Daniela Schiefeneder

Daniela Schiefeneder, Markus Haltmeier

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such emission tomography using Compton cameras. In this paper we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Because the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to numerically implement our inversion method and present numerical results.

Markus Haltmeier, Daniela Schiefeneder

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.

Markus Haltmeier, Sunghwan Moon, Daniela Schiefeneder

The Compton camera is a promising alternative to the Anger camera for imaging gamma radiation, with the potential to significantly increase the sensitivity of SPECT. Two-dimensional Compton camera image reconstruction can be implemented by inversion of the V-line transform, which integrates the emission distribution over V-lines (unions of two half-lines), that have vertices on a surrounding detector array. Inversion of the V-line transform without attenuation has recently been addressed by several authors. However, it is well known from standard SPECT that ignoring attenuation can significantly degrade the quality of the reconstructed image. In this paper we address this issue and study the attenuated V-line transform accounting for attenuation of photons in SPECT with Compton cameras. We derive an analytic inversion approach based on circular harmonics expansion, and show uniqueness of reconstruction for the attenuated V-line transform. We further develop a discrete image reconstruction algorithm based on our analytic studies, and present numerical results that demonstrate the effectiveness of our algorithm.