A Bayesian multiscale approach to joint image restoration and edge detection

This paper presents a novel wavelet-based method for simultaneous image restoration and edge detection. The Bayesian framework developed here is general enough to treat a wide class of linear inverse problems involving (white or colored) Gaussian observation noises, but we focus on convolution operators. In our new approach, a signal prior is developed by modeling the signal/image wavelet coefficients as independent Gaussian mixture random variables. We specify a uniform (non-informative) distribution on the mixing parameters, which leads to an extremely simple iterative algorithm for joint MAP restoration and edge detection. This algorithm is similar to the popular EM algorithm in that it alternates between a state estimation step and a maximization step, yet it is much simpler in each step and has a very intuitive derivation. Moreover, we show that our algorithm converges monotonically to a local maximum of the posterior distribution. Experimental results show that this new method can perform better than wavelet-vaguelette type methods that are based on linear inverse filtering followed by wavelet coefficient denoising.