A range characterization of the single-quadrant ADRT

TL;DR

This paper provides a mathematical characterization of the range of the single-quadrant ADRT, enabling stable inversion of square images through linear constraints and bijective properties.

Contribution

It introduces a linear range characterization for the single-quadrant ADRT and demonstrates stable inversion for data satisfying these constraints.

Findings

Range characterized by linear constraints

Stable inversion formula applicable to certain data

Bijection between images supported on half-strips

Abstract

This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization follows from a set of linear constraints on the codomain. We show that for data satisfying these constraints, the exact and fast inversion formula [Rim, Appl. Math. Lett. 102 106159, 2020] yields a square image in a stable manner. The range characterization is obtained by first showing that the ADRT is a bijection between images supported on infinite half-strips, then identifying the linear subspaces that stay finitely supported under the inversion formula.