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 ...

Nonlinear diffusion filtering is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a recent discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS) which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely-used explicit schemes.

. This paper gives an overview of scale-space and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate a-priori knowledge into the evolution. We sketch basic ideas behind the different filter models, discuss their theoretical foundations and scale-space properties, discrete aspects, suitable algorithms, generalizations, and applications. 1 Introduction During the last decade nonlinear diffusion filters have become a powerful and well-founded tool in multiscale image analysis. These models allow to include a-priori knowledge into the scale-space evolution, and they lead to an image simplification which simultaneously preserves or even enhances semantically important information such as edges, lines, or flow-like structures. Many papers have appeared proposing different models, investigating their theoretical foundations, and describing interesting applications. For a n...

We introduce a family of first-order multidimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" ("SIDEs"). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation [14], [15]. In an experimental section, SIDEs are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.

We discuss a semidiscrete framework for nonlinear diffusion scale-spaces, where the image is sampled on a finite grid and the scale parameter is continuous. This leads to a system of nonlinear ordinary differential equations. We investigate conditions under which one can guarantee well-posedness properties, an extremum principle, average grey level invariance, smoothing Lyapunov functionals, and convergence to a constant steady-state. These properties are in analogy to previously established results for the continuous setting. Interestingly, this semidiscrete framework helps to explain the so-called Perona-Malik paradox: The Perona-Malik equation is a forward-backward diffusion equation which is widely-used in image processing since it combines intraregional smoothing with edge enhancement. Although its continuous formulation is regarded to be ill-posed, it turns out that a spatial discretization is sufficient to create a well-posed semidiscrete diffusion scale-space. We also pro...

In most cases nonlinear diffusion filtering is implemented by means of explicit finite difference schemes. These algorithms are not very efficient, since they are only stable for small time steps. We address this problem by presenting unconditionally stable semi-implicit schemes which are based on an additive operator splitting (AOS). They are very efficient since they can be implemented by recursive filtering, and their separability allows a straightforward implementation in any dimension. We analyse their behaviour on a parallel computer and demonstrate that parallel AOS schemes on a modern shared-memory multiprocessor system with 8 processors allow a speed-up of two orders of magnitude in comparison to the widely-used explicit scheme on a single processor. c fl1997 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to server...

. Poor efficiency is a typical problem of nonlinear diffusion filtering, when the simple and popular explicit (Euler-forward) scheme is used: for stability reasons very small time step sizes are necessary. In order to overcome this shortcoming, a novel type of semi-implicit schemes is studied, so-called additive operator splitting (AOS) methods. They share the advantages of explicit and (semi-)implicit schemes by combining simplicity with absolute stability. They are reliable, since they satisfy recently established criteria for discrete nonlinear diffusion scale-spaces. Their efficiency is due to the fact that they can be separated into one-dimensional processes, for which a fast recursive algorithm with linear complexity is available. AOS schemes reveal good rotational invariance and they are symmetric with respect to all axes. Examples demonstrate that, under typical accuracy requirements, they are at least ten times more efficient than explicit schemes. 1 Introduction Although non...

. We introduce a family of first-order multi-dimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" ("SIDEs"). A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation [3, 4]. In an experimental section, SIDEs are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is demonstrated. 1 Introduction In this paper we introduce, analyze, and apply a new class of nonlinear image processing algorithms. These algorithms are motivated by the great recent interest in using evolutions specified by partial differential equations (PDE's) as image processing procedures for tas...

Automatic segmentation of stroke lesions in magnetic resonance imagery is a difficult problem because anatomical knowledge is required for the most accurate decisions. Without such knowledge, classification rules seem inconsistent. We propose a hybrid boundary and region based segmentation model built upon nonlinear scalespace and geometric active contours that captures the various segmentation rules necessary to segment lesions. After a user selects a point within damaged tissue and another point within healthy tissue, the image is examined at several levels of detail. At each such scale, the lesion is segmented several times by varying a parameter that models the range of criteria for boundaries between healthy and damaged tissue. These segmentations are collected, and the relative frequency of tissue being labeled lesion is regarded as a measure of confidence in the classification of the tissue as damaged. Experiments compare volumes and segmentations of lesions given by physicians to those given by the automatic method. Performance upper bounds are established by matching automatic segmentation parameters (scale, threshold, and/or confidence) for

Many image processing applications require to solve problems such as denoising with edge enhancement, preprocessing for segmentation, or the completion of interrupted lines. This may be accomplished by applying a suitable nonlinear anisotropic diffusion process to the image. Its diffusion tensor is adapted to the differential structure of the underlying image. Although being image enhancement tools, filters of this class are well-posed. The temporal evolution of the diffusion process creates a scale-space whose causality properties can be understood in a deterministic, stochastic, information theory based and Fourier based way. The well-posedness and scale-space results carry over to the discrete setting, which gives rise to reliable algorithms preserving all properties of the continuous framework. c fl1997 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective work...