Abstract-The scale-space technique introduced by Witkin involves generating coarser resolution images by convolving the original image with a Gaussian kernel. This approach has a major drawback: it is difficult to obtain accurately the locations of the “semantically mean-ingful ” edges at coarse scales. In this paper we suggest a new definition of scale-space, and introduce a class of algorithms that realize it using a diffusion process. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing in preference to interregion smoothing. It is shown that the “no new maxima should be generated at coarse scales ” property of conventional scale space is pre-served. As the region boundaries in our approach remain sharp, we obtain a high quality edge detector which successfully exploits global information. Experimental results are shown on a number of images. The algorithm involves elementary, local operations replicated over the image making parallel hardware implementations feasible. Index Terms-Adaptive filtering, analog VLSI, edge detection, edge enhancement, nonlinear diffusion, nonlinear filtering, parallel algo-

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...

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This article sets out a new representation of an image which is contrast independent. The image is decomposed into a tree of "shapes" based on connected components of level sets, which provides a full and non-redundant representation of the image. A fast algorithm to compute the tree, the Fast Level Lines Transform, is explained in details. Some simple and direct applications of this representation are shown. Keywords---Image representation, Image coding, Mathematical morphology, Contrast invariance. I. Introduction Image representations can be different depending on their purpose. For a deblurring, restoration, denoising purpose, the representations based on the Fourier transform are generally the best since they rely on the generation process of the image (Shannon theory), and/or on the frequency models of the degradation as for additive noise, or spurious convolution kernel. The wavelets theory, [1], [2], achieves a localization of the frequencies, and, due to the linear structur...

Multiscale image analysis has been used successfully in a number of applications to classify image features according to their relative scales. As a consequence, much has been learned about the scale-space behavior of intensity extrema, edges, intensity ridges, and grey-level blobs. In this paper, we investigate the multiscale behavior of gradient watershed regions. These regions are defined in terms of the gradient properties of the gradient magnitude of the original image. Boundaries of gradient watershed regions correspond to the edges of objects in an image. Multiscale analysis of intensity minima in the gradient magnitude image provides a mechanism for imposing a scale-based hierarchy on the watersheds associated with these minima. This hierarchy can be used to label watershed boundaries according to their scale. This provides valuable insight into the multiscale properties of edges in an image without following these curves through scale-space. In addition, the gradient watershed region hierarchy can be used for automatic or interactive image segmentation. By selecting subtrees of the region hierarchy, visually sensible objects in an image can be easily constructed.

In this paper we address the topics of scale-space and phase-based image processing in a unifying framework. In contrast to the common opinion, the Gaussian kernel is not the unique choice for a linear scale-space. Instead, we chose the Poisson kernel since it is closely related to the monogenic signal, a 2D generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. The Riesz transform itself yields the flux of the Poisson scalespace and the combination of flux and scale-space, the monogenic scale-space, provides the local features attenuation and phase-vector in scale-space. Under certain assumptions, the latter two again form a monogenic scale-space which gives deeper insight to low-level image processing. In particular, we discuss edge detection by a new approach to phase congruency and its relation to amplitude based methods, reconstruction from local amplitude and local phase, and the evaluation of the local frequency.

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...

Nonlinear diffusion processes can be found in many recent methods for image processing and computer vision. In this article, four applications are surveyed: nonlinear diffusion filtering, variational image regularization, optic flow estimation, and geodesic active contours. For each of these techniques we explain the main ideas, discuss theoretical properties and present an appropriate numerical scheme. The numerical schemes are based on additive operator splittings (AOS). In contrast to

A survey on continuous, semidiscrete and discrete well-posedness and scale-space results for a class of nonlinear diffusion filters is presented. This class does not require any monotony assumption (comparison principle) and, thus, allows image restoration as well. The theoretical results include existence, uniqueness, continuous dependence on the initial image, maximumminimum principles, average grey level invariance, smoothing Lyapunov functionals, and convergence to a constant steady state. Keywords. scale-space, nonlinear diffusion, discrete smoothing transformations. 1 Introduction In the last years nonlinear diffusion filtering has been established as a successful tool for image smoothing and restoration. Strict scale-space results have been found recently for the continuous case [16]. The goal of the present paper is to outline how they can be extended to the semidiscrete and discrete setting. This is of significant practical importance, since a scale-space representation cann...