An iterative inversion of weighted Radon transforms along hyperplanes. (10th November 2017)

• Record Type: Journal Article
• Authors: Goncharov, F O
• Is Part Of: Inverse problems. Volume 33:Number 12(2017:Dec.)
• Publication Date: 2017-11-10
• Subjects: Radon transforms -- iterative inversion -- tomography; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa91a4
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We propose iterative inversion algorithms for weighted Radon transforms R W along hyperplanes in R 3 . More precisely, expanding the weight W = W ( x, θ ), x ∈ R 3, θ ∈ S 2, into the series of spherical harmonics in θ and assuming that the zero order term w 0, 0 ( x ) ≠ 0, x ∈ R 3, we reduce the inversion of R W to solving a linear integral equation. In addition, under the assumption that the even part of W in θ (i.e. 1 2 ( W ( x, θ ) + W ( x, − θ ) ) ) is close to w 0, 0, the aforementioned linear integral equation can be solved by the method of successive approximations. Approximate inversions of R W are also given. Our results can be considered as an extension to 3D of two-dimensional results of Kunyansky (1992 Inverse Problems 8 809–19), Novikov (2014 Mosc. Math. J .14 807–23), Guillement and Novikov (2014 Inverse Problems Sci. Eng .22 787–802). In our studies we are motivated, in particular, by problems of emission tomographies in 3D. In addition, we generalize our results to the case of dimension n > 3 .