Characterization of Attenuated V-Line Transform and Direct Reconstruction from Samples

Abstract

The V-line transform (V-LT), a generalized Radon transform (RT), is widely used in Compton scattering imaging (CSI). The central mathematical problem of CSI is to reconstruct a function from its V-LT. Unlike V-LT, the attenuated V-line transform (AV-LT) accounts for the medium-induced loss of intensity, and has been demonstrated to improve image quality. The present paper makes two main contributions. (1) We establish the fundamental characterizations for AV-LT including operator properties ($L^2$-boundedness, operator norm etc.), a necessary and sufficient condition for the preservation of compact support, and sharp estimation of Sobolev smoothness. These characterizations remain unexplored, and they reveal essential differences among AV-LT, RT and V-LT. (2) We develop a direct reconstruction method based on AV-LT samples that does not require solving any equation. Our sampling formula gives exact reconstruction for band-limited functions. We also provide approximate reconstruction for functions in $H^s(\mathbb{R}^2)$ with $s>2$.