pattern matching with affine moment descriptors this paper proposes a method for matching images based on their higher order moments without knowing the point correspondences it is assumed that the disparity between the images can be explained by an affine transformation the second order statistics is used to transform the image points into canonical form which reduces the affine matching problem for determining an orthonormal transformation matrix between the two point sets next higher order complex moments are used to solve the remaining transformation these affine moment descriptors are expressed in terms of the central moments of the original data it is also shown that the resulting descriptors can be converted into affine moment invariants a general framework for deriving affine moment descriptors as well as moment invariants is described the experiments carried out with simulated data and real images indicate that the proposed method utilizing the second and third order statistics can provide good alignment results from noisy and spurious observations

probabilistic metrization of fuzzy uniformities abstract modifying the structure of fuzzy uniform spaces introduced by lowen 5 we propose a further concept of fuzzy uniformities so called fuzzy t

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

affine invariant pattern recognition using multiscale autoconvolution this paper presents a new affine invariant image transform called multiscale autoconvolution msa the proposed transform is based on a probabilistic interpretation of the image function the method is directly applicable to isolated objects and does not require extraction of boundaries or interest points and the computational load is significantly reduced using the fast fourier transform the transform values can be used as descriptors for affine invariant pattern classification and in this article we illustrate their performance in various object classification tasks as shown by a comparison with other affine invariant techniques the new method appears to be suitable for problems where image distortions can be approximated with affine transformations

parametric estimation of affine deformations of planar shapes we consider the estimation of affine transformations aligning a known 2d shape and its distorted observation the classical way to solve this registration problem is to find correspondences between the shapes and then compute the transformation parameters from these landmarks here we propose a novel approach where the exact transformation is obtained as the solution of a polynomial system of equations the method has been tested on synthetic as well as on real images and its robustness in the presence of segmentation errors and additive geometric noise has also been demonstrated we have successfully applied the method for the registration of hip prosthesis x ray images the advantage of the proposed solution is that it is fast easy to implement has linear time complexity works without established correspondences and provides an exact solution regardless of the magnitude of transformation

joint hypergraph learning and sparse regression for feature selection abstract in this paper we propose a unified framework for improved structure estimation and feature selection most existing graph based feature selection methods utilize a static representation of the structure of the available data based on the laplacian matrix of a simple graph here on the other hand we perform data structure learning and feature selection simultaneously to improve the estimation of the manifold representing the structure of the selected features we use a higher order description of the neighborhood structures present in the available data using hypergraph learning this allows those features which participate in the most significant higher order relations to be selected and the remainder discarded through a sparsification process we formulate a single objective function to capture and regularize the hypergraph weight estimation and feature selection processes finally we present an optimization algorithm to recover the hypergraph weights and a sparse set of feature selection indicators this process offers a number of advantages first by adjusting the hypergraph weights we preserve high order neighborhood relations reflected in the original data which cannot be modeled by a simple graph moreover our objective function captures the global discriminative structure of the features in the data comprehensive experiments on 9 benchmark datasets show that our method achieves statistically significant improvement over state of art feature selection methods supporting the effectiveness of the proposed method

color face image enhancement using adaptive singular value decomposition in fourier domain for face recognition face recognition is still a challenging problem because of large intra class variability small inter class variability and the presence of lighting variation to deal with these difficulties an illumination compensation method adaptive singular value decomposition in the two dimensional discrete fourier domain asvdf and an efficient brightness detector for lighting detection for face image enhancement are proposed in this paper the proposed enhancement algorithm involves three steps in the first step uniform lighting is rapidly distinguished from lateral lighting in the image by using the brightness detector which is based on the percentage ratio of pixels among the three rgb color channels asvdf is then globally performed for the uniform lighting image whereas asvdf is applied block by block for the lateral lighting image in addition to reduce computing time a region based asvdf method is introduced the asvdf method is applied to four regions of the face image experimental results for the cmu pie color feret and fei face databases show that the method considerably improves the quality of face images even lateral lighting thereby improving the accuracy of face recognition substantially adaptive singular value decomposition in the fourier domain is proposed for face recognition self adapted illumination compensation is devised to overcome lighting variation experimental results are demonstrated on cmu feret and fei databases to verify the effectiveness

enhanced local texture feature sets for face recognition under difficult lighting conditions making recognition more reliable under uncontrolled lighting conditions is one of the most important challenges for practical face recognition systems we tackle this by combining the strengths of robust illumination normalization local texture based face representations distance transform based matching kernel based feature extraction and multiple feature fusion specifically we make three main contributions 1 we present a simple and efficient preprocessing chain that eliminates most of the effects of changing illumination while still preserving the essential appearance details that are needed for recognition 2 we introduce local ternary patterns ltp a generalization of the local binary pattern lbp local texture descriptor that is more discriminant and less sensitive to noise in uniform regions and we show that replacing comparisons based on local spatial histograms with a distance transform based similarity metric further improves the performance of lbp ltp based face recognition and 3 we further improve robustness by adding kernel principal component analysis pca feature extraction and incorporating rich local appearance cues from two complementary sources gabor wavelets and lbp showing that the combination is considerably more accurate than either feature set alone the resulting method provides state of the art performance on three data sets that are widely used for testing recognition under difficult illumination conditions extended yale b cas peal r1 and face recognition grand challenge version 2 experiment 4 frgc 204 for example on the challenging frgc 204 data set it halves the error rate relative to previously published methods achieving a face verification rate of 88 1 at 0 1 false accept rate further experiments show that our preprocessing method outperforms several existing preprocessors for a range of feature sets data sets and lighting conditions

a note on fixed point theorems of hadzci c abstract in this paper we give a fixed point theorem for multi valued probabilistic contraction mappings on a menger space as a special case a fixed point theorem for multi valued contraction mappings on a fuzzy metric space is obtained the results presented improve and generalize in a sense some recent fixed point theorems proved by had ic

cross weighted moments and affine invariants for image registration and matching a framework for deriving a class of new global affine invariants for both object matching and positioning based on a novel concept of cross weighted moments with fractional weights is presented the fractional weight factor allows for a more flexible range to balance between the capability to discriminate between objects that differ only in small shape details and the sensitivity of small shape details to the presence of the noise moreover it makes it possible to arrive at low order zero order affine invariants that are more robust than those derived from higher order regular moments the affine transformation parameters are recovered from the zero and the first order cross weighted moments without requiring any feature point correspondence information the equations used to find the affine transformation parameters are linear algebraic the sensitivity of the cross weighted moment invariants to noise missing data and perspective effects is shown on real images

spherical homoscedastic distributions the equivalency of spherical and normal distributions in classification many feature representations as in genomics describe directional data where all feature vectors share a common norm in other cases as in computer vision a norm or variance normalization step where all feature vectors are normalized to a common length is generally used these representations and pre processing step map the original data from p to the surface of a hypersphere sp 1 such representations should then be modeled using spherical distributions however the difficulty associated with such spherical representations has prompted researchers to model their spherical data using gaussian distributions instead as if the data were represented in p rather than sp 1 this opens the question to whether the classification results calculated with the gaussian approximation are the same as those obtained when using the original spherical distributions in this paper we show that in some particular cases which we named spherical homoscedastic the answer to this question is positive in the more general case however the answer is negative for this reason we further investigate the additional error added by the gaussian modeling we conclude that the more the data deviates from spherical homoscedastic the less advisable it is to employ the gaussian approximation we then show how our derivations can be used to define optimal classifiers for spherical homoscedastic distributions by using a kernel which maps the original space into one where the data adapts to the spherical homoscedastic model we can derive non linear classifiers with potential applications in a large number of problems we conclude this paper by demonstrating the uses of spherical homoscedasticity in the classification of images of objects gene expression sequences and text data

an analysis of linear subspace approaches for computer vision and pattern recognition linear subspace analysis lsa has become rather ubiquitous in a wide range of problems arising in pattern recognition and computer vision the essence of these approaches is that certain structures are intrinsically or approximately low dimensional for example the factorization approach to the problem of structure from motion sfm and principal component analysis pca based approach to face recognition in lsa the singular value decomposition svd is usually the basic mathematical tool however analysis of the performance in the presence of noise has been lacking we present such an analysis here first the denoising capacity of the svd is analysed specifically given a rank r matrix corrupted by noise how much noise remains in the rank r projected version of that corrupted matrix second we study the learning capacity of the lsa based recognition system in a noise corrupted environment specifically lsa systems that attempt to capture a data class as belonging to a rank r column space will be affected by noise in both the training samples measurement noise will mean the learning samples will not produce the true subspace and the test sample which will also have measurement noise on top of the ideal clean sample belonging to the true subspace these results should help one to predict aspects of performance and to design more optimal systems in computer vision particularly in tasks such as sfm and face recognition our analysis agrees with certain observed phenomenon and these observations together with our simulations verify the correctness of our theory

illumination recovery from image with cast shadows via sparse representation in this paper we propose using sparse representation for recovering the illumination of a scene from a single image with cast shadows given the geometry of the scene the images with cast shadows can be quite complex and therefore cannot be well approximated by low dimensional linear subspaces however it can be shown that the set of images produced by a lambertian scene with cast shadows can be efficiently represented by a sparse set of images generated by directional light sources we first model an image with cast shadows composed of a diffusive part without cast shadows and a residual part that captures cast shadows then we express the problem in an l1 regularized least squares formulation with nonnegativity constraints as light has to be non negative at any point in space this sparse representation enjoys an effective and fast solution thanks to recent advances in compressive sensing in experiments on synthetic and real data our approach performs favorably in comparison with several previously proposed methods

fuzzy metric neighbourhood spaces abstract hohle founded the first coherent topological study of menger s probabilistic metric spaces by endowing them with fuzzy t uniformities we continue hohle s work in the special case when the triangular norm t used is min that is when the resulting probabilistic metric uniformities are lowen fuzzy uniformities in this case we call the induced fuzzy topological spaces fuzzy pseudo metric neighbourhood spaces we introduce axioms of n regularity and n normality which are satisfied in fuzzy pseudo metric neighbourhood spaces we introduced a lowen fuzzy uniform topology on the fuzzy real line ri called the n euclidean topology by means of a fuzzy probabilistic metric on r i called the axiom of n complete regularity which characterizes the lowen fuzzy uniform topologies

achieving robust face recognition from video by combining a weak photometric model and a learnt generic face invariant 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 the central contribution is an illumination invariant which we show to be suitable for recognition from video of loosely constrained head motion in particular there are three contributions i we show how a photometric model of image formation can be combined with a statistical model of generic face appearance variation to exploit the proposed invariant and generalize in the presence of extreme illumination changes ii we introduce a video sequence re illumination algorithm to achieve fine alignment of two video sequences and iii we use the smoothness of geodesically local appearance manifold structure and a robust same identity likelihood to achieve robustness to unseen head poses we describe a fully automatic recognition system based on the proposed method and an extensive evaluation on 323 individuals and 1474 video sequences with extreme illumination pose and head motion variation our system consistently achieved a nearly perfect recognition rate over 99 7 on all four databases

relations among some low rank subspace recovery models recovering intrinsic low dimensional subspaces from data distributed on them is a key preprocessing step to many applications in recent years a lot of work has modeled subspace recovery as low rank minimization problems we find that some representative models such as robust principal component analysis r pca robust low rank representation r lrr and robust latent low rank representation r latlrr are actually deeply connected more specifically we discover that once a solution to one of the models is obtained we can obtain the solutions to other models in closed form formulations since r pca is the simplest our discovery makes it the center of low rank subspace recovery models our work has two important implications first r pca has a solid theoretical foundation under certain conditions we could find globally optimal solutions to these low rank models at an overwhelming probability although these models are nonconvex second we can obtain significantly faster algorithms for these models by solving r pca first the computation cost can be further cut by applying low complexity randomized algorithms for example our novel filtering algorithm to r pca although for the moment the formal proof of our filtering algorithm is not yet available experiments verify the advantages of our algorithm over other state of the art methods based on the alternating direction method

analytic pca construction for theoretical analysis of lighting variability in images of a lambertian object we analyze theoretically the subspace best approximating images of a convex lambertian object taken from the same viewpoint but under different distant illumination conditions we analytically construct the principal component analysis for images of a convex lambertian object explicitly taking attached shadows into account and find the principal eigenmodes and eigenvalues with respect to lighting variability our analysis makes use of an analytic formula for the irradiance in terms of spherical harmonic coefficients of the illumination and shows under appropriate assumptions that the principal components or eigenvectors are identical to the spherical harmonic basis functions evaluated at the surface normal vectors our main contribution is in extending these results to the single viewpoint case showing how the principal eigenmodes and eigenvalues are affected when only a limited subset the upper hemisphere of normals is available and the spherical harmonics are no longer orthonormal over the restricted domain our results are very close both qualitatively and quantitatively to previous empirical observations and represent the first essentially complete theoretical explanation of these observations

clustering with hypergraphs the case for large hyperedges the extension of conventional clustering to hypergraph clustering which involves higher order similarities instead of pairwise similarities is increasingly gaining attention in computer vision this is due to the fact that many clustering problems require an affinity measure that must involve a subset of data of size more than two in the context of hypergraph clustering the calculation of such higher order similarities on data subsets gives rise to hyperedges almost all previous work on hypergraph clustering in computer vision however has considered the smallest possible hyperedge size due to a lack of study into the potential benefits of large hyperedges and effective algorithms to generate them in this paper we show that large hyperedges are better from both a theoretical and an empirical standpoint we then propose a novel guided sampling strategy for large hyperedges based on the concept of random cluster models our method can generate large pure hyperedges that significantly improve grouping accuracy without exponential increases in sampling costs we demonstrate the efficacy of our technique on various higher order grouping problems in particular we show that our approach improves the accuracy and efficiency of motion segmentation from dense long term trajectories

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

on the wiberg algorithm for matrix factorization in the presence of missing components this paper considers the problem of factorizing a matrix with missing components into a product of two smaller matrices also known as principal component analysis with missing data pcamd the wiberg algorithm is a numerical algorithm developed for the problem in the community of applied mathematics we argue that the algorithm has not been correctly understood in the computer vision community although there are many studies in our community almost every one of which refers to the wiberg study as far as we know there is no literature in which the performance of the wiberg algorithm is investigated or the detail of the algorithm is presented in this paper we present derivation of the algorithm along with a problem in its implementation that needs to be carefully considered and then examine its performance the experimental results demonstrate that the wiberg algorithm shows a considerably good performance which should contradict the conventional view in our community namely that minimization based algorithms tend to fail to converge to a global minimum relatively frequently the performance of the wiberg algorithm is such that even starting with random initial values it converges in most cases to a correct solution even when the matrix has many missing components and the data are contaminated with very strong noise our conclusion is that the wiberg algorithm can also be used as a standard algorithm for the problems of computer vision

principal angles separate subject illumination spaces in ydb and cmu pie the theory of illumination subspaces is well developed and has been tested extensively on the yale face database b ydb and cmu pie pie data sets this paper shows that if face recognition under varying illumination is cast as a problem of matching sets of images to sets of images then the minimal principal angle between subspaces is sufficient to perfectly separate matching pairs of image sets from nonmatching pairs of image sets sampled from ydb and pie this is true even for subspaces estimated from as few as six images and when one of the subspaces is estimated from as few as three images if the second subspace is estimated from a larger set 10 or more this suggests that variation under illumination may be thought of as useful discriminating information rather than unwanted noise

what is the set of images of an object under all possible illumination conditions the appearance of an object depends on both the viewpoint from which it is observed and the light sources by which it is illuminated if the appearance of two objects is never identical for any pose or lighting conditions then in theory the objects can always be distinguished or recognized the question arises what is the set of images of an object under all lighting conditions and pose in this paper we consider only the set of images of an object under variable illumination including multiple extended light sources and shadows we prove that the set of n pixel images of a convex object with a lambertian reflectance function illuminated by an arbitrary number of point light sources at infinity forms a convex polyhedral cone in r n and that the dimension of this illumination cone equals the number of distinct surface normals furthermore the illumination cone can be constructed from as few as three images in addition the set of n pixel images of an object of any shape and with a more general reflectance function seen under all possible illumination conditions still forms a convex cone in r n extensions of these results to color images are presented these results immediately suggest certain approaches to object recognition throughout we present results demonstrating the illumination cone representation

fixed point theorems for multivalued mappings in probabilistic metric spaces abstract in this paper we prove a fixed point theorem for multivalued mappings in probabilistic metric spaces some applications in fuzzy metric and random normed spaces are given

inverse rendering of lambertian surfaces using subspace methods we propose a vector space approach for inverse rendering of a lambertian convex object with distant light sources in this problem the texture of the object and arbitrary lightings are both to be recovered from multiple images of the object and its 3d model our work is motivated by the observation that all possible images of a lambertian object lie around a low dimensional linear subspace spanned by the first few spherical harmonics the inverse rendering can therefore be formulated as a matrix factorization in which the basis of the subspace is encoded in a spherical harmonic matrix s associated with the object s geometry a necessary and sufficient condition on s for unique factorization is derived with an introduction to a new notion of matrix rank called nonseparable full rank a singular value decomposition based algorithm for exact factorization in the noiseless case is introduced in the presence of noise two algorithms namely alternating and optimization based are proposed to deal with two different types of noise a random sample consensus based algorithm is introduced to reduce the size of the optimization problem which is equal to the number of pixels in each image implementations of the proposed algorithms are done on a real data set

combined affine geometric transformations and spatially dependent radiometric deformations a decoupled linear estimation framework this paper considers the problem of registering two observations of the same object where the observations differ due to a combined effect of an affine geometric transformation and nonuniform illumination changes the problem of deriving new representations of the observations that are both invariant to geometric transformations and linear in the illumination model is analyzed in this framework we present a novel method for linear estimation of illumination changes in an affine invariant manner thus decoupling the original problem into two simpler ones the computational complexity of the method is low as it requires no more than solving a linear set of equations the prior step of illumination estimation is shown to improve the accuracy of state of the art registration techniques by a factor of two

continuity of fuzzy multiplication in the n euclidean space abstract the n euclidean space m i is a fuzzy normed vector space whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers the fuzzy multiplication on the fuzzy real line easily extends to a multiplication on m i we show that under that multiplication m i is a fuzzy normed algebra in consequence fuzzy multiplication is shown to be continuous in the n euclidean topology on m i

the causal perturbation expansion revisited rescaling the interacting dirac sea the causal perturbation expansion defines the dirac sea in the presence of a time dependent external field it yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces after giving a self contained introduction to the ideas and techniques we show that this operator is in general not idempotent we modify the standard construction by a rescaling procedure giving a projector on the generalized negative energy subspace the resulting rescaled causal perturbation expansion uniquely defines the fermionic projector in terms of a series of distributional solutions of the dirac equation the technical core of the paper is to work out the combinatorics of the expansion in detail it is also shown that the fermionic projector with interaction can be obtained from the free projector by a unitary transformation we finally analyze the consequences of the rescaling procedure on the light cone expansion

what are textons textons refer to fundamental micro structures in natural images and videos and are considered as the atoms of pre attentive human visual perception julesz 1981 unfortunately the word texton remains a vague concept in the literature for lack of a good mathematical model in this article we first present a three level generative image model for learning textons from texture images in this model an image is a superposition of a number of image bases selected from an over complete dictionary including various gabor and laplacian of gaussian functions at various locations scales and orientations these image bases are in turn generated by a smaller number of texton elements selected from a dictionary of textons by analogy to the waveform phoneme word hierarchy in speech the pixel base texton hierarchy presents an increasingly abstract visual description and leads to dimension reduction and variable decoupling by fitting the generative model to observed images we can learn the texton dictionary as parameters of the generative model then the paper proceeds to study the geometric dynamic and photometric structures of the texton representation by further extending the generative model to account for motion and illumination variations 1 for the geometric structures a texton consists of a number of image bases with deformable spatial configurations the geometric structures are learned from static texture images 2 for the dynamic structures the motion of a texton is characterized by a markov chain model in time which sometimes can switch geometric configurations during the movement we call the moving textons as motons the dynamic models are learned using the trajectories of the textons inferred from video sequence 3 for photometric structures a texton represents the set of images of a 3d surface element under varying illuminations and is called a lighton in this paper we adopt an illumination cone representation where a lighton is a texton triplet for a given light source a lighton image is generated as a linear sum of the three texton bases we present a sequence of experiments for learning the geometric dynamic and photometric structures from images and videos and we also present some comparison studies with k mean clustering sparse coding independent component analysis and transformed component analysis we shall discuss how general textons can be learned from generic natural images

light cone expansion of the dirac sea in the presence of chiral and scalar potentials we study the dirac sea in the presence of external chiral and scalar pseudoscalar potentials in preparation a method is developed for calculating the advanced and retarded green s functions in an expansion around the light cone for this we first expand all feynman diagrams and then explicitly sum up the perturbation series the light cone expansion expresses the green s functions as an infinite sum of line integrals over the external potential and its partial derivatives the dirac sea is decomposed into a causal and a noncausal contribution the causal contribution has a light cone expansion which is closely related to the light cone expansion of the green s functions it describes the singular behavior of the dirac sea in terms of nested line integrals along the light cone the noncausal contribution on the other hand is to every order in perturbation theory a smooth function in position space

appearance characterization of linear lambertian objects generalized photometric stereo and illumination invariant face recognition traditional photometric stereo algorithms employ a lambertian reflectance model with a varying albedo field and involve the appearance of only one object in this paper we generalize photometric stereo algorithms to handle all appearances of all objects in a class in particular the human face class by making use of the linear lambertian property a linear lambertian object is one which is linearly spanned by a set of basis objects and has a lambertian surface the linear property leads to a rank constraint and consequently a factorization of an observation matrix that consists of exemplar images of different objects e g faces of different subjects under different unknown illuminations integrability and symmetry constraints are used to fully recover the subspace bases using a novel linearized algorithm that takes the varying albedo field into account the effectiveness of the linear lambertian property is further investigated by using it for the problem of illumination invariant face recognition using just one image attached shadows are incorporated in the model by a careful treatment of the inherent nonlinearity in lambert s law this enables us to extend our algorithm to perform face recognition in the presence of multiple illumination sources experimental results using standard data sets are presented

compact probabilistic metrics on bounded closed intervals of distribution functions a menger space a special type of probabilistic metric spaces is said to be compact if its strong uniformity is compact we construct in a natural way menger t metrics on the set of distribution functions one for each copula t a special type of continuous t norms we show that on each bounded closed interval of distribution functions the strong uniformities of all our spaces are induced by the modified levy metric hence they are compact we establish an alternative description and a number of good properties of our menger spaces which may render them well behaved examples for workers in the field

completeness of the standard fuzzy real line for fns abstract we consider a special class of fuzzy metric neighbourhood spaces where the fuzzy distance is induced by a fuzzy pseudo norm this class is contained in lowen s well known category fns of fuzzy neighbourhood spaces we define the fuzzy limit of a sequence a fuzzy cauchy sequence and fuzzy convergence in fuzzy metric neighbourhood spaces we prove that the set of all nonnegative fuzzy real numbers r i under the n euclidean metric the standard fuzzy real line for fns is complete but that the smallest fuzzy neighbourhood real algebra m i including r i is not complete we also characterize the fuzzy cauchy sequences in m i in terms of their corresponding real sequences of cuts

fuzzy inner product spaces abstract the n euclidean space m i is a fuzzy pseudo normed algebra whose underlying set is the smallest real vector space including all nonnegative fuzzy real numbers we introduce axioms for a fuzzy inner product on a unitary m i module with values in m i we establish the fuzzy schwarz inequality for fuzzy inner products we introduce the cartesian product of fuzzy inner product spaces and investigate the relations between their underlying topologies we conclude the research by supplying two interesting examples of fuzzy inner product spaces

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

segmented linear subspaces for illumination robust face recognition all images of a convex lambertian surface captured with a fixed pose under varying illumination are known to lie in a convex cone in the image space that is called the illumination cone since this cone model is too complex to be built in practice researchers have attempted to approximate it with simpler models in this paper we propose a segmented linear subspace model to approximate the cone our idea of segmentation is based on the fact that the success of low dimensional linear subspace approximations of the illumination cone increases if the directions of the surface normals get close to each other hence we propose to cluster the image pixels according to their surface normal directions and to approximate the cone with a linear subspace for each of these clusters separately we perform statistical performance evaluation experiments to compare our system to other popular systems and demonstrate that the performance increase we obtain is statistically significant

perturbative description of the fermionic projector normalization causality and furry s theorem the causal perturbation expansion of the fermionic projector is performed with a contour integral method different normalization conditions are analyzed it is shown that the corresponding light cone expansions are causal in the sense that they only involve bounded line integrals for the resulting loop diagrams we prove a generalized furry theorem

shadow graphs and 3d texture reconstruction we present methods for recovering surface height fields such as geometric details of 3d textures by incorporating shadow constraints we introduce shadow graphs which give a new graph based representation for shadow constraints it can be shown that the shadow graph alone is sufficient to solve the shape from shadow problem from a dense set of images shadow graphs provide a simpler and more systematic approach to represent and integrate shadow constraints from multiple images to recover height fields from a sparse set of images we propose a method for integrated shadow and shading constraints previous shape from shadow algorithms do not consider shading constraints while shape from shading usually assumes there is no shadow our method is based on collecting a set of images from a fixed viewpoint as a known light source changes its position it first builds a shadow graph from shadow constraints from which an upper bound for each pixel can be derived if the height values of a small number of pixels are initialized correctly finally a constrained optimization procedure is designed to make the results from shape from shading consistent with the height bounds derived from the shadow constraints our technique is demonstrated on both synthetic and real imagery

face recognition from a single training image under arbitrary unknown lighting using spherical harmonics in this paper we propose two novel methods for face recognition under arbitrary unknown lighting by using spherical harmonics illumination representation which require only one training image per subject and no 3d shape information our methods are based on the result which demonstrated that the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace we provide two methods to estimate the spherical harmonic basis images spanning this space from just one image our first method builds the statistical model based on a collection of 2d basis images we demonstrate that by using the learned statistics we can estimate the spherical harmonic basis images from just one image taken under arbitrary illumination conditions if there is no pose variation compared to the first method the second method builds the statistical models directly in 3d spaces by combining the spherical harmonic illumination representation and a 3d morphable model of human faces to recover basis images from images across both poses and illuminations after estimating the basis images we use the same recognition scheme for both methods we recognize the face for which there exists a weighted combination of basis images that is the closest to the test face image we provide a series of experiments that achieve high recognition rates under a wide range of illumination conditions including multiple sources of illumination our methods achieve comparable levels of accuracy with methods that have much more onerous training data requirements comparison of the two methods is also provided

bayesian object localisation in images a bayesian approach to intensity based object localisation is presented that employs a learned probabilistic model of image filter bank output applied via monte carlo methods to escape the inefficiency of exhaustive search an adequate probabilistic account of image data requires intensities both in the foreground i e over the object and in the background to be modelled some previous approaches to object localisation by monte carlo methods have used models which we claim do not fully address the issue of the statistical independence of image intensities it is addressed here by applying to each image a bank of filters whose outputs are approximately statistically independent distributions of the responses of individual filters over foreground and background are learned from training data these distributions are then used to define a joint distribution for the output of the filter bank conditioned on object configuration and this serves as an observation likelihood for use in probabilistic inference about localisation the effectiveness of probabilistic object localisation in image clutter using bayesian localisation is illustrated because it is a monte carlo method it produces not simply a single estimate of object configuration but an entire sample from the posterior distribution for the configuration this makes sequential inference of configuration possible two examples are illustrated here coarse to fine scale inference and propagation of configuration estimates over time in image sequences

a fixed point theorem in probabilistic metric spaces and an application abstract the notion of a c contraction type multivalued mapping is introduced this notion is a generalization of the notion of c contraction introduced by t l hicks univ u novom sadu zb rad prirod mat fak ser mat 13 1983 63 72 a fixed point theorem for c contraction is proved an application on the existence of a random fixed point for random operator f m m where m d is a separable metric space and a m a measure space with a decomposable measure of nsa type is given

a flag representation for finite collections of subspaces of mixed dimensions abstract given a finite set of subspaces of r n perhaps of differing dimensions we describe a flag of vector spaces i e a nested sequence of vector spaces that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation the utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions when the set of subspaces all have the same dimension d the flag mean is related to several commonly used subspace representations for instance the d dimensional subspace in the flag corresponds to the extrinsic manifold mean when the set of subspaces is both well clustered and equidimensional of dimension d then the d dimensional component of the flag provides an approximation to the karcher mean an intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of multiset canonical correlation analysis two examples utilizing the carnegie mellon university pose illumination and expression database cmu pie serve as visual illustrations of the algorithm

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

separation axioms and the fuzzy real lines we show that for each lattice l which is completely distributive and equipped with an order reversing involution the canonical topology of the fuzzy real line l is uniformizable in the sense of hutton furthermore this uniformity induces a pseudometric which induces the canonical topology consequently l satisfies the higher order separation axioms of hutton and sarkar more generally we answer most of the open questions concerning l and its subspaces with respect to each of the separation axiom schemes of hutton and sarkar answer some of these same questions for the stratified fuzzy real line c l and its subspaces and analyze c l and its subspaces with respect to the separation axioms of rodabaugh

on photometric issues in 3d visual recognition from asingle 2d image we describe the problem of recognition under changing illumination conditions and changing viewing positions from a computational and human vision perspective on the computational side we focus on the mathematical problems of creating an equivalence class for images of the same 3d object undergoing certain groups of transformations mostly those due to changing illumination and briefly discuss those due to changing viewing positions the computational treatment culminates in proposing a simple scheme for recognizing via alignment an image of a familiar object taken from a novel viewing position and a novel illumination condition on the human vision aspect the paper is motivated by empirical evidence inspired by mooney images of faces that suggest a relatively high level of visual processing is involved in compensating for photometric sources of variability and furthermore that certain limitations on the admissible representations of image information may exist the psychophysical observations and the computational results that follow agree in several important respects such as the same apparent limitations on image representations

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

decoupled linear estimation of affine geometric deformations and nonlinear intensity transformations of images we consider the problem of registering two observations on an arbitrary object where the two are related by a geometric affine transformation of their coordinate systems and by a nonlinear mapping of their intensities more generally the framework is that of jointly estimating the geometric and radiometric deformations relating two observations on the same object we show that the original high dimensional nonlinear and nonconvex search problem of simultaneously recovering the geometric and radiometric deformations can be represented by an equivalent sequence of two linear systems a solution of this sequence yields an exact explicit and efficient solution to the joint estimation problem

affine invariant features from the trace transform the trace transform is a generalization of the radon transform that allows one to construct image features that are invariant to a chosen group of image transformations in this paper we propose a methodology and appropriate functionals that can be computed from the image function and which can be used to calculate features invariant to the group of affine transforms we demonstrate the usefulness of the constructed image descriptors in retrieving images from an image database and compare it with relevant state of the art object retrieval methods

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

the urysohn lemma for fuzzy neighbourhood spaces abstract the framework fns of lowen s fuzzy neighbourhood spaces is a special environment for the study of fuzzy topology it encompasses its own metrics uniformities and proximities among other interesting constructions this is a fresh and fast growing field of research it promises to develop quickly to a calculus which combines depth with applicability within this framework we establish the urysohn lemma for the mashhour morsi axiom of n normality the formulation employs the mashhour morsi n euclidean space as the suitable fuzzy analogue of the real line the proof makes use of the artico moresco fuzzy proximities the lowen fuzzy uniformities and the mashhour morsi axiom of n complete regularity

particle filter with a mode tracker for visual tracking across illumination changes in this correspondence our goal is to develop a visual tracking algorithm that is able to track moving objects in the presence of illumination variations in the scene and that is robust to occlusions we treat the illumination and motion x y translation and scale parameters as the unknown state sequence the observation is the entire image and the observation model allows for occasional occlusions modeled as outliers the nonlinearity and multimodality of the observation model necessitate the use of a particle filter pf due to the inclusion of illumination parameters the state dimension increases thus making regular pfs impractically expensive we show that the recently proposed approach using a pf with a mode tracker can be used here since even in most occlusion cases the posterior of illumination conditioned on motion and the previous state is unimodal and quite narrow the key idea is to importance sample on the motion states while approximating importance sampling by posterior mode tracking for estimating illumination experiments demonstrate the advantage of the proposed algorithm over existing pf based approaches for various face and vehicle tracking we are also able to detect illumination model changes e g those due to transition from shadow to sunlight or vice versa by using the generalized expected log likelihood statistics and successfully compensate for it without ever loosing track

fuzzy t neighbourhood spaces part 3 t separation axioms we address the problem of identifying a useful set of mutually compatible separation axioms for each category t fns of t neighbourhood spaces it should contain one of t complete regularity that characterizes t uniformizability as indeed we achieve here this seems to necessitate the parameterization of most separation axioms by the triangular norm t with the exception of the two axioms t0 and r0 which have categorical definitions valid in any topological category we introduce t separation axioms in terms of t and fuzzy closure operators only making them meaningful for the whole of fts we then characterize them within t fns in terms of t fuzzy closures of crisp fuzzy subsets and t neighbourhoods we study our t axioms and establish some entailments between them our t complete regularity commits us to investigate a t real line which is a t neighbourhood space induced by a certain probabilistic t metric due to hohle fuzzy sets and systems 1 1978 311

manifold manifold distance and its application to face recognition with image sets in this paper we address the problem of classifying image sets for face recognition where each set contains images belonging to the same subject and typically covering large variations by modeling each image set as a manifold we formulate the problem as the computation of the distance between two manifolds called manifold manifold distance mmd since an image set can come in three pattern levels point subspace and manifold we systematically study the distance among the three levels and formulate them in a general multilevel mmd framework specifically we express a manifold by a collection of local linear models each depicted by a subspace mmd is then converted to integrate the distances between pairs of subspaces from one of the involved manifolds we theoretically and experimentally study several configurations of the ingredients of mmd the proposed method is applied to the task of face recognition with image sets where identification is achieved by seeking the minimum mmd from the probe to the gallery of image sets our experiments demonstrate that as a general set similarity measure mmd consistently outperforms other competing nondiscriminative methods and is also promisingly comparable to the state of the art discriminative methods

face relighting from a single image under arbitrary unknown lighting conditions in this paper we present a new method to modify the appearance of a face image by manipulating the illumination condition when the face geometry and albedo information is unknown this problem is particularly difficult when there is only a single image of the subject available recent research demonstrates that the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace using a spherical harmonic representation moreover morphable models are statistical ensembles of facial properties such as shape and texture in this paper we integrate spherical harmonics into the morphable model framework by proposing a 3d spherical harmonic basis morphable model shbmm the proposed method can represent a face under arbitrary unknown lighting and pose simply by three low dimensional vectors i e shape parameters spherical harmonic basis parameters and illumination coefficients which are called the shbmm parameters however when the image was taken under an extreme lighting condition the approximation error can be large thus making it difficult to recover albedo information in order to address this problem we propose a subregion based framework that uses a markov random field to model the statistical distribution and spatial coherence of face texture which makes our approach not only robust to extreme lighting conditions but also insensitive to partial occlusions the performance of our framework is demonstrated through various experimental results including the improved rates for face recognition under extreme lighting conditions

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

a fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces in this paper we shall generalize a fixed point theorem for multivalued mappings in probabilistic metric spaces proved in j math anal appl 202 1996 433 449 as a corollary a fixed point result in menger spaces s f t where t is a strict t norm is obtained some special cases where t is a member of dombi schweizer sklar and aczel alsina families of t norms are investigated an application in fuzzy metric spaces is presented

groupwise geometric and photometric direct image registration image registration consists in estimating geometric and photometric transformations that align two images as best as possible the direct approach consists in minimizing the discrepancy in the intensity or color of the pixels the inverse compositional algorithm has been recently proposed by baker et al for the direct estimation of groupwise geometric transformations it is efficient in that it performs several computationally expensive calculations at a pre computation phase photometric transformations act on the value of the pixels they account for effects such as lighting change jointly estimating geometric and photometric transformations is thus important for many tasks such as image mosaicing we propose an algorithm to jointly estimate groupwise geometric and photometric transformations while preserving the efficient pre computation based design of the original inverse compositional algorithm it is called the dual inverse compositional algorithm it uses different approximations than the simultaneous inverse compositional algorithm and handles groupwise geometric and global photometric transformations its name stems from the fact that it uses an inverse compositional update rule for both the geometric and the photometric transformations we demonstrate the proposed algorithm and compare it to previous ones on simulated and real data this shows clear improvements in computational efficiency and in terms of convergence

hallucinating optimal high dimensional subspaces abstract linear subspace representations of appearance variation are pervasive in computer vision this paper addresses the problem of robustly matching such subspaces computing the similarity between them when they are used to describe the scope of variations within sets of images of different possibly greatly so scales a naive solution of projecting the low scale subspace into the high scale image space is described first and subsequently shown to be inadequate especially at large scale discrepancies a successful approach is proposed instead it consists of i an interpolated projection of the low scale subspace into the high scale space which is followed by ii a rotation of this initial estimate within the bounds of the imposed downsampling constraint the optimal rotation is found in the closed form which best aligns the high scale reconstruction of the low scale subspace with the reference it is compared to the method is evaluated on the problem of matching sets of i face appearances under varying illumination and ii object appearances under varying viewpoint using two large data sets in comparison to the naive matching the proposed algorithm is shown to greatly increase the separation of between class and within class similarities as well as produce far more meaningful modes of common appearance on which the match score is based