Curvature Estimates for Submanifolds in Warped Products

We give estimates of the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

Construction of Hamiltonian-Minimal Lagrangian submanifolds in Complex Euclidean Space

We describe several families of Lagrangian submanifolds in complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

Equivariant Nielsen root theory for G-maps

Let X be a compact Hausdorff space, Y be a connected topological manifold, f : X -> Y be a map between closed manifolds and a is an element of Y. The vanishing of the Nielsen root number N(f; a) implies that f is homotopic to a root free map h, i.e., h similar to f and h(-1) (a) = empty set. In this paper, we prove an equivariant analog of this result for G-maps between G-spaces where G is a finite group. (C) 2010 Elsevier B.V. All rights reserved.

Unfolding a face : from singular to manifold

Face recognition from a single image remains an important task in many practical applications and a significant research challenge. Some of the challenges are inherent to the problem, for example due to changing lighting conditions. Others, no less significant, are of a practical nature – face recognition algorithms cannot be assumed to operate on perfect data, but rather often on data that has already been subject to pre-processing errors (e.g. localization and registration errors). This paper introduces a novel method for face recognition that is both trained and queried using only a single image per subject. The key concept, motivated by abundant prior work on face appearance manifolds, is that of face part manifolds – it is shown that the appearance seen through a sliding window overlaid over an image of a face, traces a trajectory over a 2D manifold embedded in the image space. We present a theoretical argument for the use of this representation and demonstrate how it can be effectively exploited in the single image based recognition. It is shown that while inheriting the advantages of local feature methods, it also implicitly captures the geometric relationship between discriminative facial features and is naturally robust to face localization errors. Our theoretical arguments are verified in an experimental evaluation on the Yale Face Database.

Face recognition with image sets using manifold density divergence

In many automatic face recognition applications, a set of a person's face images is available rather than a single image. In this paper, we describe a novel method for face recognition using image sets. We propose a flexible, semi-parametric model for learning probability densities confined to highly non-linear but intrinsically low-dimensional manifolds. The model leads to a statistical formulation of the recognition problem in terms of minimizing the divergence between densities estimated on these manifolds. The proposed method is evaluated on a large data set, acquired in realistic imaging conditions with severe illumination variation. Our algorithm is shown to match the best and outperform other state-of-the-art algorithms in the literature, achieving 94% recognition rate on average.

Automatic cast listing in feature-length films with Anisotropic Manifold Space

Our goal is to automatically determine the cast of a feature-length film. This is challenging because the cast size is not known, with appearance changes of faces caused by extrinsic imaging factors (illumination, pose, expression) often greater than due to differing identities. The main contribution of this paper is an algorithm for clustering over face appearance manifolds. Specifically: (i) we develop a novel algorithm for exploiting coherence of dissimilarities between manifolds, (ii) we show how to estimate the optimal dataset-specific discriminant manifold starting from a generic one, and (iii) we describe a fully automatic, practical system based on the proposed algorithm. The performance of the system is evaluated on well-known featurelength films and situation comedies on which it is shown to produce good results.

Learning over sets using boosted manifold principal angles (BoMPA)

In this paper we address the problem of classifying vector sets. We motivate and introduce a novel method based on comparisons between corresponding vector subspaces. In particular, there are two main areas of novelty: (i) we extend the concept of principal angles between linear subspaces to manifolds with arbitrary nonlinearities; (ii) it is demonstrated how boosting can be used for application-optimal principal angle fusion. The strengths of the proposed method are empirically demonstrated on the task of automatic face recognition (AFR), in which it is shown to outperform state-of-the-art methods in the literature.

A pose-wise linear illumination manifold model for face recognition using video

The objective of this work is to recognize faces using video sequences both for training and novel input, in a realistic, unconstrained setup in which lighting, pose and user motion pattern have a wide variability and face images are of low resolution. There are three major areas of novelty: (i) illumination generalization is achieved by combining coarse histogram correction with fine illumination manifold-based normalization; (ii) pose robustness is achieved by decomposing each appearance manifold into semantic Gaussian pose clusters, comparing the corresponding clusters and fusing the results using an RBF network; (iii) a fully automatic recognition system based on the proposed method is described and extensively evaluated on 600 head motion video sequences with extreme illumination, pose and motion pattern variation. On this challenging data set our system consistently demonstrated a very high recognition rate (95% on average), significantly outperforming state-of-the-art methods from the literature.

An illumination invariant face recognition system for access control using video

Illumination and pose invariance are the most challenging aspects of face recognition. In this paper we describe a fully automatic face recognition system that uses video information to achieve illumination and pose robustness. In the proposed method, highly nonlinear manifolds of face motion are approximated using three Gaussian pose clusters. Pose robustness is achieved by comparing the corresponding pose clusters and probabilistically combining the results to derive a measure of similarity between two manifolds. Illumination is normalized on a per-pose basis. Region-based gamma intensity correction is used to correct for coarse illumination changes, while further refinement is achieved by combining a learnt linear manifold of illumination variation with constraints on face pattern distribution, derived from video. Comparative experimental evaluation is presented and the proposed method is shown to greatly outperform state-of-the-art algorithms. Consistent recognition rates of 94-100% are achieved across dramatic changes in illumination.

Infrared face recognition: a comprehensive review of methodologies and databases

Automatic face recognition is an area with immense practical potential which includes a wide range of commercial and law enforcement applications. Hence it is unsurprising that it continues to be one of the most active research areas of computer vision. Even after over three decades of intense research, the state-of-the-art in face recognition continues to improve, benefitting from advances in a range of different research fields such as image processing, pattern recognition, computer graphics, and physiology. Systems based on visible spectrum images, the most researched face recognition modality, have reached a significant level of maturity with some practical success. However, they continue to face challenges in the presence of illumination, pose and expression changes, as well as facial disguises, all of which can significantly decrease recognition accuracy. Amongst various approaches which have been proposed in an attempt to overcome these limitations, the use of infrared (IR) imaging has emerged as a particularly promising research direction. This paper presents a comprehensive and timely review of the literature on this subject. Our key contributions are (i) a summary of the inherent properties of infrared imaging which makes this modality promising in the context of face recognition; (ii) a systematic review of the most influential approaches, with a focus on emerging common trends as well as key differences between alternative methodologies; (iii) a description of the main databases of infrared facial images available to the researcher; and lastly (iv) a discussion of the most promising avenues for future research. © 2014 Elsevier Ltd.