Analysis of 3D SPECT image reconstruction and its extension to ultrasonic diffraction tomography

We present a detailed investigation of the theory for image reconstruction in 3D SPECT with uniform attenuation and distance-dependent spatial resolution. In particular, we present an alternative approach for derivation of relationships between the ideal and modified sinograms, and we investigate systematically the implication of the spatial resolution function for these relationships as well as on the ideal-sinogram estimation methods that are developed from these relationships. In our theory, the optical transfer function (OTF) of the resolution function is required to have a specific form. This form is satisfied by the OTF of the 3D Cauchy (or 3D Gaussian) exactly (or approximately). In this work, we develop a formula for converting a general OTF into that specific form so that our theory can be applied. Finally, we point out that the theory developed for SPECT can readily be generalized to 2D ultrasound diffraction tomography