. 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...

The completion of interrupted lines or the enhancement of flow-like structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the so-called interest operator (second-moment matrix, structure tensor). An m-dimensional formulation of this method is analysed with respect to its well-posedness and scale-space properties. An efficient scheme is presented which uses a stabilization by a semi-implicit additive operator splitting (AOS), and the scale-space behaviour of this method is illustrated by applying it to both 2-D and 3-D images.

We investigate the "deep structure" of a scale-space image. The emphasis is on topology, i.e. we concentrate on critical points---points with vanishing gradient---and top-points---critical points with degenerate Hessian---and monitor their displacements, respectively generic morsifications in scalespace. Relevant parts of catastrophe theory in the context of the scale-space paradigm are briefly reviewed, and subsequently rewritten into coordinate independent form. This enables one to implement topological descriptors using a conveniently defined, global coordinate system. 1 Introduction 1.1 Historical Background A fairly well understood way to endow an image with a topology is to embed it into a one-parameter family of images known as a "scale-space image". The parameter encodes "scale" or "resolution" (coarse/fine scale means low/high resolution, respectively). Among the simplest is the linear or Gaussian scale-space model. Proposed by Iijima [13] in the context of pattern recogniti...

. 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 investigate the deep structure of a scale space image. We concentrate on scale space critical points -- points with vanishing gradient with respect to both spatial and scale direction. We show that these points are always saddle points. They turn

Abstract. The behaviour of critical points of Gaussian scale-space images is mainly described by their creation and annihilation. In existing literature these events are determined in so-called canonical coordinates. A description in a user-defined Cartesian coordinate system is stated, as well as the results of a straightforward implementation. The location of a catastrophe can be predicted with subpixel accuracy. An example of an annihilation is given. Also an upper bound is derived for the area where critical points can be created. Experimental data of an MR, a CT, and an artificial noise image satisfy this result. 1

In the past decades linear scale-space theory was derived on the basis of various axiomatics. In this paper we revisit these axioms and show that they merely coincide with the following physical principles, namely that the image domain is a Galilean space, that the total energy exchange between a region and its surrounding is preserved under linear filtering and that the physical observables should be invariant under the group of similarity transformations. These observables are elements of the similarity jet spanned by natural coordinates and differential energies read out by a vision system. Furthermore, linear scale-space theory is extended to spatio-temporal images on bounded and curved domains. Our theory permits a delay-operation at the present moment which is in agreement with the motion detection model of Reichardt. In this respect our theory deviates from that of Koenderink which requires additional syntactical operators to realise such a delay-operation. Finally, the semi-d...

In order to investigate the deep structure of Gaussian scale space images, one needs to understand the behaviour of spatial critical points under the influence of blurring. We show how the mathematical framework of catastrophe theory can be used to describe and model the behaviour of critical point trajectories when various different types of generic events, viz. annihilations and creations of pairs of spatial critical points, (almost) coincide. Although such events are non-generic in mathematical sense, they are not unlikely to be encountered in practice. Furthermore the behaviour leads to the observation that fine-to-coarse tracking of critical points doesn't suffice, since trajectories can form closed loops in scale space. The modelling of the trajectories include these loops. We apply the theory to an artificial image and a simulated MR image and show the occurrence of the described behaviour.

This paper proposes a general algebraic construction technique for image scale-spaces. The basic idea is to first downscale the image by some factor using an invertible scaling, then apply an image operator (linear or morphological) at a unit scale, and finally resize the image to its original scale. It is then required that the resulting one-parameter family of image operators satisfies the semigroup property. Such an approach encompasses linear as well as nonlinear (morphological) operators. Furthermore, there exists some freedom as to which semigroup operation on the scale- (or time-) axis is being chosen. Particular attention is given to additive and supremal semigroups. A large part of the paper is devoted to morphological scale-spaces, in particular to scale-spaces associated with an erosion or an opening. In these cases, classical tools from convex analysis, such as the (Young-Fenchel) conjugate, play an important role. 1991 Mathematics Subject Classification: 68U10, 52A41. Key...

In a recent work, J. J. Koenderink and A. J. Van Doorn considered a family of three intertwined scale-spaces coined the locally orderless image (LOI) (1999, J. Comput. Vision, 31 (2/3), 159–168). The LOI represents the image, observed at inner scale σ, as a local histogram with bin-width β, at each location, with a Gaussian-shape region of interest of extent α. LOIs form a natural and elegant extension of scalespace theory, show causal consistency, and enable the smooth transition between pixels, histograms, and isophotes. The aim of this work is to demonstrate the wide applicability and versatility of LOIs. We present applications for a range of image processing tasks, including new nonlinear diffusion schemes, adaptive histogram equalization and variations, several methods for noise and scratch removal, texture rendering, classification, and segmentation. C ○ 2000 Academic Press Key Words: scale-space; local histograms.