform is closely related to the boundary rigidity problem [60,68] and the spectral rigidity of closed Riemannian manifolds [22,23,56]. Other recent considerations include generalizations of many existing results to some classes of open Riemannian manifolds [18,21,24,44] and to the matrix weighted ray transforms [38,59] as well as their statistical analysis [45,46].…”

A geometric based preprocessing for weighted ray transforms with applications in SPECT

“…Weighted Radon transforms defined in (1), (2) and some of their generalizations are wellknown in many domains of pure mathematics: in theory of groups ([G+59], [G+62], [HC58a], [He65], [He99], [Il16]), harmonic analysis ( [St82], [St91]), PDEs ( [Be84], [Jo55]), integral geometry ( [Sh12], [Pa+16]), microlocal analysis ( [Be84], [Qu+14], [Qu+18]) and can be also of self-interest ( [Qu80], [Fri+08], [Bo11], [Il19]). At the same time, transformations P W (and a lot less R W ) are used as an important tool in many computerized tomographies ( [Qu83], [Na86], [Mi+87], [Ku14], [No02], [Qu06], [De07], [Ngu+09], [BJ11], [MiDeP11]).…”