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.

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.

We analyze a variational approach to image segmentation that is based on a strictly convex non-quadratic cost functional. The smoothness term combines a standard first-order measure for image regions with a total-variation based measure for signal transitions. Accordingly, the costs associated with "discontinuities" are given by the length of level lines and local image contrast. For real images, this provides a reasonable approximation of the variational model of Mumford and Shah that has been suggested as a generic approach to image segmentation. The global properties of the convex variational model are favorable to applications: Uniqueness of the solution, continuous dependence of the solution on both data and parameters, consistent and efficient numerical approximation of the solution with the FEM-method. Various global and local properties of the convex variational model are analyzed and illustrated with numerical examples. Apart from the favorable global properties, the approach ...

Differential methods are frequently used techniques for optic flow computations. They can be classified into local methods such as the Lucas-Kanade technique or Bigfin's structure tensor method, and into global methods such as the Horn-Schunck approach and its modifications.

fait un grand honneur. Grace `a Madame Doina Cioranescu, j'ai pu 'etudier en France, `a l'Universit'e de Nice, au Laboratoire de Math'ematiques Jean-Alexandre Dieudonn'e. Elle a consacr'e beaucoup de son temps et de son 'energie pour organiser le programme europ'een MATAROU (Math'ematiques Appliqu'ees en Roumanie) dont j'ai benefici'e comme boursi`ere TEMPUS. Sans elle, cette th`ese n'aurait pas exist'e. Je voudrais remercier Monsieur Michel Barlaud et Monsieur Rachid Deriche, qui m'ont fait l'honneur de faire partie de jury de ma th`ese. Je les remercie aussi pour les discussions que nous avons eues et les conseils qu'ils m'ont donn'es, `a des diff'erentes occasions. Je remercie aussi Monsieur Mihail Megan, qui a accept'e d'etre directeur de th`ese de la partie roumaine et membre du jury. Pour m'avoir permis d'utiliser dans les meilleurs conditions le support informatique indispensable `a mon travail, je tiens `a remercier Jean-Marc Lacroix et Bernard L'Homme. Ils ont toujours r'epond

In this paper we address the problem of estimating and analyzing the motion in image sequences showing fluid phenomenon. Due to the great deal of spatial and temporal distortions that luminance patterns exhibit in images of fluid, standard techniques from Computer Vision, originally designed for quasi-rigid motions with stable salient features, are not well adapted in this context. In that prospect, we investigate a dedicated energybased motion estimator. The considered functional includes an original data model relying on the continuity equation of fluid mechanics. This new data model, which is specifically designed to be embedded in a multiresolution framework, is associated to an original divcurl type regularization. The optimization of the global energy function is solved within an efficient multigrid scheme. The performances of the resulting fluid flow estimator are demonstrated both on synthetic and real (meteorological) image sequences.