## An EM Algorithm for Wavelet-Based Image Restoration (2002)

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Citations: | 235 - 20 self |

### BibTeX

@MISC{Figueiredo02anem,

author = {Mario A.T. Figueiredo and Robert D. Nowak},

title = {An EM Algorithm for Wavelet-Based Image Restoration},

year = {2002}

}

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### Abstract

This paper introduces an expectation-maximization (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 the wavelet coecients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require very demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation oered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an ecient iterative process requiring O(N log N) operations per iteration. Thus, it is the rst 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. Moreover, our new approach outperforms several of the best existing methods in benchmark tests, and in some cases is also much less computationally demanding.

### Citations

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Citation Context ...he underlying priors/models of real world images. Although wavelets have also been shown to be effective in image restoration problems (see [3], [4], [8], [9], [17], [18], [22], [28], [29], [33], and =-=[34]), m-=-ajor difficulties arise • unlike alone, is not block-circulant, thus it is not diagonalized by the DFT; • unlike alone, is not orthogonal, thus precluding efficient coefficient-wise rules. (6) (7)... |