Variational regularization of the weighted conical Radon transform

Abstract

Recovering a function from integrals over conical surfaces recently got significant interest. It is relevant for emission tomography with Compton cameras and other imaging applications. In this paper, we consider the weighted conical Radon transform with vertices on the sphere. Opposed to previous works on conical Radon transform, we allow a general weight depending on the distance of the integration point from the vertex. As first main result, we show uniqueness of inversion for that transform. To stably invert the weighted conical Radon transform, we use general convex variational regularization. We present numerical minimization schemes based on the Chambolle-Pock primal dual algorithm. Within this framework, we compare various regularization terms, including non-negativity constraints, $H^1$-regularization and total variation regularization. Compared to standard quadratic Tikhonov regularization, TV-regularization is demonstrated to significantly increase the reconstruction quality from conical Radon data.

References in corpus (5)

• Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm
• Exact inversion of the conical Radon transform with a fixed opening angle
• Inversion formulas for the broken-ray Radon transform
• Detecting small low emission radiating sources
• Artifacts in the inversion of the broken ray transform in the plane

Cited by in corpus (1)

• On a three dimensional Compton scattering tomography system with fixed source