Abstract. The Bayesian approach and especially the maximum a posteriori (MAP) estimator is most widely used to solve various problems in signal and image processing, such as denoising and deblurring, zooming, and reconstruction. The reason is that it provides a coherent statistical framework to combine observed (noisy) data with prior information on the unknown signal or image which is optimal in a precise statistical sense. This paper presents an objective critical analysis of the MAP approach. It shows that the MAP solutions substantially deviate from both the data-acquisition model and the prior model that underly the MAP estimator. This is explained theoretically using several analytical properties of the MAP solutions and is illustrated using examples and experiments. It follows that the MAP approach is not relevant in the applications where the data-observation and the prior models are accurate. The construction of solutions (estimators) that respect simultaneously two such models remains an open question. We address a wide variety of inverse problems where an estimatex ∈ R p of an original (unknown) X = x (e.g. an image, a signal or some parameters) is recovered from a realization 1 of noisy data Y = y ∈ R q using a statistical model for their production as well as a prior model for the original X. Typical applications are signal and image restoration, segmentation, motion estimation, sequence processing, color reproduction, optical imaging, tomography, seismic and nuclear imaging, and many others. The likelihood function f Y|X (y|x)-the distribution for the observed data Y = y given an original X = x-is governed by physical considerations concerned with the data-acquisition device. By far the most common models are of the form where A : R p → R q is a linear operator (e.g. a blurring kernel, a Fourier or a Radon transform, or a subsampling operator) and N is additive noise which is independent of X. If the noise samples N i , 1 ≤ i ≤ q are independent and identically distributed 2000 Mathematics Subject Classification: Primary: 62H99, 68U99; Secondary: 49K99.
However, Nikolova REF pointed out from the viewpoint of MAP that the solutions of the TVL1 model substantially deviate from both the data-acquisition model and the prior model, and Minru Bai [23] further pointed out that the TVL1 model does not perform well at the sparsity of Kx − f and there are many biased estimates produced by the TVL1 model.
Model distortions in Bayesian MAP reconstruction
Inverse Problems and Imaging
Mathematics