Interpolating orientation fields : an axiomatic approach

International audienc

Total variation regularization for manifold-valued data

We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $\ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer

A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices

Lissage anisotrope Doppler par extension de l'opérateur de diffusion de Laplace-Beltrami dans le cadre de la géométrie Riemannienne de l'information de Chentsov

- Nous traitons le problème du débruitage spatial de l'estimation des paramètres de lois statistiques grâce au flot géométrique non-linéaire de Laplace-Beltrami. Cet opérateur , récemment introduit par N. Sochen et R. Kimmel [2] en traitement d'image, est étendu au cas d'un espace hybride formé des coordonnées spatiales (géométrie Euclidienne de la mesure) et des paramètres des lois statistiques (géométrie de l'information de Chentsov dont la métrique est donnée par la matrice d'information de Fisher). Cet opérateur EDP agissant intrinsèquement sur la variété entropique possède la propriété de conserver les discontinuités, ce qui permet de lisser les estimateurs des paramètres de lois statistiques de façon anisotrope. Nous appliquons ce formalisme au lissage spatial de paramètres statistiques de processus autorégressifs via la métrique de Siegel, en déduisant l'expression des opérateurs agissant sur les coefficients de réflexion qui permettent de lisser anisotropiquement des spectres Doppler. Que ce soit, pour la théorie des « Espaces d'Echelle », pour la métrique de Chentsov des densités gaussiennes, ou pour les modèles AR, le modèle de géométrie hyperbolique de Poincaré dans le disque unité semble jouer un rôle central. Our problem adresses the spatial denoising of statistical parameters estimation via the géométrie non-linear Laplace-Beltrami Flow based on the embedding space geometry deduced from Information geometry. Input of our problem is a map W: Σ→M where Σ is a n-dimensional Riemannian manifold and X is the embedding of this manifold in a space which is hybrid space of spatial coordinates and statistical parameters coordinates (Chentsov space of Information geometry deduced from Fisher metric). Then, our problem is solved viewing parameters space of Chentsov as embedding maps, that flow toward minimal surfaces.This spatial denoising, recently introduced by N. Sochen & R. Kimmel [2] in Image processing, and presently extended for spatial statistical parameters estimation denoising preserves discontinuities (The operator has the good behaviour that the projection is edge preserving.). After the definition of the Laplace-Beltrami flow for any statistical parameters space based on Fisher metric, we provide different applications for anisotropic spatial smoothing of Gaussian law parameters and reflection coefficient in Doppler analysis. In case of Doppler analysis, the hybrid space is the Embedding Siegel space. In all cases, Poincaré's hyperbolic geometry model in unit disk seems to play a central role

Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging

Mumford-Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even more involved since typical features of vector spaces are not available. In this paper, we propose algorithms for Mumford-Shah and for Potts regularization of manifold-valued signals and images. For the univariate problems, we derive solvers based on dynamic programming combined with (convex) optimization techniques for manifold-valued data. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging), we show that our algorithms compute global minimizers for any starting point. For the multivariate Mumford-Shah and Potts problems (for image regularization) we propose a splitting into suitable subproblems which we can solve exactly using the techniques developed for the corresponding univariate problems. Our method does not require any a priori restrictions on the edge set and we do not have to discretize the data space. We apply our method to diffusion tensor imaging (DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum