Inversion of the attenuated V-line transform for SPECT with Compton cameras

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.