Luminita A. Vese

Simultaneous structure and texture image inpainting

An algorithm for the simultaneous filling-in of texture and structure in regions of missing image information is presented in this paper. The basic idea is to first decompose the image into the sum of two functions with different basic characteristics, and then reconstruct each one of these functions separately with structure and texture filling-in algorithms. The first function used in the decomposition is of bounded variation, representing the underlying image structure, while the second function captures the texture and possible noise. The region of missing information in the bounded variation image is reconstructed using image inpainting algorithms, while the same region in the texture image is filled-in with texture synthesis techniques. The original image is then reconstructed adding back these two sub-images. The novel contribution of this paper is then in the combination of these three previously developed components, image decomposition with inpainting and texture synthesis, which permits the simultaneous use of filling-in algorithms that are suited for different image characteristics. Examples on real images show the advantages of this proposed approach.

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing

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∈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.

Active Contours without Edges for Vector-Valued Images

In this paper, we propose an active contour algorithm for object detection in vector-valued images (such as RGB or multispectral). The model is an extension of the scalar Chan?Vese algorithm to the vector-valued case 1]. The model minimizes a Mumford?Shah functional over the length of the contour, plus the sum of the fitting error over each component of the vector-valued image. Like the Chan?Vese model, our vector-valued model can detect edges both with or without gradient. We show examples where our model detects vector-valued objects which are undetectable in any scalar representation. For instance, objects with different missing parts in different channels are completely detected (such as occlusion). Also, in color images, objects which are invisible in each channel or in intensity can be detected by our algorithm. Finally, the model is robust with respect to noise, requiring no a priori denoising step.

Image Decomposition and Restoration Using Total Variation Minimization and the H1

In this paper, we propose a new model for image restoration and image decomposition into cartoon and texture, based on the total variation minimization of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268], and on oscillatory functions, which follows results of Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2002]. This paper also continues the ideas introduced by the authors in a previous work on image decomposition models into cartoon and texture [L. Vese and S. Osher, J. Sci. Comput., to appear]. Indeed, by an alternative formulation, an initial image f is decomposed here 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 is modeled by an oscillatory function, bounded in the norm dual to

A Variational Method in Image Recovery

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A level set algorithm for minimizing the Mumford-Shah functional in image processing

. After some transformation, the resulting PDE is of fourth order, envolving the Laplacian of the curvature of level lines. Fina...

Image Denoising and Decomposition with Total Variation Minimization and Oscillatory Functions

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.

Fast Cartoon + Texture Image Filters

We show how the piecewise-smooth Mumford-Shah segmentation problem can be solved using the level set method of Osher and Sethian (1988). The obtained algorithm can be simultaneously used to denoise, segment, detect-extract edges, and perform active contours. The proposed model is also a generalisation of a previous active contour model without edges, proposed by the authors in Chan et al., (2001), and of its extension to the case with more than two segments for piecewise-constant segmentation Chan et al., (2000). Based on the four color theorem, we can assume that in general, at most two level set functions are sufficient to detect and represent distinct objects of distinct intensities, with triple junctions, or T-junctions.

Numerical Methods for p -Harmonic Flows and Applications to Image Processing

In this paper, we propose a new variational model for image denoising and decomposition, witch combines the total variation minimization model of Rudin, Osher and Fatemi from image restoration, with spaces of oscillatory functions, following recent ideas introduced by Meyer. The spaces introduced here are appropriate to model oscillatory patterns of zero mean, such as noise or texture. Numerical results of image denoising, image decomposition and texture discrimination are presented, showing that the new models decompose better a given image, possible noisy, into cartoon and oscillatory pattern of zero mean, than the standard ones. The present paper develops further the models previously introduced by the authors in Vese and Osher (Modeling textures with total variation minimization and oscillating patterns in image processing, UCLA CAM Report 02-19, May 2002, to appear in Journal of Scientific Computing, 2003). Other recent and related image decomposition models are also discussed.

Nonlocal Mumford-Shah Regularizers for Color Image Restoration

Can images be decomposed into the sum of a geometric part and a textural part? In a theoretical breakthrough, [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. Providence, RI: American Mathematical Society, 2001] proposed variational models that force the geometric part into the space of functions with bounded variation, and the textural part into a space of oscillatory distributions. Meyers models are simple minimization problems extending the famous total variation model. However, their numerical solution has proved challenging. It is the object of a literature rich in variants and numerical attempts. This paper starts with the linear model, which reduces to a low-pass/high-pass filter pair. A simple conversion of the linear filter pair into a nonlinear filter pair involving the total variation is introduced. This new-proposed nonlinear filter pair retains both the essential features of Meyers models and the simplicity and rapidity of the linear model. It depends upon only one transparent parameter: the texture scale, measured in pixel mesh. Comparative experiments show a better and faster separation of cartoon from texture. One application is illustrated: edge detection.