Image restoration under wavelet-domain priors: an expectation-maximization approach

This paper describes an expectation-maximization (EM) algorithm for wavelet-based image restoration (deconvolution). The observed image is assumed to be a convolved (e.g., blurred) and noisy version of the original image. Regularization is achieved by using a complexity penalty/prior in the wavelet domain, taking advantage of the well known sparsity of wavelet representations. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator in the discrete Fourier domain. The algorithm alternates between an FFT-based E-step and a DWT-based M-step, resulting in a very efficient iterative process requiring O(N log N) operations per iteration (where N stands for the number of pixels). The algorithm, which also estimates the noise variance, is called WAFER, standing for wavelet and Fourier EM restoration. The conditions for convergence of the proposed algorithm are also presented.