An analysis of the role of positivity and mixture model constraints in poisson deconvolution problems

We consider a class of estimation problems in which data of a Poisson character are related by a linear model to a target function that satisfies certain physical constraints. The classic example of this situation is the reconstruction problem of positron emission tomography (PET). There the function of interest satisfies positivity constraints. This article examines the impact of such constraints by comparing simple unconstrained reconstruction methods with constrained alternatives based on maximum likelihood (ML) and least squares (LS) formulations. Data from a series of numerical experiments are presented to quantify the significance of constraints. Although these experiments show that constraints are important, the differences between ML and LS based implementations of constraints are quite small. Thus, in order to evaluate the impact of constraints, it appears to be sufficient to focus on comparing constrained versus unconstrained implementations of LS. This simplifies the analysis of constraints considerably. A perturbation analysis technique is proposed to summarize the impact of constraints in terms of a single relative efficiency measure. The predictions obtained by this analysis are found to be in good agreement with experimental da.