A Loss Function Model for the Restoration of Grey Level Images

Common loss functions used for the restoration of grey scale images include the zero-one loss and the sum of squared errors. The corresponding estimators, the posterior mode and the posterior marginal mean, are optimal Bayes estimators with respect to their way of measuring the loss for different error configurations. However, both these loss functions have a fundamental weakness: the loss does not depend on the spatial structure of the errors. This is important because a systematic structure in the errors can lead to misinterpretation of the estimated image. We propose a new loss function that also penalizes strong local sample covariance in the error and we discuss how the optimal Bayes estimator can be estimated using a two-step Markov chain Monte Carlo and simulated annealing algorithm. We present simulation results for some artificial data which show improvement with respect to small structures in the image.