The thin plate as a regularizer in Bayesian SPECT reconstruction

The attractions of the maximum-likelihood (ML) method for SPECT reconstruction, namely accurate system and noise modelling, are tempered by its unstable nature. The variety of Bayesian MAP approaches proposed in recent years can both stabilize the reconstructions and lead to better bias and variance. The authors' own work (see IEEE Trans. on Medical Imaging, vol. MI-14, no. 4, p. 669-80, Dec. 1995) used a nonquadratic prior (the weak plate) that imposed piecewise smoothness on the first derivative of the solution to achieve results superior to those obtained using a 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. Here, the authors 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. The authors used an EM-OSL (one-step-late) algorithm to compute MAP solutions with both these priors, and found that the thin plate exhibits better bias behaviour than the membrane at little cost in variance. Also, both priors outperform ML-EM in terms of bias and variance if practical stopping criteria are used for ML-EM. 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 late model leads to improved reconstructions in edge regions.