Parameter selection for a Markovian signal reconstruction with edge detection

We propose a method for the parameter selection for a Bayesian reconstruction of 1D or 2D signals, constituted by locally homogeneous regions, from incomplete and noisy projection data. A piecewise Gaussian Markov model (PG MM), defined as a sum of truncated quadratic potential functions, is used to regularise the reconstruction, which is otherwise ill-posed. This model is called the weak string in 1D and the weak membrane in 2D. The posterior energy is highly non-convex and the MAP estimator is piecewise continuous; the model parameters play a particularly decisive role. The resolution of the reconstruction-the finest recoverable features-is determined jointly by the parameters and the observation model. On the other hand, we propose a method for the determination of the parameters in order to reach, or at least to approach as closely as possible, a desired resolution. This model needs the evaluation of several posterior edge detection thresholds