Abstract—Many image processing problems are ill posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. In this paper, we first give conditions for the design of such an edge-preserving regularization. Under these conditions, we show that it is possible to introduce an auxiliary variable whose role is twofold. First, it marks the discontinuities and ensures their preservation from smoothing. Second, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This leads to the definition of an original reconstruction algorithm, called ARTUR. Some theoretical properties of ARTUR are discussed. Experimental results illustrate the behavior of the algorithm. These results are shown in the field of tomography, but this method can be applied in a large number of applications in image processing. I.

. We consider least-squares problems where the coefficient matrices A; b are unknown-butbounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Least-squares, uncertainty, robustness, second-order cone...

We propose, analyze and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also isotropic forms of total variation discretizations. The per-iteration computational complexity of the algorithm is three Fast Fourier Transforms (FFTs). We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or q-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the Lagged Diffusivity algorithm for total-variation-based deblurring. Some extensions of our algorithm are also discussed.

In this paper, we propose the application of the hierarchical Bayesian paradigm to the image restoration problem. We derive expressions for the iterative evaluation of the two hyperparameters applying the evidence and maximum a posteriori (MAP) analysis within the hierarchical Bayesian paradigm. We show analytically that the analysis provided by the evidence approach is more realistic and appropriate than the MAP approach for the image restoration problem. We furthermore study the relationship between the evidence and an iterative approach resulting from the set theoretic regularization approach for estimating the two hyperparameters, or their ratio, defined as the regularization parameter. Finally the proposed algorithms are tested experimentally.

In this paper, we present a new approach to the recovering of dipole magnitudes in a distributed source model for magnetoencephalographic (MEG) and electroencephalographic (EEG) imaging. This method consists in introducing spatial and temporal a priori information as a cure to this ill-posed inverse problem. A nonlinear spatial regularization scheme allows the preservation of dipole moment discontinuities between some a priori noncorrelated sources, for instance, when considering dipoles located on both sides of a sulcus. Moreover, we introduce temporal smoothness constraints on dipole magnitude evolution at time scales smaller than those of cognitive processes. These priors are easily integrated into a Bayesian formalism, yielding a maximum a posteriori (MAP) estimator of brain electrical activity. Results from EEG simulations of our method are presented and compared with those of classical quadratic regularization and a now popular generalized minimum-norm technique called low-resolution electromagnetic tomography (LORETA).

In this paper the problem of restoring an image distorted by a linear space-invariant (LSI) point-spread function (psf) which is not exactly known is formulated as the solution of a perturbed set of linear equations. The regularized constrained total least-squares (RCTLS) method is used to solve this set of equations. Using the diagonalization properties of the discrete Fourier transform (DFT) for circulant matrices, the RCTLS estimate is computed in the DFT domain. This significantly reduces the computational cost of this approach and makes its implementation possible even for large images. An error analysis of the RCTLS estimate, based on the mean-squared-error (MSE) criterion is performed to verify its superiority over the constrained total least-squares (CTLS) estimate. Numerical experiments for different psf errors are performed to test the RCTLS estimator for this problem. Objective and visual comparisons are presented with the linear minimum mean-squared-error (LMMSE) and the re...

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite

this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two ways: (a) by a constraint ,operator which incorporates properties of the response of the hu- man visual system into the restoration process and (b) by a weight matrix which assigns greater importance for the deconvolution process to areas of high spatial activity than to areas of low spatial activity. Different degrees of trust are assigned to the various distorted images depending on the amounts of noise. The proposed algorithm is general and can be used for any type of linear distortion and constraint operators. It can also be used to restore signals other than images. Experimental results obtained by an iterative implementation of the proposed algorithms are pre- sented. c 1990 Academic Press, Inc

The satellite image deconvolution problem is ill-posed and must be regularized. Herein, we use an edge-preserving regularization model using a ' function, involving two hyperparameters. Our goal is to estimate the optimal parameters in order to automatically reconstruct images. We propose to use the Maximum Likelihood Estimator (MLE), applied to the observed image. We need sampling from prior and posterior distributions. Since the convolution prevents from using standard samplers, we have developed a modied Geman-Yang algorithm, using an auxiliary variable and a cosine transform. We present a Markov Chain Monte Carlo Maximum Likelihood (MCMCML) technique which is able to simultaneously achieve the estimation and the reconstruction.

Blind linear system identication consists in estimating the parameters of a linear timeinvariant