Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the ...

Abstract. In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L p norm in general and an L1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non-flat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.

A multigrid anisotropic diffusion algorithm for image processing is presented. The multigrid implementation provides an efficient hierarchical relaxation method that facilitates the application of anisotropic diffusion to time-critical processes. Through a multigrid V-cycle, the anisotropic diffusion equations are successively transferred to coarser grids and used in a coarseto -fine error correction scheme. When a coarse grid with a trivial solution is reached, the coarse grid estimates of the residual error can be propagated to the original grid and used to refine the solution. The main benefits of the multigrid approach are rapid intraregion smoothing and reduction of artifacts due to the elimination of low-frequency error. In the paper, the theory of multigrid anisotropic diffusion is developed. Then, the intergrid transfer functions, relaxation techniques, diffusion coefficients, and boundary conditions are discussed. The analysis includes the examination of the storage requirements, the computational cost, and the solution quality. Finally, experimental results are reported that demonstrate the effectiveness of the multigrid approach.

A novel approach for color image denoising is proposed in this paper. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color pixel is considered as an-dimensional vector. The vectors' direction, a unit vector, gives the chromaticity, while the magnitude represents the pixel brightness. The chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data such as chromaticity. Both isotropic and anisotropic diffusion flows are presented for this-dimensional chromaticity diffusion flow. The brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement. We present the underlying theory, a number of examples, and briefly compare with the current literature. Index Terms---Brightness, chromaticity, color image denoising, directions, harmonic maps, isotropic and anisotropic diffusion, liquid crystals, partial differential equations. I.

In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to regularize directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L2 norm, and edge preserving diffusion, obtained from an L1 norm. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.

Abstract. We propose and analyze extensions of the Mumford-Shah functional for color images. Our main motivation is the concept of images as surfaces. We also review most of the relevant theoretical background and computer vision literature. Keywords: color, Mumford-Shah functional, segmentation, variational methods.

Nonlinear anisotropic diffusion filtering is a procedure based on nonlinear evolution partial differential equations which seeks to improve images qualitatively by removing noise while preserving details and even enhancing edges. However, well known implementations are sensitive to parameters which are necessarily tuned to sharpen a narrow range of edge slopes; otherwise, edges are either blurred or staircased. In this work, nonlinear anisotropic diffusion filters have been developed which sharpen edges over a wide range of slope scales and which reduce noise conservatively with dissipation purely along feature boundaries. Specifically, the range of sharpened edge slopes is widened as backward diffusion normal to level sets is balanced with forward diffusion tangent to level sets. Also, noise is reduced by selectively altering the balance toward diminishing normal backward diffusion and particularly toward total variation filtering. The theoretical motivation for the proposed filters is presented together with computational results comparing them with other nonlinear anisotropic diffusion filters on both synthetic images and magnetic resonance images.

We investigate how the Perona-Malik scheme evolves piecewise smooth initial data in one dimension. By scaling a natural parameter that appears in the scheme in an appropriate way with respect to the grid size, we obtain a meaningful continuum limit. The resulting evolution can be seen as the gradient flow for an energy, just as the discrete evolutions are gradient flows for discrete energies. It involves, except at special isolated times, solving a system of heat equations coupled to each other through nonlinear boundary conditions. At the special times, the solutions experience gradient blowup; nevertheless, there is a natural continuation for the solutions beyond these singular times.

In this paper, a diffusion-based curvelet shrinkage is proposed for discontinuity-preserving de-noising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme. In order to suppress the pseudo-Gibbs and curvelet-like artifacts, the conventional shrinkage results are further processed by a projected total variation diffusion, in which only the in-significant curvelet coefficients or high-frequency part of the signal are changed by use of a constrained projection. Numerical experiments from piecewise-smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components, in comparison to some existing methods.

this article, we present a variational approach to MAP estimation with a more qualitative and tutorial emphasis. The key idea behind this approach is to use geometric insight in helping construct regularizing functionals and avoiding a subjective choice of a prior in MAP estimation. Using tools from robust statistics and information theory, weshow that we can extend this strategy and develop two gradient descent flows for image denoising with a demonstrated performance