On Radon transforms on compact Lie groups

Abstract

Abstract. We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to S 1 nor to S 3 . This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from S 1 .

Fourier Analysis of Periodic Radon Transforms

“…Since the second term of (80) is independent of f , it can be neglected in the minimization problem (8). On the other hand,…”

“…The X-ray transform and tensor tomography on T n has been applied to other integral geometry problems. These examples include the broken ray transform on boxes [7], the geodesic ray transform on Lie groups [8], tensor tomography on periodic slabs [10], and the ray transforms on Minkowski tori [9]. We expect that the d-plane Radon transform on T n has applications in similar and generalized geometric problems as well, but have not studied this possibility any further.…”

A support theorem for the X-ray transform on manifolds with plane covers

“…See, e.g., [19], for a survey with a list of open problems and [16,17] for extensions of these methods to Cartan-Hadamard manifolds. In this paper we will only consider non-compact manifolds, but there are also various results about the X-ray and related transforms for compact homogeneous spaces, see [9, section IV•1] and references in its bibliographic notes and, e.g., [4,10] for other recent results.…”

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]).…”