A continuation method for emission tomography

One approach to improved reconstructions in emission tomography has been the incorporation of additional source information via Gibbs priors that assume a source f that is piecewise smooth. A natural Gibbs prior for expressing such constraints is an energy function E(f,l) defined on binary valued line processes l as well as f. MAP estimation leads to the difficult problem of minimizing a mixed (continuous and binary) variable objective function. Previous approaches have used Gibbs 'potential' functions, φ(fv) and φ(fh), defined solely on spatial derivatives, fv and fh, of the source. These φ functions implicitly incorporate line processes, but only in an approximate manner. The correct φ function, φ*, consistent with the use of line processes, leads to difficult minimization problems. In this work, the authors present a method wherein the correct φ* function is approached through a sequence of smooth φ functions. This is the essence of a continuation method in which the minimum of the energy function corresponding to one member of the φ function sequence is used as an initial condition for the minimization of the next, less approximate, stage. The continuation method is implemented using a GEM-ICM procedure. Simulation results show improvement using the continuation method relative to using φ* alone, and to conventional EM reconstructions