On artifacts in limited data spherical Radon transform: curved observation surface

• Published in: Inverse problems Vol. 32; no. 1; pp. 15012 - 15043
• Main Authors: Barannyk, Lyudmyla L, Frikel, Jürgen, Nguyen, Linh V
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.01.2016
• Subjects: limited data problem; Mathematical analysis; Mathematical models; microlocal analysis; pseudodifferential and Fourier integral operators; Radon; Reconstruction; Singularities; Smoothing; spherical Radon transform; Three dimensional; tomography
• ISSN: 0266-5611; 1361-6420
• DOI: 10.1088/0266-5611/32/1/015012

We study the limited data problem of the spherical Radon transform in two and three-dimensional spaces with general acquisition surfaces. In such situations, it is known that the application of filtered-backprojection reconstruction formulas might generate added artifacts and degrade the quality of reconstructions. In this article, we explicitly analyze a family of such inversion formulas, depending on a smoothing function that vanishes to order k on the boundary of the acquisition surfaces. We show that the artifacts are k orders smoother than their generating singularity. Moreover, in two-dimensional space, if the generating singularity is conormal satisfying a generic condition then the artifacts are even orders smoother than the generating singularity. Our analysis for three-dimensional space contains an important idea of lifting up space. We also explore the theoretical findings in a series of numerical experiments. Our experiments show that a good choice of the smoothing function leads to a significant improvement of reconstruction quality.