The hyperbolic X-ray transform: new range characterizations, mapping properties and functional relations

TL;DR

This paper introduces new mathematical characterizations and properties of the X-ray transform on the Poincaré disk, enhancing understanding of its invertibility and boundary behavior through advanced functional analysis.

Contribution

It provides novel range descriptions, singular value decompositions, and boundary analysis for the X-ray transform on the Poincaré disk, extending recent Euclidean disk results.

Findings

New singular value decompositions for the X-ray transform

Range characterizations and boundary behavior analysis

Surjectivity results for the backprojection operator

Abstract

We derive new singular value decompositions and range characterizations for the X-ray transform on the Poincar\'e disk, intertwining relations with distinguished differential operators of wedge type, and a surjectivity result for the backprojection operator. New functional settings are found, which allow to sharply understand boundary behavior issues and invertibility settings. The approach mainly exploits analogous results obtained only recently in the Euclidean disk, together with the projective equivalence between the two models.