Scattered PET data for attenuation-map reconstruction in PET/MRI: Fundamentals

It is well-known that attenuation correction is important in Positron Emission Tomography (PET), and particularly challenging when the latter is combined with Magnetic Resonance Imaging (MRI). Recently, we have presented techniques to use scattered PET coincidences for reconstruction of attenuation maps. These techniques primarily rely on scatter-to-attenuation back-projection of scattered coincidences on potential scattering locations in image space. In this article, we present some of the fundamentals not included in, but useful for comprehending these previous publications. This comprises examples illustrating the need for additional information in PET-based PET attenuation correction (in particular, sinogram and image-space ambiguities), a scatter coordinate system used for expressing integral terms, and the derivation of a single-scatter measurement equation similar to those found earlier by other authors.