The spherical Radon transform with centers on cylindrical surfaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:448

Article

Haltmeier, Markus¹  Moon, Sunghwan² 

[1] Univ Innsbruck, Dept Math, Technikestr 13, A-6020 Innsbruck, Austria

[2] Ulsan Natl Inst Sci & Technol, Dept Math Sci, Ulsan 44919, South Korea

关键词: Spherical means;    Radon transform;    Inversion;    Reconstruction formula;   

DOI  :  10.1016/j.jmaa.2016.11.022

来源: Elsevier

【 摘 要 】

Recovering a function from its spherical Radon transform with centers of spheres of integration restricted to a hypersurface is at the heart of several modern imaging technologies, including SAR, ultrasound imaging, and photo- and thermoacoustic tomography. In this paper we study an inversion of the spherical Radon transform with centers of integration restricted to cylindrical surfaces of the form Gamma x R-m, where r is a hypersurface in R-n. We show that this transform can be decomposed into two lower dimensional spherical Radon transforms, one with centers on r and one with a planar center-set in Rm+1. Together with explicit inversion formulas for the spherical Radon transform with a planar center-set and existing algorithms for inverting the spherical Radon transform with a center-set Gamma, this yields reconstruction procedures for general cylindrical domains. In the special case of spherical or elliptical cylinders we obtain novel explicit inversion formulas. For three spatial dimensions, these inversion formulas can be implemented efficiently by backprojection type algorithms only requiring Omicron(N-4/3) floating point operations, where N is the total number of unknowns to be recovered. We present numerical results demonstrating the efficiency of the derived algorithms. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free