Inversion of the exponential X-ray transform. II: Numerics

The exponential X-ray transform arises in single photon emission computed tomography and is defined on functions on the plane by ??_μf(φ,x) = ∫f (x + tφ)e^μt where μ is a constant. In MMAS(10), 561–574, 1988], we derived analytical formulae for filters K corresponding to a general point spread function E that can be used to invert the exponential X-ray transform via a filtered backprojection algorithm. Here, we use those formulae to derive expressions suitable for numerical computation of the filters corresponding to a specific family of bandlimited point spread functions and give the results of reconstructions of a mathematical phantom using these filters. Also included is an analogue of the Shepp–Logan ellipse theorem, IEEE Trans. Nucl. Sci. (21), 21–43, 1974], for the exponential X-ray transform.