The cone-beam transform and spherical convolution operators. (17th August 2018)

• Record Type: Journal Article
• Authors: Quellmalz, Michael; Hielscher, Ralf; Louis, Alfred K
• Is Part Of: Inverse problems. Volume 34:Number 10(2018:Oct.)
• Publication Date: 2018-08-17
• Subjects: cone-beam transform -- singular value decomposition -- spherical convolution -- Funk–Radon transform -- Radon transform; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aad679
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis (2016 Inverse Problems 32 115005) states reconstruction formulas based on a new generalized Funk–Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk–Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev (2015 J. Inverse Ill-Posed Problems 23 173–85).