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 ...

We extend the geometric framework introduced in Sochen et al. (IEEE Trans. on Image Processing, 7(3):310–318, 1998) for image enhancement. We analyze and propose enhancement techniques that selectively smooth images while preserving either the multi-channel edges or the orientation-dependent texture features in them. Images are treated as manifolds in a feature-space. This geometrical interpretation lead to a general way for grey level, color, movies, volumetric medical data, and color-texture image enhancement. We first review our framework in which the Polyakov action from high-energy physics is used to develop a minimization procedure through a geometric flow for images. Here we show that the geometric flow, based on manifold volume minimization, yields a novel enhancement procedure for color images. We apply the geometric framework and the general Beltrami flow to feature-preserving denoising of images in various spaces. Next, we introduce a new method for color and texture enhancement. Motivated by Gabor’s geometric image sharpening method (Gabor, Laboratory Investigation, 14(6):801–807, 1965), we present a geometric sharpening procedure for color images with texture. It is based on inverse diffusion across the multi-channel edge, and diffusion along the edge.

A simplified color image formation model is used to construct an algorithm for image reconstruction from CCD sensors samples. The proposed method involves two successive steps. The first is motivated by Cok’s template matching technique, while the second step uses steerable inverse diffusion in color. Classical linear signal processing techniques tend to oversmooth the image and result in noticeable color artifacts along edges and sharp features. The question is how should the different color channels support each other to form the best possible reconstruction. Our answer is to let the edges support the color information, and the color channels support the edges, and thereby achieve better perceptual results than those that are bounded by the sampling theoretical limit.

Signal and image enhancement is considered in the context of a new type of diffusion process that simultaneously enhances, sharpens and denoises images. The nonlinear diffusion coefficient is locally adjusted according to image features such as edges, textures and moments. As such it can switch the diffusion process from a forward to a backward (inverse) mode according to a given set of criteria. This results in a forward-and-backward (FAB) adap- tive diffusion process that enhances features while locally denoising smoother segments of the signal or image. The proposed method, using the FAB process, is applied in a super-resolution scheme.

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 present a framework for enhancing images while preserving either the edge or the orientation-dependent texture information present in them. We do this by treating images as manifolds in a feature-space. This geometrical interpretation leads to a natural way for grey level, color, movies, volumetric medical data, and color-texture image enhancement. Following this, we invoke the Polyakov action from high-energy physics, and develop a minimization procedure through a geometric flow. This flow, based on manifold volume minimization yields a natural enhancement procedure. We apply this framework to edgepreserving denoising of grey value and color images, for volumetric medical data, and orientation-preserving flows for grey level and color texture images. 1 Introduction In this paper, we present a general framework for processing images of various types like grey scale, color, and those that have orientation-dependent information such as textures. We do this by treating images as emb...

In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher's contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the Engquist-Osher scheme, TVD schemes, entropy conditions, ENO and WENO schemes and numerical schemes for Hamilton-Jacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to Hamilton-Jacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.

: In this paper, we develop a common cellular neural network framework for various adaptive nonlinear filters based on robust statistic and geometry-driven diffusion paradigms. The base models of both approaches are defined as difference-controlled nonlinear CNN templates while the self-adjusting property is ensured by simple analogic (analog and logic) CNN algorithms. Two adaptive strategies are shown for the order statistic class. When applied to the images distorted by impulse noise both give more visually pleasing results with lower frequency weighted mean square error than the median base model. Generalizing a variational approach we derive the constrained anisotropic diffusion, where the output of the geometry-driven diffusion model is forced to stay close to a pre-defined morphological constraint. We propose a coarse-grid CNN approach that is capable of calculating an acceptable noise-level estimate (proportional to the variance of the Gaussian noise) and controlling t...

Abstract We recently realized that the Marr-Hildreth edges, computed as the zero crossings of the image Laplacian, can be viewed as optimal edge integration curves solving a geometric variational problem. We used this observation to derive a new set of edge integration and object segmentation procedures. Here we show that the edge detectors proposed by Haralick, and subsequently claimed to be optimal, in some sense (based on 1D criteria) by Canny, and then numerically and computationally enhanced by Deriche, can also be interpreted as optimal edge contours whose normals align with the image gradient field, and further satisfy a topological uniformity measure inside a closed region defined by the contour. The combination of these two measures yield a robust edge detection/integration procedure, as well as better geometric curve evolution based segmentation procedures. In other words we provide 2D variational reasoning for the classical Haralick/Canny/Deriche-like edge detectors. We show how to use this new formulation for introducing novel geometric segmentation by curve evolution.

In this work we have tried to solve several problems that involve scalar and vectorial image information lying on the plane or on 3D surfaces. We will present and discuss four novel algorithms devised to solve those problems: tracking of objects in movies, tracking of regions on deforming 3D surfaces, restoration of damaged pictures, removal of objects from images, and solving of PDE's on implicit surfaces for denoising, texture synthesis and flow visualization. Examples are shown and future lines of research are suggested, both to improve performance and to extend the present techniques to new applications.