Analytical reconstruction for multi-segment slant hole SPECT

In this paper, we present a ´fully 3D analytical reconstruction algorithm´ derived for the 2D parallel X-ray projections, whose imaging geometry can be divided into a set of circular arcs. The derived algorithm was applied to a novel multisegment slant hole (MSSH) single photon emission computed tomography (SPECT) system (Bal et al 2002) developed in our lab. For MSSH SPECT the acquisition geometry can be represented as four circular arcs, on the unit sphere of projection directions, where the central two arcs overlap to form a semi great circle while the other two arcs are parallel to and separated by a from the central semi circle. This MSSH geometry represents an over-determined system and hence the 3D filtered backprojection (FBP) algorithm can have an infinite number of valid filters. Assuming all projections have similar noise levels, the optimal FBP filter is found by taking central sections through the inverse of the 3D transfer function of the backprojection process. We have followed this (usual) path to derive the FBP filter for our MSSH geometry. The new contribution is the intermediate results of the filter for an arbitrary circular arc on the unit sphere of less than 360°. With our closed-form expression a large class of geometries can be constructed provided the geometry represents a collection of circular (partial) arcs. We have implemented the MSSH filter and run idealized simulations as well as physical phantom studies to verify the 3D FBP filter.