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We propose here a class of restoration algorithms for color images, based upon the Mumford-Shah (MS) model and nonlocal image information. The Ambrosio-Tortorelli and Shah elliptic approximations are defined to work in a small local neighborhood, which are sufficient to denoise smooth regions with sharp boundaries. However, texture is nonlocal in nature and requires semilocal/non-local information for efficient image denoising and restoration. Inspired from recent works (nonlocal means of Buades, Coll, Morel, and nonlocal total variation of Gilboa, Osher), we extend the local Ambrosio-Tortorelli and Shah approximations to MS functional (MS) to novel nonlocal formulations, for better restoration of fine structures and texture. We present several applications of the proposed nonlocal MS regularizers in image processing such as color image denoising, color image deblurring in the presence of Gaussian or impulse noise, color image inpainting, color image super-resolution, and color filter array demosaicing. In all the applications, the proposed nonlocal regularizers produce superior results over the local ones, especially in image inpainting with large missing regions. We also prove several characterizations of minimizers based upon dual norm formulations.

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. Meyer's 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 Meyer's 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.

We denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical imaging. Diffusion imaging is a relatively new and powerful method to measure the 3D profile of water diffusion at each point. This can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. However, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter preselected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. We propose a variational method to denoise HARDI data by denoising the spherical Apparent Diffusion Coefficient (sADC), a field of radial functions derived from the data. We use vectorial total variation regularization, an L1 data fidelity term and the logarithmic barrier function in the minimization. We present experiments of denoising synthetic and real HARDI data.

The implicit framework of the level-set method has several advantages when tracking propagating fronts. Indeed, the evolving contour is embedded in a higher dimensional level-set function and its evolution can be phrased in terms of a Eulerian formulation. The ability of this intrinsic method to handle topological changes (merging and breaking) makes it useful in a wide range of applications (fluid mechanics, computer vision) and particularly in image segmentation, the main subject of this paper. Nevertheless, in some applications, this topological flexibility turns out to be undesirable: for instance, when the shape to be detected has a known topology, or when the resulting shape must be homeomorphic to the initial one. The necessity of designing topology-preserving processes arises in medical imaging, for example, in the human cortex reconstruction. It is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved. Therefore, we propose in this paper a segmentation model based on an implicit level-set formulation and on the geodesic active contours, in which a topological constraint is enforced.

Abstract: In this paper, we present a new non-parametric combined segmentation and registration method. The shapes to be registered are implicitly modeled with level set functions and the problem is cast as an optimization one, combining a matching criterion based on the active contours without edges for segmentation (Chan and Vese, 2001) and a nonlinear-elasticity-based smoother on the displacement vector field. This modeling is twofold: first, registration is jointly performed with segmentation since guided by the segmentation process; it means that the algorithm produces both a smooth mapping between the two shapes and the segmentation of the object contained in the reference image. Secondly, the use of a nonlinear-elasticity-type regularizer allows large deformations to occur, which makes the model comparable in this point with the viscous fluid registration method. In the theoretical minimization problem we introduce, the shapes to be matched are viewed as Ciarlet–Geymonat materials. We prove the existence of minimizers of the introduced functional and derive an approximated problem based on the Saint Venant–Kirchhoff stored energy for the numerical implementation and solved by an augmented Lagrangian technique. Several applications are proposed to demonstrate the potential of this method to both segmentation of one single image and to registration between two images. [Copyright &y& Elsevier]

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u 0 into a sum u+ v. Here u∈ BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H − s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images. [ABSTRACT FROM AUTHOR]

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 in-painting 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 in-painting 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. [ABSTRACT FROM AUTHOR]

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 Legendre­Fenchel 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. [ABSTRACT FROM AUTHOR]