eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

class based feature matching across unrestricted transformations we develop a novel method for class based feature matching across large changes in viewing conditions the method is based on the property that when objects share a similar part the similarity is preserved across viewing conditions given a feature and a training set of object images we first identify the subset of objects that share this feature the transformation of the feature s appearance across viewing conditions is determined mainly by properties of the feature rather than of the object in which it is embedded therefore the transformed feature will be shared by approximately the same set of objects based on this consistency requirement corresponding features can be reliably identified from a set of candidate matches unlike previous approaches the proposed scheme compares feature appearances only in similar viewing conditions rather than across different viewing conditions as a result the scheme is not restricted to locally planar objects or affine transformations the approach also does not require examples of correct matches we show that by using the proposed method a dense set of accurate correspondences can be obtained experimental comparisons demonstrate that matching accuracy is significantly improved over previous schemes finally we show that the scheme can be successfully used for invariant object recognition

remarque sur le probleme de yamabe x g designant une variete localement conformement plate on etudie avec une approche toute nouvelle la fonction de green de l operateur laplacien conforme sur x les resultats obtenus nous permettent de montrer que les fonctions locales e r2 1 n2 prolongees par leurs constantes naturelles fournissent l inegalite 14n n 2 n2n du theoreme fondamental de aubin theoreme 1 page 280 de t aubin j funct anal 32 1979 148 174 let x g be a locally conformally flat manifold in this paper we study with a very new approach the green function of the conformal laplacian operator on x then with the results obtained we prove that the local functions e r2 1 n2 continued by their natural constants lead to the inequality 14n n 2 n2n of the fundamental theorem of aubin theorem 1 p 280 of t aubin j funct anal 32 1979 148 174

space and time bounds on indexing 3d models from 2d images model based visual recognition systems often match groups of image features to groups of model features to form initial hypotheses which are then verified in order to accelerate recognition considerably the model groups can be arranged in an index space hashed offline such that feasible matches are found by indexing into this space for the case of 2d images and 3d models consisting of point features bounds on the space required for indexing and on the speedup that such indexing can achieve are demonstrated it is proved that even in the absence of image error each model must be represented by a 2d surface in the index space this places an unexpected lower bound on the space required to implement indexing and proves that no quantity is invariant for all projections of a model into the image theoretical bounds on the speedup achieved by indexing in the presence of image error are also determined and an implementation of indexing for measuring this speedup empirically is presented it is found that indexing can produce only a minimal speedup on its own however when accompanied by a grouping operation indexing can provide significant speedups that grow exponentially with the number of features in the groups

bayesian nonparametric estimation of continuous monotone functions with applications to dose response analysis in this article we consider monotone nonparametric regression in a bayesian framework the monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions and a general random probability measure is assumed as the prior for the mixing distribution we investigate the choice of the underlying parametric distribution function and find that the two sided power distribution function is well suited both from a computational and mathematical point of view the model is motivated by traditional nonlinear models for dose response analysis and provides possibilities to elicitate informative prior distributions on different aspects of the curve the method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose response analysis

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

a non compactness result on the fractional yamabe problem in large dimensions abstract let x n 1 g be an n 1 dimensional asymptotically hyperbolic manifold with conformal infinity m n h the fractional yamabe problem addresses to solve p g h u c u n 2 n 2 u 0 on m where c r and p g h is the fractional conformal laplacian whose principal symbol is the laplace beltrami operator on m in this paper we construct a metric on the half space x r n 1 which is conformally equivalent to the unit ball for which the solution set of the fractional yamabe equation is non compact provided that n 24 for 0 and n 25 for 1 where 0 1 is a certain transition exponent the value of turns out to be approximately 0 940197

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

recognizing imprecisely localized partially occluded and expression variant faces from a single sample per class the classical way of attempting to solve the face or object recognition problem is by using large and representative data sets in many applications though only one sample per class is available to the system in this contribution we describe a probabilistic approach that is able to compensate for imprecisely localized partially occluded and expression variant faces even when only one single training sample per class is available to the system to solve the localization problem we find the subspace within the feature space e g eigenspace that represents this error for each of the training images to resolve the occlusion problem each face is divided into k local regions which are analyzed in isolation in contrast with other approaches where a simple voting space is used we present a probabilistic method that analyzes how good a local match is to make the recognition system less sensitive to the differences between the facial expression displayed on the training and the testing images we weight the results obtained on each local area on the basis of how much of this local area is affected by the expression displayed on the current test image

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

singular value decomposition and local near neighbors for face recognition under varying illumination illumination processing is a challenging task in face recognition although numerous techniques have been proposed to tackle this problem none of them can achieve high performance under different illumination variations in this paper we propose three closely related illumination processing techniques to address different illumination variations first we employ singular value decomposition svd bases with frequency interpretation and corresponding contribution coefficients i e normalized nonlinear singular values to obtain the high frequency facial features of a face image in logarithm domain which is termed as the high frequency svd face hfsvd face moreover we propose the adaptive hfsvd face ahfsvd face which adaptively selects a special nonlinear parameter to generate hfsvd face according to the face image illumination level the illumination level of a face image is estimated by its singular values second we construct the local near neighbor feature lnn feature based on the local region which is an illumination invariant measure then the discriminative weights dws and gaussian weights gws respectively combine with several lnn features to generate the fusion lnn feature flnn feature which can be named as dwflnn feature and gwlnn feature the bipolar sigmoidal function removes the high frequency interference of flnn feature to form lnn face including dwlnn face and gwlnn face third we develop the h l svd as a preprocessing technique which consists of the normalized low frequency components and the corrected high frequency components divided by svd and then the illumination insensitive measure can be extracted from h l svd by lnn face to robustly tackle the face image with severe illumination variations the performances of the proposed methods are verified using yale b cmu pie lfw and our self built driver face databases the experimental results indicate that the proposed methods can obtain high performances under different illumination variations and outperform several state of the art approaches which are proposed to address face recognition under varying illumination hfsvd face consists of svd bases and normalized nonlinear singular values ahfsvd face is the hfsvd face with a special nonlinear parameter lnn face is extracted from several local regions

the han li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds abstract the han li conjecture states that let m g 0 be an n dimensional n 3 smooth compact riemannian manifold with boundary having positive generalized yamabe constant and c be any real number then there exists a conformal metric of g 0 with scalar curvature 1 and boundary mean curvature c combining with z c han and y y li s results we answer this conjecture affirmatively except for the case that n 8 the boundary is umbilic the weyl tensor of m vanishes on the boundary and has an interior non zero point

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

discrimination on the grassmann manifold fundamental limits of subspace classifiers we derive fundamental limits on the reliable classification of linear and affine subspaces from noisy linear features drawing an analogy between discrimination among subspaces and communication over vector wireless channels we define two shannon inspired characterizations of asymptotic classifier performance first we define the classification capacity which characterizes the necessary and sufficient conditions for vanishing misclassification probability as the signal dimension the number of features and the number of subspaces to be discriminated all approach infinity second we define the diversity discrimination tradeoff which by analogy with the diversity multiplexing tradeoff of fading vector channels characterizes relationships between the number of discernible subspaces and the misclassification probability as the feature noise power approaches zero we derive upper and lower bounds on these quantities which are tight in many regimes numerical results including a face recognition application validate the results in practice

the effect of linear perturbations on the yamabe problem in conformal geometry the compactness conjecture asserts that the set of yamabe metrics on a smooth compact aspherical riemannian manifold left m g right is compact established in the locally conformally flat case by schoen lecture notes in mathematics vol 1365 pp 120 154 springer berlin 1989 surveys pure application and mathematics 52 longman science technology pp 311 320 harlow 1991 and for n le 24 by khuri marques schoen j differ geom 81 1 143 196 2009 it has revealed to be generally false for n ge 25 as shown by brendle j am math soc 21 4 951 979 2008 and brendle marques j differ geom 81 2 225 250 2009 a stronger version of it the compactness under perturbations of the yamabe equation is addressed here with respect to the linear geometric potential frac n 2 4 n 1 mathrm scal g mathrm scal g being the scalar curvature of left m g right we show that a priori l infty bounds fail for linear perturbations on all manifolds with n ge 4 as well as a priori gradient l 2 bounds fail for non locally conformally flat manifolds with n ge 6 and for locally conformally flat manifolds with n ge 7 in several situations the results are optimal our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g

face recognition the problem of compensating for changes in illumination direction a face recognition system must recognize a face from a novel image despite the variations between images of the same face a common approach to overcoming image variations because of changes in the illumination conditions is to use image representations that are relatively insensitive to these variations examples of such representations are edge maps image intensity derivatives and images convolved with 2d gabor like filters here we present an empirical study that evaluates the sensitivity of these representations to changes in illumination as well as viewpoint and facial expression our findings indicated that none of the representations considered is sufficient by itself to overcome image variations because of a change in the direction of illumination similar results were obtained for changes due to viewpoint and expression image representations that emphasized the horizontal features were found to be less sensitive to changes in the direction of illumination however systems based only on such representations failed to recognize up to 20 percent of the faces in our database humans performed considerably better under the same conditions we discuss possible reasons for this superiority and alternative methods for overcoming illumination effects in recognition

some results about the existence of critical points for the willmore functional using a perturbative approach it is shown existence and multiplicity of critical points for the willmore functional in ambient manifold mathbb r 3 g epsilon where g epsilon is a metric close and asymptotic to the euclidean one with the same technique it is proved a non existence result in general riemannian manifold m g of dimension three