An iterative inversion of weighted Radon transforms along hyperplanes

F O Goncharov (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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) is not zero at all 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), 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.

Mots clés: weighted Radon transforms; tomography

Domaines: Physique mathématique [math-ph]

F O Goncharov. An iterative inversion of weighted Radon transforms along hyperplanes. 2016. ⟨hal-01405387v2⟩