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

This paper deals with establishing relations between a number of widely-used nonlinear filters for digital image processing. We cover robust statistical estimation with (local) M-estimators, local mode filtering in image or histogram space, bilateral filtering, nonlinear diffusion, and regularisation approaches. Although these methods originate in different mathematical theories, we show that their implementation reveals a highly similar structure. We demonstrate that all these methods can be cast into a unified framework of functional minimisation combining nonlocal data and nonlocal smoothness terms. This unification contributes to a better understanding of the individual methods, and it opens the way to new techniques combining the advantages of known filters. Keywords: image analysis, M-estimators, mode filtering, nonlinear diffusion, bilateral filter, regularisation

Abstract. Variational methods for optic flow computation have the reputation of producing good results at the expense of being too slow for real-time applications. We show that real-time variational computation of optic flow fields is possible when appropriate methods are combined with modern numerical techniques. We consider the CLG method, a recent variational technique that combines the quality of the dense flow fields of the Horn and Schunck approach with the noise robustness of the Lucas–Kanade method. For the linear system of equations resulting from the discretised Euler–Lagrange equations, we present a fast full multigrid scheme in detail. We show that under realistic accuracy requirements this method is 175 times more efficient than the widely used Gauß-Seidel algorithm. On a 3.06 GHz PC, we have computed 27 dense flow fields of size 200 × 200 pixels within a single second. 1

We present a generative approach to model-based motion segmentation by incorporating a statistical shape prior into a novel variational segmentation method. The shape prior statistically encodes a training set of object outlines presented in advance during a training phase. In a region

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Based on a geometric interpretation of the optic flow constraint equation, we propose a conditional probability on the spatio-temporal image gradient. We consistently derive a variational approach for the segmentation of the image domain into regions of homogeneous motion.

We present a novel variational approach to dense motion estimation of highly non-rigid structures in image sequences. Our representation of the motion vector field is based on the extended Helmholtz Decomposition into its principal constituents: The laminar flow and two potential functions related to the solenoidal and irrotational flow, respectively. The potential functions, which are of primary interest for flow pattern analysis in numerous application fields like remote sensing or fluid mechanics, are directly estimated from image sequences with a variational approach. We use regularizers with derivatives up to third order to obtain unbiased high–quality solutions. Computationally, the approach is made tractable by means of auxiliary variables. The performance of the approach is demonstrated with ground-truth experiments and real-world data.

Abstract. We introduce a new approach to modelling gradient flows of contours and surfaces. While standard variational methods (e.g. level sets) compute local interface motion in a differential fashion by estimating local contour velocity via energy derivatives, we propose to solve surface evolution PDEs by explicitly estimating integral motion of the whole surface. We formulate an optimization problem directly based on an integral characterization of gradient flow as an infinitesimal move of the (whole) surface giving the largest energy decrease among all moves of equal size. We show that this problem can be efficiently solved using recent advances in algorithms for global hypersurface optimization [4, 2, 11]. In particular, we employ the geo-cuts method [4] that uses ideas from integral geometry to represent continuous surfaces as cuts on discrete graphs. The resulting interface evolution algorithm is validated on some 2D and 3D examples similar to typical demonstrations of levelset methods. Our method allows for computation of gradient flows for hypersurfaces with respect to a fairly general class of continuous functionals and it is flexible with respect to distance metrics on the space of contours/surfaces. Our preliminary tests for standard L2 distance metric demonstrate numerical stability, topological changes and an absence of any oscillatory motion. Index Terms — front propagation, gradient flows for hypersurfaces, min-cut/max-flow algorithms on graphs, geo-cuts, integral geometry. 2 1

We present a variational method for the segmentation of piecewise ane ow elds. Compared to other approaches to motion segmentation, we minimize a single energy functional both with respect to the motion models in the separate regions and with respect to the shape of the separating contour. In the manner of region competition, the evolution of the segmenting contour is driven by a force which aims at maximizing a homogeneity measure with respect to the estimated motion in the adjoining regions.

Variational methods for optic flow computation have the reputation of producing good results at the expense of being too slow for realtime applications. We show that real-time variational computation of optic flow fields is possible when appropriate methods are combined with modern numerical techniques. We consider the CLG method, a recent variational technique that combines the quality of the dense flow fields of the Horn and Schunck approach with the noise robustness of the Lucas–Kanade method. For the linear system of equations resulting from the discretised Euler–Lagrange equations, we present different multigrid schemes in detail. We show that under realistic accuracy requirements they are up to 247 times more efficient than the widely used Gauß-Seidel algorithm. On a 3.06 GHz PC, we have computed 40 dense flow fields of size 200 × 200 pixels within a single second.