An EM algorithm for wavelet-based image restoration

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 the 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, the resulting criteria are solved approximately or require 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 the Fourier domain. Thus, it is a general-purpose approach to wavelet-based image restoration with computational complexity comparable to that of standard wavelet denoising schemes or of frequency domain deconvolution methods. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(N log N) operations per iteration. 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 performs competitively with, in some cases better than, the best existing methods in benchmark tests.

A coordinate gradient descent method for ℓ 1-regularized convex minimization

“…and {J k } is chosen by the Gauss-Southwell-r rule (12) or the GaussSouthwell-q rule (13). Then {x k } converges to some x * ∈ X * .…”

“…Following [14], we considered three standard benchmark problems summarized in Table 7, all based on the well-known Cameraman image [12,13]. The description of A = RW is found in the end of Subsection 4.3.…”

“…First, we designed and implemented an efficient and robust method (CGD method), which can be viewed as a hybrid of the block coordinate descent with FPC/IST, for solving the convex 1 -minimization problems (1). In particular, we demonstrate its effectiveness on large-scale problems (2) arising from applications such as compressed sensing [4,5,9,37,38] and image deconvolution [12,13,36], and (4) arising from data classification. At each iteration of this method, we replace f in F ρ by a strictly convex quadratic approximation and apply block coordinate descent to generate a feasible descent direction.…”

“…But the theoretical convergence properties have not been analyzed. This method and the FPC method are closely related to the iterative shrinkage/thresholding (IST) methods [8,12,13]. The FPC method and the IST methods use the same form of subproblem (3) to generate a sequence of iterates x k with a more conservative choice of η k where η k is fixed at a constant value.…”

“…This method is simple, highly parallelizable, and is suited for solving largescale problems. We choose the coordinate block according to the Gauss-Southwell (GS) type rules; see (12) and (13). We choose the stepsize according to an Armijo type rule for the general problem (1) and (4); see (8) and Section 5.…”