A filtering approach to image reconstruction in 3D SPECT

We present a new approach to three-dimensional (3D) image reconstruction us ing analytical inversion of the exponential divergent beam transform, which can serve as a mathematical model for cone-beam 3D SPECT imaging. We apply a circular cone-beam scan and assume constant attenuation inside a convex area with a known boundary, which is satisfactory in brain imaging. The rec onstruction problem is reduced to an image restoration problem characterize d by a shift-variant point spread function which is given analytically. The method requires two computation steps: backprojection and filtering. The m odulation transfer function (MTF) of the filter is derived by means of an o riginal methodology using the 2D Laplace transform. The filter is implement ed in the frequency domain and requires 2D Fourier transform of transverse slices. In order to obtain a shift-invariant cone-beam projection-backproje ction operator we resort to an approximation, assuming that the collimator has a relatively large focal length. Nevertheless, numerical experiments de monstrate surprisingly good results for detectors with relatively short foc al lengths. The use of a wavelet-based filtering algorithm greatly improves the stability to Poisson noise.