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.

Robust learning of discriminative projection for multicategory classification on the Stiefel manifold

Learning a robust projection with a small number of training samples is still a challenging problem in face recognition, especially when the unseen faces have extreme variation in pose, illumination, and facial expression. To address this problem, we propose a framework formulated under statistical learning theory that facilitates robust learning of a discriminative projection. Dimensionality reduction using the projection matrix is combined with a linear classifier in the regularized framework of lasso regression. The projection matrix in conjunction with the classifier parameters are then found by solving an optimization problem over the Stiefel manifold. The experimental results on standard face databases suggest that the proposed method outperforms some recent regularized techniques when the number of training samples is small.

Supervised subspace learning with multi-class Lagrangian SVM on the Grassmann manifold

Learning robust subspaces to maximize class discrimination is challenging, and most current works consider a weak connection between dimensionality reduction and classifier design. We propose an alternate framework wherein these two steps are combined in a joint formulation to exploit the direct connection between dimensionality reduction and classification. Specifically, we learn an optimal subspace on the Grassmann manifold jointly minimizing the classification error of an SVM classifier. We minimize the regularized empirical risk over both the hypothesis space of functions that underlies this new generalized multi-class Lagrangian SVM and the Grassmann manifold such that a linear projection is to be found. We propose an iterative algorithm to meet the dual goal of optimizing both the classifier and projection. Extensive numerical studies on challenging datasets show robust performance of the proposed scheme over other alternatives in contexts wherein limited training data is used, verifying the advantage of the joint formulation.

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Study guide for TAFE management unit in Victoria.

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.

A manifold approach to face recognition from low quality video across illumination and pose using implicit super-resolution

We consider the problem of matching a face in a low resolution query video sequence against a set of higher quality gallery sequences. This problem is of interest in many applications, such as law enforcement. Our main contribution is an extension of the recently proposed Generic Shape-Illumination Manifold (gSIM) framework. Specifically, (i) we show how super-resolution across pose and scale can be achieved implicitly, by off-line learning of subsampling artefacts; (ii) we use this result to propose an extension to the statistical model of the gSIM by compounding it with a hierarchy of subsampling models at multiple scales; and (iii) we describe an extensive empirical evaluation of the method on over 1300 video sequences – we first measure the degradation in performance of the original gSIM algorithm as query sequence resolution is decreased and then show that the proposed extension produces an error reduction in the mean recognition rate of over 50%.

Face Recognition from Video Using the Generic Shape-Illumination Manifold

In spite of over two decades of intense research, illumination and pose invariance remain prohibitively challenging aspects of face recognition for most practical applications. The objective of this work is to recognize faces using video sequences both for training and recognition 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. In particular there are three areas of novelty: (i) we show how a photometric model of image formation can be combined with a statistical model of generic face appearance variation, learnt offline, to generalize in the presence of extreme illumination changes; (ii) we use the smoothness of geodesically local appearance manifold structure and a robust same-identity likelihood to achieve invariance to unseen head poses; and (iii) we introduce an accurate video sequence “reillumination” algorithm to achieve robustness to face motion patterns in video. We describe a fully automatic recognition system based on the proposed method and an extensive evaluation on 171 individuals and over 1300 video sequences with extreme illumination, pose and head motion variation. On this challenging data set our system consistently demonstrated a nearly perfect recognition rate (over 99.7%), significantly outperforming state-of-the-art commercial software and methods from the literature

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.