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This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and v, where u holds the geometrical information and v holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L 2 , G, L 1 and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and v may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV − L 1 minimization problem, and third we introduce the theory and design tools for the TV-Gabor model.

Following a recent work by Y. Meyer, decomposition models into a geometrical component and a textured component have recently been proposed in image processing. In such approaches, negative Sobolev norms have seemed to be useful to modelize oscillating patterns. In this paper, we compare the properties of various norms that are dual of Sobolev or Besov norms. We then propose a decomposition model which splits an image into three components: a first one containing the structure of the image, a second one the texture of the image, and a third one the noise. Our decomposition model relies on the use of three different semi-norms: the total variation for the geometrical component, a negative Sobolev norm for the texture, and a negative Besov norm for the noise. We illustrate our study with numerical examples.

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International audienceThis article introduces a generalization of the discrete optimal transport, with applications to color image manipulations. This new formulation includes a relaxation of the mass conservation constraint and a regularization term. These two features are crucial for image processing tasks where one must take into account families of multimodal histograms with large mass variation across modes. The corresponding relaxed and regularized transportation problem is the solution of a convex optimization problem. Depending on the regularization used, this minimization can be solved using standard linear programming methods or first order proximal splitting schemes. The resulting transportation plan can be used as a color transfer map, which is robust to mass variation across image color palettes. Furthermore, the regularization of the transport plan helps remove colorization artifacts due to noise amplification. We also extend this framework to compute the barycenter of distributions. The barycenter is the solution of an optimization problem, which is separately convex with respect to the barycenter and the transportation plans, but not jointly convex. A block coordinate descent scheme converges to a stationary point of the energy. We show that the resulting algorithm can be used for color normalization across several images. The relaxed and regularized barycenter defines a common color palette for those images. Applying color transfer toward this average palette performs a color normalization of the input images

This paper deals with first-order numerical schemes for image restoration. These schemes rely on a dualitybased algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able to improve contemporary schemes. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples. We make some comparisons with a class of efficient algorithms (proved to be optimal among first-order numerical schemes) recently introduced by Y. Nesterov.

This paper is concerned with the convergence of over-relaxations of FB algorithm (in particular FISTA), in the case when proximal maps and/or gradients are computed with a possible error. We show that provided these errors are small enough, then the algorithm still converges to a minimizer of the functional, and with a speed of convergence (in terms of values of the functional) that remains the same as in the noise free case. We also show that larger errors can be allowed, using a lower over-relaxation than FISTA. This still leads to the convergence of iterates, and with ergodic convergence speed faster than the classical FB algorithm and FISTA.

International audienceThis paper provides a new method to colorize gray-scale images. While the computation of the luminance channel is directly performed by a linear transformation, the colorization process is an ill-posed problem that requires some priors. In the literature two classes of approach exist. The first class includes manual methods that need the user to manually add colors on the image to colorize. The second class includes exemplar-based approaches where a color image, with a similar semantic content, is provided as input to the method. These two types of priors have their own advantages and drawbacks. In this paper, a new variational framework for exemplar-based colorization is proposed. A nonlocal approach is used to find relevant color in the source image in order to suggest colors on the gray-scale image. The spatial coherency of the result as well as the final color selection is provided by a nonconvex variational framework based on a total variation. An efficient primal-dual algorithm is provided, and a proof of its convergence is proposed. In this work, we also extend the proposed exemplar-based approach to combine both exemplar-based and manual methods. It provides a single framework that unifies advantages of both approaches. Finally, experiments and comparisons with state-of-the-art methods illustrate the efficiency of our proposal. 1. Introduction. The colorization of a gray-scale image consists of adding color information to it. It is useful in the entertainment industry to make old productions more attractive. The reverse operation is based on perceptual assumptions and is today an active research area [28], [13], [37]. Colorization can also be used to add information in order to help further analysis of the image by a user (e.g., sensor fusion [43]). It can also be used for art restoration ; see, e.g., [17] or [41]. It is an old subject that began with the ability of screens and devices to display colors. A seminal approach consists in mapping each level of gray into a color-space [18]. Nevertheless, all colors cannot be recovered without an additional prior. In the existing approaches, priors can be added in two ways: with a direct addition of color o