The thin plate as a regularizer in Bayesian SPECT reconstruction

Bayesian MAP (maximum a posteriori) methods for SPECT reconstruction can both stabilize reconstructions and lead to better bias and variance relative to ML methods. In previous work, a nonquadratic prior (the weak plate) that imposed piecewise smoothness on the first derivative of the solution led to much improved bias/variance behavior relative to results obtained using a more conventional nonquadratic prior (the weak membrane) that imposed piecewise smoothness of the zeroth derivative. By relaxing the requirement of imposing spatial discontinuities and using instead a quadratic (no discontinuities) smoothing prior, algorithms become easier to analyze, solutions easier to compute, and hyperparameter calculation becomes less of a problem. In this work, we investigated whether the advantages of weak plate relative to weak membrane are retained when non-piecewise quadratic versions-the thin plate and membrane priors-are used. We compared, with three different phantoms, the bias/variance behavior of three approaches: (1) FBP with membrane and thin plate implemented as smoothing filters, (2) ML-EM with two stopping criteria, and (3) MAP with thin plate and membrane priors. In cases (1) and (3), the thin plate always led to better bias behavior at comparable variance relative to membrane priors/filters. Also, approaches (1) and (3) outperformed ML-EM at both stopping criteria. The net conclusion is that, while quadratic smoothing priors are not as good as piecewise versions, the simple modification of the membrane model to the thin plate model leads to improved bias behavior.