Mathematical formulation of the potato peeler perspective

In parallel beam computed tomography, the measured projections at conjugate views are mathematically identical. This symmetry can be exploited for either reducing the scanning angle, or the size of the detector arrays. In single-photon emission computed tomography (SPECT), because the gamma-rays in the conjugate views suffer different photon attenuation, the measured projections at conjugate views are generally different. Therefore, it had been often considered that projections over 360 degrees and the whole detector face are required for exactly reconstructing the distributions of gamma-ray emitters. In the case of uniform attenuation, it has been shown that redundant information in the sinogram can be exploited to perform reconstruction from short-scan data. For the case of non-uniform attenuation we have recently proposed the "potato-peeler" perspective to reveal redundant information in the sinogram data, and that only "half-data" is needed to reconstruct a unique image. In this work, we present a preliminary investigation on the extension of the mathematical formulation of the potato-peeler perspective to cases when there are a finite number of discontinuities in the activity distribution.