1,450 research outputs found
Estimating the mean manifold of a deformable object from noisy observations
Assume we have a set of noisy observations (for example, images) of different objects, each undergoing a different geometric deformation, yet all the deformations belong to the same family. As a result of the action of these deformations, the set of different observations on each object is generally a manifold in the ambient space of observations. It has been shown, [1], that in the absence of noise, in those cases where the set of deformations admits a finite-dimensional representation, the universal manifold embedding (UME) provides a mapping from the space of observations to a low dimensional linear space. The manifold corresponding to each object is mapped to a distinct linear subspace of Euclidean space, and the dimension of the subspace is the same as that of the manifold. In the presence of noise, different observations are mapped to different subspaces. In this paper we derive a method for “averaging” the different subspaces, obtained from different observations made on the same object, in order to estimate the mean representation of the object manifold. The mean manifold representation is then employed to minimize the effects of noise in matched manifold detectors and to improve the separability of data sets in the context of object detection and classification
Warped Kaluza-Klein Dark Matter
Warped compactifications of type IIB string theory contain natural dark
matter candidates: Kaluza-Klein modes along approximate isometry directions of
long warped throats. These isometries are broken by the full compactification,
including moduli stabilization; we present a thorough survey of Kaluza-Klein
mode decay rates into light supergravity modes and Standard Model particles. We
find that these dark matter candidates typically have lifetimes longer than the
age of the universe. Interestingly, some choices for embedding the Standard
Model in the compactification lead to decay rates large enough to be observed,
so this dark matter sector may provide constraints on the parameter space of
the compactification.Comment: 37pp; v2. references, minor clarificatio
Effective versions of the positive mass theorem
The study of stable minimal surfaces in Riemannian -manifolds
with non-negative scalar curvature has a rich history. In this paper, we prove
rigidity of such surfaces when is asymptotically flat and has horizon
boundary. As a consequence, we obtain an effective version of the positive mass
theorem in terms of isoperimetric or, more generally, closed volume-preserving
stable CMC surfaces that is appealing from both a physical and a purely
geometric point of view. We also include a proof of the following conjecture of
R. Schoen: An asymptotically flat Riemannian -manifold with non-negative
scalar curvature that contains an unbounded area-minimizing surface is
isometric to flat .Comment: All comments welcome! The final version has appeared in Invent. Mat
Recognising facial expressions in video sequences
We introduce a system that processes a sequence of images of a front-facing human face and recognises a set of facial expressions. We use an efficient appearance-based face tracker to locate the face in the image sequence and estimate the deformation of its non-rigid components. The tracker works in real-time. It is robust to strong illumination changes and factors out changes in appearance caused by illumination from changes due to face deformation. We adopt a model-based approach for facial expression recognition. In our model, an image of a face is represented by a point in a deformation space. The variability of the classes of images associated to facial expressions are represented by a set of samples which model a low-dimensional manifold in the space of deformations. We introduce a probabilistic procedure based on a nearest-neighbour approach to combine the information provided by the incoming image sequence with the prior information stored in the expression manifold in order to compute a posterior probability associated to a facial expression. In the experiments conducted we show that this system is able to work in an unconstrained environment with strong changes in illumination and face location. It achieves an 89\% recognition rate in a set of 333 sequences from the Cohn-Kanade data base
Slopes of smooth curves on Fano manifolds
Ross and Thomas introduced the concept of slope stability to study
K-stability, which has conjectural relation with the existence of constant
scalar curvature K\"ahler metric. This paper presents a study of slope
stability of Fano manifolds of dimension with respect to smooth
curves. The question turns out to be easy for curves of genus and the
interest lies in the case of smooth rational curves. Our main result classifies
completely the cases when a polarized Fano manifold is not slope
stable with respect to a smooth curve. Our result also states that a Fano
threefold with Picard number 1 is slope stable with respect to every smooth
curve unless is the projective space.Comment: 13 pages, Theorems in the original version were modified. This paper
will be published in the Bulletin of the London Mathematical Societ
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