Beyond Gauss: Image-Set Matching on the Riemannian Manifold of PDFs

State-of-the-art image-set matching techniques typically implicitly model each image-set with a Gaussian distribution. Here, we propose to go beyond these representations and model image-sets as probability distribution functions (PDFs) using kernel density estimators. To compare and match image-sets, we exploit Csiszar f-divergences, which bear strong connections to the geodesic distance defined on the space of PDFs, i.e., the statistical manifold. Furthermore, we introduce valid positive definite kernels on the statistical manifolds, which let us make use of more powerful classification schemes to match image-sets. Finally, we introduce a supervised dimensionality reduction technique that learns a latent space where f-divergences reflect the class labels of the data. Our experiments on diverse problems, such as video-based face recognition and dynamic texture classification, evidence the benefits of our approach over the state-of-the-art image-set matching methods

Adaptive algorithms for identification of symmetric and positive definite matrices

Adaptive estimation and identification algorithms involving unknown symmetric and positive definite (SPD) matrix-valued parameters are ubiquitous in engineering applications. The problem of estimating the noise covariance matrices in estimation algorithms is considered first. An adaptive Kalman filter to estimate the noise covariance matrix of the noises entering a linear time invariant system is introduced first. The convergence of the estimates as well as the states is guaranteed with mild assumptions on the system. Conditions of estimability of the noise covariance matrix are discussed. The generalization of the adaptive Kalman fitler to the linear time varying case is introduced next. To maintain positive definiteness of the noise covariance estimates a differential geometric approach is adopted. The geometry of the manifold of SPD matrices is used to develop a Riemannian optimization based adaptive Kalman filter that ensure positive definiteness of the estimate. The convergence of the Riemannian optimization-based estimate and the adaptive Kalman filter is established under mild conditions of uniform observability and uniform controllability of the system. An adaptive control problem with an unknown SPD matrix is considered next. A novel projection scheme is introduced that ensures that the estimates of the unknown SPD matrix are SPD. Adaptive update laws for identifying the SPD matrix are also presented. The adaptive control laws are shown to globally stabilize systems in problems such as the adaptive angular velocity tracking, adaptive attitude control, and the adaptive trajectory tracking of robotic manipulators with parameter uncertainties within the generalized mass matrix. In general, such a method can be applied to estimation of symmetric matrices with eigenvalue constraints.Aerospace Engineerin

The Role of Riemannian Manifolds in Computer Vision: From Coding to Deep Metric Learning

A diverse number of tasks in computer vision and machine learning enjoy from representations of data that are compact yet discriminative, informative and robust to critical measurements. Two notable representations are offered by Region Covariance Descriptors (RCovD) and linear subspaces which are naturally analyzed through the manifold of Symmetric Positive Definite (SPD) matrices and the Grassmann manifold, respectively, two widely used types of Riemannian manifolds in computer vision. As our first objective, we examine image and video-based recognition applications where the local descriptors have the aforementioned Riemannian structures, namely the SPD or linear subspace structure. Initially, we provide a solution to compute Riemannian version of the conventional Vector of Locally aggregated Descriptors (VLAD), using geodesic distance of the underlying manifold as the nearness measure. Next, by having a closer look at the resulting codes, we formulate a new concept which we name Local Difference Vectors (LDV). LDVs enable us to elegantly expand our Riemannian coding techniques to any arbitrary metric as well as provide intrinsic solutions to Riemannian sparse coding and its variants when local structured descriptors are considered. We then turn our attention to two special types of covariance descriptors namely infinite-dimensional RCovDs and rank-deficient covariance matrices for which the underlying Riemannian structure, i.e. the manifold of SPD matrices is out of reach to great extent. %Generally speaking, infinite-dimensional RCovDs offer better discriminatory power over their low-dimensional counterparts. To overcome this difficulty, we propose to approximate the infinite-dimensional RCovDs by making use of two feature mappings, namely random Fourier features and the Nystrom method. As for the rank-deficient covariance matrices, unlike most existing approaches that employ inference tools by predefined regularizers, we derive positive definite kernels that can be decomposed into the kernels on the cone of SPD matrices and kernels on the Grassmann manifolds and show their effectiveness for image set classification task. Furthermore, inspired by attractive properties of Riemannian optimization techniques, we extend the recently introduced Keep It Simple and Straightforward MEtric learning (KISSME) method to the scenarios where input data is non-linearly distributed. To this end, we make use of the infinite dimensional covariance matrices and propose techniques towards projecting on the positive cone in a Reproducing Kernel Hilbert Space (RKHS). We also address the sensitivity issue of the KISSME to the input dimensionality. The KISSME algorithm is greatly dependent on Principal Component Analysis (PCA) as a preprocessing step which can lead to difficulties, especially when the dimensionality is not meticulously set. To address this issue, based on the KISSME algorithm, we develop a Riemannian framework to jointly learn a mapping performing dimensionality reduction and a metric in the induced space. Lastly, in line with the recent trend in metric learning, we devise end-to-end learning of a generic deep network for metric learning using our derivation

Multi-Modal Similarity Learning for 3D Deformable Registration of Medical Images

Alors que la perspective de la fusion d images médicales capturées par des systèmes d imageries de type différent est largement contemplée, la mise en pratique est toujours victime d un obstacle théorique : la définition d une mesure de similarité entre les images. Des efforts dans le domaine ont rencontrés un certain succès pour certains types d images, cependant la définition d un critère de similarité entre les images quelle que soit leur origine et un des plus gros défis en recalage d images déformables. Dans cette thèse, nous avons décidé de développer une approche générique pour la comparaison de deux types de modalités donnés. Les récentes avancées en apprentissage statistique (Machine Learning) nous ont permis de développer des solutions innovantes pour la résolution de ce problème complexe. Pour appréhender le problème de la comparaison de données incommensurables, nous avons choisi de le regarder comme un problème de plongement de données : chacun des jeux de données est plongé dans un espace commun dans lequel les comparaisons sont possibles. A ces fins, nous avons exploré la projection d un espace de données image sur l espace de données lié à la seconde image et aussi la projection des deux espaces de données dans un troisième espace commun dans lequel les calculs sont conduits. Ceci a été entrepris grâce à l étude des correspondances entre les images dans une base de données images pré-alignées. Dans la poursuite de ces buts, de nouvelles méthodes ont été développées que ce soit pour la régression d images ou pour l apprentissage de métrique multimodale. Les similarités apprises résultantes sont alors incorporées dans une méthode plus globale de recalage basée sur l optimisation discrète qui diminue le besoin d un critère différentiable pour la recherche de solution. Enfin nous explorons une méthode qui permet d éviter le besoin d une base de données pré-alignées en demandant seulement des données annotées (segmentations) par un spécialiste. De nombreuses expériences sont conduites sur deux bases de données complexes (Images d IRM pré-alignées et Images TEP/Scanner) dans le but de justifier les directions prises par nos approches.Even though the prospect of fusing images issued by different medical imagery systems is highly contemplated, the practical instantiation of it is subject to a theoretical hurdle: the definition of a similarity between images. Efforts in this field have proved successful for select pairs of images; however defining a suitable similarity between images regardless of their origin is one of the biggest challenges in deformable registration. In this thesis, we chose to develop generic approaches that allow the comparison of any two given modality. The recent advances in Machine Learning permitted us to provide innovative solutions to this very challenging problem. To tackle the problem of comparing incommensurable data we chose to view it as a data embedding problem where one embeds all the data in a common space in which comparison is possible. To this end, we explored the projection of one image space onto the image space of the other as well as the projection of both image spaces onto a common image space in which the comparison calculations are conducted. This was done by the study of the correspondences between image features in a pre-aligned dataset. In the pursuit of these goals, new methods for image regression as well as multi-modal metric learning methods were developed. The resulting learned similarities are then incorporated into a discrete optimization framework that mitigates the need for a differentiable criterion. Lastly we investigate on a new method that discards the constraint of a database of images that are pre-aligned, only requiring data annotated (segmented) by a physician. Experiments are conducted on two challenging medical images data-sets (Pre-Aligned MRI images and PET/CT images) to justify the benefits of our approach.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF

Conjugate gradient algorithm for efficient covariance tracking with Jensen‐Bregman LogDet metric

Region covariance descriptor that fuses multiple features compactly has proven to be very effective for visual tracking. While working effectively, the exhaustive global search strategy of covariance tracking is still inefficient, and there is much room for improvement. It may cause inconsecutive tracking trajectory and distraction. A suitable region similarity metric for covariance matching between the candidate object region and a given appearance template is of much importance. However, the computational burden of the metric, especially for large matrices under Riemannian space, may hinder its application in gradient‐based algorithms. In this study, the authors propose an algorithm which, by minimising the metric function, exploits an efficient conjugate gradient method to iteratively search the best matched candidate, and determines the search step size by non‐monotonic liner strategy. Then, an inferential reasoning in view of new efficient metric is derived for the gradient‐based algorithm. The authors test the proposed tracking method on test baseline dataset. Both quantitative and qualitative results demonstrate the effectiveness of the proposed algorithm compared with other state‐of‐the‐art methods