Image Restoration Using the EM Algorithm and Wavelet-Based Complexity Regularization
Figueiredo, Mario; Nowak, Robert David
This paper introduces an <i>expectation-maximization</i> (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in terms of teh wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, teh resulting criteria are solved approximately or requre very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in teh Fourier domain. The algorithm alternates between an E-step based on teh fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring <i>O</i>(<i>N</i>log<i>N</i>) operations per iteration. Thus, it is the first image restoration algorithm that optimizes a wavelet-based penalized likelihood criterion and has computational complexity comparable to that of standard wavelet denoising or frequency domain deconvolution methods. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Morever, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.