Abstract. We focus in this paper on some reconstruction/restoration methods whose aim is to improve the resolution of digital images. The main point here is to study the ability of such methods to preserve one-dimensional (1D) structures. Indeed, such structures are important since they are often carried by the image "edges." First we focus on linear methods, give a general framework to design them, and show that the preservation of 1D structures pleads in favor of the cancellation of the periodization of the image spectrum. More precisely, we show that preserving 1D structures implies the linear methods to be written as a convolution of the "sinc interpolation." As a consequence, we cannot cope linearly with Gibbs effects, sharpness of the results, and the preservation of the 1D structure. Second, we study variational nonlinear methods and, in particular, the one based on total variation. We show that this latter permits us to avoid these shortcomings. We also prove the existence and consistency of an approximate solution to this variational problem. At last, this theoretical study is highlighted by experiments, both on synthetic and natural images, which show the effects of the described methods on images as well as on their spectrum.

In this paper, we present a general framework for image restoration; despite its simplicity, certain variational and certain wavelet approaches can be formulated within this framework. This permits the construction of a natural model, with only one parameter, which has the advantages of both approaches. We give a mathematical analysis of this model, describe our algorithm and illustrate this by some experiments.

The dictionary learning problem aims at finding a dictionary of atoms that best represents an image according to a given objective. The most usual objective consists of representing an image or a class of images sparsely. Most algorithms performing dictionary learning iteratively estimate the dictionary and a sparse representation of images using this dictionary. Dictionary learning has led to many state of the art algorithms in image processing. However, its numerical complexity restricts its use to atoms with a small support since the computations using the constructed dictionaries require too much resources to be deployed for large scale applications.In order to alleviate these issues, this paper introduces a new strategy to learn dictionaries composed of atoms obtained as a composition of K convolutions with S-sparse kernels. The dictionary update step associated with this strategy is a non-convex optimization problem. We reformulate the problem in order to reduce the number of its irrelevant stationary points and introduce a Gauss-Seidel type algorithm, referred to as Alternative Least Square Algorithm, for its resolution. The search space of the considered optimization problem is of dimension KS, which is typically smaller than the size of the target atom and is much smaller than the size of the image. The complexity of the algorithm is linear with regard to the size of the image.Our experiments show that we are able to approximate with a very high accuracy many atoms such as modified DCT, curvelets, sinc functions or cosines when K is large (say K = 10). We also argue empirically that, maybe surprisingly, the algorithm generally converges to a global minimum for large values of K and S.

Abstract. This paper deals with two complementary methods in noisy image deblurring: a nonlinear shrinkage of wavelet-packets coefficients called FCNR and Rudin-Osher-Fatemi's variational method. The FCNR has for objective to obtain a restored image with a white noise. It will prove to be very efficient to restore an image after an invertible blur but limited in the opposite situation. Whereas the Total Variation based method, with its ability to reconstruct the lost frequencies by interpolation, is very well adapted to non-invertible blur, but that it tends to erase low contrast textures. This complementarity is highlighted when the methods are applied to the restoration of satellite SPOT images.

In this paper, we seek a solution to linear inverse problems arising in image restoration in terms of a recently posed optimization problem which combines total variation minimization and wavelet-thresholding ideas. The resulting nonlinear programming task is solved via a dual Uzawa method in its general form, leading to an efficient and general algorithm which allows for very good structure-preserving reconstructions. Along with a theoretical study of the algorithm, the paper details some aspects of the implementation, discusses the numerical convergence and eventually displays a few images obtained for some difficult restoration tasks.