Jean-François Aujol

Structure-Texture Image Decomposition--Modeling, Algorithms, and Parameter Selection

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 ρ, where u holds the geometrical information and ρ 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: L2, G, L1 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 ρ may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV − L1 minimization problem, and third we introduce the theory and design tools for the TV-Gabor model.

A Variational Approach to Removing Multiplicative Noise

This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, we prove the existence of a minimizer and we show the capability of our model on some numerical examples. We study the associated evolution problem, for which we derive existence and uniqueness results for the solution. We prove the convergence of an implicit scheme to compute the solution.

Image Decomposition into a Bounded Variation Component and an Oscillating Component

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.

Dual Norms and Image Decomposition Models

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.

Some First-Order Algorithms for Total Variation Based Image Restoration

This paper deals with first-order numerical schemes for image restoration. These schemes rely on a duality-based 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.

Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization

We investigate how TV regularization naturally recognizes the scale of individual features of an image, and we show how this perception of scale depends on the amount of regularization applied to the image. We give an automatic method driven by the geometry of the image for finding the minimum value of the regularization parameter needed to remove all features below a user-chosen threshold. We explain the relation of Meyers G norm to the perception of scale, which provides a more intuitive understanding of this norm. We consider other applications of this ability to recognize scale, including the multiscale effects of TV regularization and the rate of loss of image features of various scales as a function of increasing amounts of regularization. Several numerical results are given.

Color image decomposition and restoration

Meyer has recently introduced an image decomposition model to split an image into two components: a geometrical component and a texture (oscillatory) component. Inspired by his work, numerical models have been developed to carry out the decomposition of gray scale images. In this paper, we propose a decomposition algorithm for color images. We introduce a generalization of Meyer’s G norm to RGB vectorial color images, and use Chromaticity and Brightness color model with total variation minimization. We illustrate our approach with numerical examples.

Adaptive regularization of the NL-means : Application to image and video denoising

Image denoising is a central problem in image processing and it is often a necessary step prior to higher level analysis such as segmentation, reconstruction, or super-resolution. The nonlocal means (NL-means) perform denoising by exploiting the natural redundancy of patterns inside an image; they perform a weighted average of pixels whose neighborhoods (patches) are close to each other. This reduces significantly the noise while preserving most of the image content. While it performs well on flat areas and textures, it suffers from two opposite drawbacks: it might over-smooth low-contrasted areas or leave a residual noise around edges and singular structures. Denoising can also be performed by total variation minimization-the Rudin, Osher and Fatemi model-which leads to restore regular images, but it is prone to over-smooth textures, staircasing effects, and contrast losses. We introduce in this paper a variational approach that corrects the over-smoothing and reduces the residual noise of the NL-means by adaptively regularizing nonlocal methods with the total variation. The proposed regularized NL-means algorithm combines these methods and reduces both of their respective defaults by minimizing an adaptive total variation with a nonlocal data fidelity term. Besides, this model adapts to different noise statistics and a fast solution can be obtained in the general case of the exponential family. We develop this model for image denoising and we adapt it to video denoising with 3D patches.

EXEMPLAR-BASED INPAINTING FROM A VARIATIONAL POINT OF VIEW

Among all methods for reconstructing missing regions in a digital image, the so-called exemplar-based algorithms are very efficient and often produce striking results. They are based on the simple idea—initially used for texture synthesis—that the unknown part of an image can be reconstructed by simply pasting samples extracted from the known part. Beyond heuristic considerations, there have been very few contributions in the literature to explain from a mathematical point of view the performances of these purely algorithmic and discrete methods. More precisely, a recent paper by Levina and Bickel [Ann. Statist., 34 (2006), pp. 1751–1773] provides a theoretical explanation of their ability to recover very well the texture, but nothing equivalent has been done so far for the recovery of geometry. Our purpose in this paper is twofold: (1) to propose well-posed variational models in the continuous domain that can be naturally associated to exemplar-based algorithms; (2) to investigate their ability to recons...

A Bias-Variance Approach for the Nonlocal Means

This paper deals with the parameter choice for the nonlocal means (NLM) algorithm. After basic computations on toy models highlighting the bias of the NLM, we study the bias-variance trade-off of this filter so as to highlight the need of a local choice of the parameters. Relying on Steins unbiased risk estimate, we then propose an efficient algorithm to locally set these parameters, and we compare this method with the NLM with optimal global parameter.