We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.

Total Variation (TV) methods are very effective for recovering "blocky", possibly discontinuous, images from noisy data. A fixed point algorithm for minimizing a TV-penalized least squares functional is presented and compared with existing minimization schemes. A variant of the cell-centered finite difference multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems. Numerical results are presented for one- and two-dimensional examples; in particular, the algorithm is applied to actual data obtained from confocal microscopy.

Abstract. The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove image fine structures. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the “method noise, ” defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods are compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of L 2 distances of the denoised version to the original image. The most powerful evaluation method seems, however, to be the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method.

This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u + v, where u E BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u + v, but we also show how the method can be used for texture discrimination and texture segmentation.

Inspired by the recent work of Bertalmio et al. on digital inpaintings [SIGGRAPH 2000], we develop general mathematical models for local inpaintings of nontexture images. On smooth regions, inpaintings are connected to the harmonic and biharmonic extensions, and inpainting orders are analyzed. For inpaintings involving the recovery of edges, we study a variational model that is closely connected to the classical total variation (TV) denoising model of Rudin, Osher, and Fatemi [PhSG D, 60 (1992), pp. 259--268]. Other models are also discussed based on the Mumford--Shah regularity [Comm. Pure Appl. Mathq XLII (1989), pp. 577--685] and curvature driven di#usions (CDD) of Chan and Shen [J. Visual Comm. Image Rep., 12 (2001)]. The broad applications of the inpainting models are demonstrated through restoring scratched old photos, disocclusion in vision analysis, text removal, digital zooming, and edge-based image coding.

This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the Legendre-Fenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of half-quadratic problems easier to solve numerically. First we prove an existence and uniqueness result, and then we describe the algorithm for computing the solution and we give a proof of convergence. Finally, we present some experimental results for synthetic and real images.

In this paper, we propose a new model for image restoration and decomposition, based on the total variation minimization of Rudin-Osher-Fatemi, and of the results of Y. Meyer on oscillatory functions. An initial image f is decomposed into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component by an oscillatory function, with bounded H-1 norm. After some transformation, the resulting PDE is of fourth order. Finally, image decomposition, alenoising and aleblurring numerical results are shown.

Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filter---the digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational optimization problem, which differs from most statistical filters. Applications are given in the denoising of one-dimensional (1-D) signals, two-dimensional (2-D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.

We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S¹, the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line RP¹ and the chromaticity feature (from color images) that lives on the unit sphere S². In this paper, we apply the variational method to denoise and restore general nonflat image features. Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed. Riemannian objects such as metric, distance and Levi-Civita connection play important roles in the models. Computational algorithms are also developed for the resulting nonlinear equations. The mathematical framework can be applied to restoring general nonflat data outside the scope of image processing and computer vision.

Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced by Mumford [21] to computer vision as a prior curve model. By functionalizing the elastica energy, Masnou and Morel [19] proposed an elastica based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation, and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [2] and Chan and Shen [6, 7]. A computational scheme based on numerical PDEs is presented, which allows the handling of topologically complex inpainting domains.