In this paper we show that a classic optical ow technique by Nagel and Enkelmann (1986) can be regarded as an early anisotropic diusion method with a diusion tensor. We introduce three improvements into the model formulation that (i) avoid inconsistencies caused by centering the brightness term and the smoothness term in dierent images, (ii) use a linear scale-space focusing strategy from coarse to ne scales for avoiding convergence to physically irrelevant local minima, and (iii) create an energy functional that is invariant under linear brightness changes. Applying a gradient descent method to the resulting energy functional leads to a system of diusion{reaction equations. We prove that this system has a unique solution under realistic assumptions on the initial data, and we present an ecient linear implicit numerical scheme in detail. Our method creates ow elds with 100 % density over the entire image domain, it is robust under a large range of parameter variations, and it c...

Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for data-driven and flow-driven, isotropic and anisotropic, as well as spatial and spatio-temporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are well-posed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flow-driven regularizers is identified, and a design criterion is proposed for constructing anisotropic flow-driven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.

For strongly undirected anisotropic processes such as coherence-enhancing diffusion...

The goal of this paper is to present segmentation algorithms which combine regularization by nonlinear partial differential equations (PDEs) with a watershed transformation with region merging. We develop efficient algorithms for two wellfounded PDE methods. They use an additive operator splitting (AOS) leading to recursive and separable filters. Further speed-up can be obtained by embedding AOS schemes into a pyramid framework. Examples are presented which demonstrate that the preprocessing by these PDE techniques eases and stabilizes the segmentation. The typical CPU time for segmenting a 256 2 image on a workstation is less than 2 seconds. Key Words: Nonlinear diffusion, additive operator splitting, Gaussian pyramid, watershed segmentation, region merging CR Subject Classification: I.4.6, I.4.3, I.4.4. 1 Introduction Segmentation is one of the bottlenecks of many image analysis and computer vision tasks ranging from medical image processing to robot navigation. Ideally it sho...

ing it is crucial to use numerical schemes with highly accurate directional behaviour. To this end, we introduce a novel algorithm for coherence-enhancing anisotropic diusion. It applies recently discovered dierentiation lters with optimal rotation invariance [11], and comes down to an explicit scheme on a 5#5 stencil. By comparing it with several common algorithms we demonstrate its superior behaviour regarding rotation invariance and avoidance of blurring artifacts (dissipativity). Wealsoshow that the new scheme is more than three times more ecient than common explicit schemes on 3 # 3 stencils. It does not require to solve linear systems of equations, and it can be easily implemented in any dimension. ######### Low-level vision, diusion ltering, scale-spaces, performance evaluation, rotation invariance, fast algorithms 1 Introduction In this paper we present and evaluate a novel algorithm for coherence