An iterative inversion of weighted Radon transforms along hyperplanes

TL;DR

This paper develops iterative algorithms for inverting weighted Radon transforms in three dimensions, extending previous 2D results and applicable to 3D emission tomography, by reducing the problem to solving linear integral equations.

Contribution

It introduces a new iterative inversion method for weighted Radon transforms in 3D, generalizing earlier 2D approaches and applicable to higher dimensions.

Findings

Linear integral equations can be solved via successive approximations.

Approximate inversions of the weighted Radon transform are provided.

Results extend 2D inversion techniques to 3D and higher dimensions.

Abstract

We propose iterative inversion algorithms for weighted Radon transforms $R_W$ along hyperplanes in $R^3$. More precisely, expandingthe weight $W = W (x, \theta), x \in R^3 , \theta \in S^2$ , into the series of spherical harmonics in $\theta$ and assuming that the zero order term $w_{0,0}(x)$ is not zero at any $x \in R^3$ , 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 $\theta$ (i.e., $1/2(W (x, \theta) + W (x, -\theta))$) 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), Novikov (2014), Guillement, Novikov (2014). 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 $n > 3$.