input output stabilization of linear systems on spl zopf a formal framework is set up for the discussion of generalized autoregressive with external input models of the form ay bu where a and b are linear operators with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers different notions of stability are explored and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context various problems inherent in using spl zopf as the time axis are analyzed in this operatorial framework

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

texture segmentation using independent scale component wise riemannian covariance gaussian mixture model in kl measure based multi scale nonlinear structure tensor space this paper proposes a novel texture segmentation approach using independent scale component wise riemannian covariance gaussian mixture model icrgmm in kullback leibler kl measure based multi scale nonlinear structure tensor msnst space we use the independent scale distribution and full covariance structure to replace the covariant scale distribution and 1d variance structure used in our previous research to construct the optimal full covariance structure we define the full covariance on kl euclidean log euclidean and riemannian gradient mappings and compare their performances the comparison experiments demonstrate that the riemannian gradient mapping leads to its optimum properties over other choices when constructing the full covariance to estimate and update the statistical parameters more accurately the component wise expectation maximization for mixtures cem 2 algorithm is proposed instead of the originally used k means algorithm the superiority of the proposed icrgmm has been demonstrated based on texture clustering and graph cuts based texture segmentation using a large number of synthesis texture images and real natural scene textured images and further analyzed in terms of error ratio and modified f measure respectively

worst case input output identification we consider worst case l1 identification of causal linear shift invariant systems from time series many results are given on general aspects of identification algorithm performance existence of optimal algorithms robust convergence and input experiment design the identification methodology studied here is compatible with the modelling requirements of modern robust control design

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

lattice tiling and density conditions for subspace gabor frames abstract the well known density theorem in time frequency analysis establishes the connection between the existence of a gabor frame g g a b e 2 i b m x g x a n m n z d for l 2 r d and the density of the time frequency lattice a z d b z d this is also tightly related to lattice tiling and packing in this paper we investigate the density theorem for gabor systems in l 2 s with s being an a z d periodic subset of r d we characterize the existence of a gabor frame for l 2 s in terms of a condition that involves the haar measure of the group generated by a z d and b t 1 z d this new characterization is used to recover the density theorem and several related known results in the literature additionally we apply this approach to obtain the density theorems for multi windowed and super gabor frames for l 2 s

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

the existence of subspace wavelet sets let h be a reducing subspace of l2 rd that is a closed subspace of l2 rd with the property that f amt l h for any f h m z and l zd where a is a d d expansive matrix it is known that h is a reducing subspace if and only if there exists a measurable subset m of rd such that atm m and f h l2 rd m under some given conditions of m it is known that there exist a dilation subspace wavelet sets iwith respect to h in this paper we prove that this holds in general

single and multiple object tracking using log euclidean riemannian subspace and block division appearance model object appearance modeling is crucial for tracking objects especially in videos captured by nonstationary cameras and for reasoning about occlusions between multiple moving objects based on the log euclidean riemannian metric on symmetric positive definite matrices we propose an incremental log euclidean riemannian subspace learning algorithm in which covariance matrices of image features are mapped into a vector space with the log euclidean riemannian metric based on the subspace learning algorithm we develop a log euclidean block division appearance model which captures both the global and local spatial layout information about object appearances single object tracking and multi object tracking with occlusion reasoning are then achieved by particle filtering based bayesian state inference during tracking incremental updating of the log euclidean block division appearance model captures changes in object appearance for multi object tracking the appearance models of the objects can be updated even in the presence of occlusions experimental results demonstrate that the proposed tracking algorithm obtains more accurate results than six state of the art tracking algorithms

structural identification of muitivariabie systems by the eigenvector method abstract the structural identification of linear muitivariabie systems can be performed on the basis of the predicted percentage of reconstruction errors ppcre associated to a sequence of increasing complexity models this paper extends some previous results describing the properties of the ppcre for canonical models whose parameters are estimated by means of the eigenvector method

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

guaranteed stability with subspace methods abstract we show how stability of models can be guaranteed when using the class of identification algorithms which have become known as subspace methods in many of these methods the a matrix is obtained or can be obtained as the product of a shifted matrix with a pseudo inverse we show that whenever the shifted matrix is formed by introducing one block of zeros in the appropriate position then a stable model results the cost of this is some possibly large distortion of the results but in some applications that is outweighed by the advantage of guaranteed stability

a bilinear differential forms approach to parametric structured state space modelling we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

vector valued weak gabor dual frames on discrete periodic sets the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

pedestrian detection via classification on riemannian manifolds we present a new algorithm to detect pedestrian in still images utilizing covariance matrices as object descriptors since the descriptors do not form a vector space well known machine learning techniques are not well suited to learn the classifiers the space of d dimensional nonsingular covariance matrices can be represented as a connected riemannian manifold the main contribution of the paper is a novel approach for classifying points lying on a connected riemannian manifold using the geometry of the space the algorithm is tested on inria and daimlerchrysler pedestrian datasets where superior detection rates are observed over the previous approaches

gabor super frames with hermite functions we investigate vector valued gabor frames sometimes called gabor superframes based on hermite functions hn let h h0 h1 hn be the vector of the first n 1 hermite functions we give a complete characterization of all lattices lambda subseteq mathbb r 2 such that the gabor system rm e 2 pi i lambda 2 t bf h t lambda 1 lambda lambda 1 lambda 2 in lambda is a frame for l 2 mathbb r mathbb c n 1 as a corollary we obtain sufficient conditions for a single hermite function to generate a gabor frame and a new estimate for the lower frame bound the main tools are growth estimates for the weierstrass function a new type of interpolation problem for entire functions on the bargmann fock space and structural results about vector valued gabor frames

state space modeling of two dimensional vector exponential trajectories we solve two problems in modeling polynomial vector exponential trajectories dependent on two independent variables in the first one we assume that the data generating system has no inputs and we compute a state representation of the most powerful unfalsified model for this data in the second instance we assume that the data generating system is controllable and quarter plane causal and we compute a roesser input state output model we provide procedures for solving these identification problems both based on the factorization of constant matrices directly constructed from the data from which state trajectories can be computed

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

construction and reconstruction of tight framelets and wavelets via matrix mask functions abstract the paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis gmra we show the existence of a scaling vector of a gmra such that its first component exhausts the spectrum of the core space near the origin the corresponding low pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector we also prove a generalization of the unitary extension principle for an infinite number of generators this results in the construction scheme for tight framelets using low pass and high pass matrix masks generalizing the classical mra constructions we prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets as an illustration we exhibit a pathwise connected class of non msf non mra wavelets sharing the same wavelet dimension function

efficient clustering on riemannian manifolds reformulating computer vision problems over riemannian manifolds has demonstrated superior performance in various computer vision applications this is because visual data often forms a special structure lying on a lower dimensional space embedded in a higher dimensional space however since these manifolds belong to non euclidean topological spaces exploiting their structures is computationally expensive especially when one considers the clustering analysis of massive amounts of data to this end we propose an efficient framework to address the clustering problem on riemannian manifolds this framework implements random projections for manifold points via kernel space which can preserve the geometric structure of the original space but is computationally efficient here we introduce three methods that follow our framework we then validate our framework on several computer vision applications by comparing against popular clustering methods on riemannian manifolds experimental results demonstrate that our framework maintains the performance of the clustering whilst massively reducing computational complexity by over two orders of magnitude in some cases highlightswe propose a kernelised random projection framework for clustering manifold points we present three projection methods conforming to our proposed framework we contrast our proposal to clustering methods on manifolds in various vision tasks we show the proposal obtain significant speed up whilst maintaining the performance we analyse the parameters contributing to the speed up

from time series to linear system part ii exact modelling abstract in the second part of this paper the problem of finding an exact model for a q dimensional infinite time series is considered first a mathematical vocabulary for discussing exact modelling is developed it is then shown how the results of part i guarantee the existence of a most powerful ar model for an observed time series two algorithms for obtaining such an ar model are subsequently derived one of these algorithms gives a shortest lag input output model the problem of obtaining a minimal state space realization of the observed time series is also considered in order to do that realization theory based on the truncated behaviour is developed as an extensive example the classical situation with impulse response measurements is discussed

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

face recognition on large scale video in the wild with hybrid euclidean and riemannian metric learning face recognition on large scale video in the wild is becoming increasingly important due to the ubiquity of video data captured by surveillance cameras handheld devices internet uploads and other sources by treating each video as one image set set based methods recently have made great success in the field of video based face recognition in the wild world videos often contain extremely complex data variations and thus pose a big challenge of set modeling for set based methods in this paper we propose a novel hybrid euclidean and riemannian metric learning herml method to fuse multiple statistics of image set specifically we represent each image set simultaneously by mean covariance matrix and gaussian distribution which generally complement each other in the aspect of set modeling however it is not trivial to fuse them since mean covariance matrix and gaussian model typically lie in multiple heterogeneous spaces equipped with euclidean or riemannian metric therefore we first implicitly map the original statistics into high dimensional hilbert spaces by exploiting euclidean and riemannian kernels with a logdet divergence based objective function the hybrid kernels are then fused by our hybrid metric learning framework which can efficiently perform the fusing procedure on large scale videos the proposed method is evaluated on four public and challenging large scale video face datasets extensive experimental results demonstrate that our method has a clear superiority over the state of the art set based methods for large scale video based face recognition highlightsrepresent image set by mean covariance and gaussian for discriminant information heterogeneous euclidean and riemannian kernels are exploited and fused clearly clear superiority over state of the art set based methods is achieved in testing

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

on the choice of inputs in identification for robust control the thesis that noisy identification has close ties to the study of the singular value decomposition of perturbed matrices is investigated in particular by assuming an upper bound on the norm of the perturbation one can obtain a convex parametrization of an uncertain family of systems which contains the system generating the data in this approach the second smallest singular value s of an appropriately defined data matrix becomes a quantity of importance as it provides an upper bound for the size of the uncertain family this yields a new tool leading to the design of input functions which are optimal or persistently exciting from the point of view of identification for robust control

continuity of latent variable models the authors study the continuity of the behavior of dynamical systems as a function of the parameters in their behavioral equations a system is defined in terms of its behavior and continuity requires that this behavior be continuous in the limit the problem is demonstrated by means of an example involving an rlc circuit whose port behavior exhibits a surprising discontinuity as a function of the numerical values of the elements in the circuit the main result states that a system described by means of difference equations involving manifest external and latent internal variables will have a continuous behavior in the limit if the limit system is observable

the subspace nevanlinna interpolation problem and the most powerful unfalsified model a generalization of the tangential nevanlinna interpolation problem will be studied from a behavioral point of view necessary and sufficient conditions for its solvability and a characterization of all its solutions are derived these results are obtained by associating to the interpolation data a behavior that enjoys a special structure

characterizing humans on riemannian manifolds in surveillance applications head and body orientation of people is of primary importance for assessing many behavioral traits unfortunately in this context people are often encoded by a few noisy pixels so that their characterization is difficult we face this issue proposing a computational framework which is based on an expressive descriptor the covariance of features covariances have been employed for pedestrian detection purposes actually a binary classification problem on riemannian manifolds in this paper we show how to extend to the multiclassification case presenting a novel descriptor named weighted array of covariances especially suited for dealing with tiny image representations the extension requires a novel differential geometry approach in which covariances are projected on a unique tangent space where standard machine learning techniques can be applied in particular we adopt the campbell baker hausdorff expansion as a means to approximate on the tangent space the genuine geodesic distances on the manifold in a very efficient way we test our methodology on multiple benchmark datasets and also propose new testing sets getting convincing results in all the cases

hessian eigenmaps locally linear embedding techniques for high dimensional data we describe a method for recovering the underlying parametrization of scattered data m i lying on a manifold m embedded in high dimensional euclidean space the method hessian based locally linear embedding derives from a conceptual framework of local isometry in which the manifold m viewed as a riemannian submanifold of the ambient euclidean space r n is locally isometric to an open connected subset of euclidean space r d because does not have to be convex this framework is able to handle a significantly wider class of situations than the original isomap algorithm the theoretical framework revolves around a quadratic form h f m h f m f dm defined on functions f m r here hf denotes the hessian of f and h f averages the frobenius norm of the hessian over m to define the hessian we use orthogonal coordinates on the tangent planes of m the key observation is that if m truly is locally isometric to an open connected subset of r d then h f has a d 1 dimensional null space consisting of the constant functions and a d dimensional space of functions spanned by the original isometric coordinates hence the isometric coordinates can be recovered up to a linear isometry our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the laplacian eigenmaps framework where we substitute a quadratic form based on the hessian in place of one based on the laplacian

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

totally dissipative systems in a totally dissipative behavior all non trivial trajectories dissipate energy a characterization of such behaviors is given in terms of properties of the one and two polynomial matrices associated with the supply rate and with their kernel and image representation

a general hankel norm approximation scheme for linear recursive filtering let x t and y t be stochastic processes which are weakly stationary and stationarily correlated we consider the problem of finding an approximate recursive low dimensional filter of x t based on the observation of the past of y t using hankel norm techniques several estimation problems have been investigated in the past using these techniques we present here a general framework which includes many of these approaches as special cases we also discuss some new applications the approximate filter so constructed allows for an a priori bound on the estimation error

an algorithm for constructing a minimal partial realization in the multivariable case it is known that the berlekamp massey algorithm from information theory can be used to compute scalar minimal partial realizations recently it has been interpreted in terms of exact modeling of behaviors in this paper we extend these results to the multivariable case we put the behavioral theory of exact modeling to work in deriving an extension of the berlekamp massey algorithm for computing multivariable minimal partial realizations

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

rational time frequency super gabor frames and their duals abstract this paper addresses general super gabor systems with rational time frequency product lattices a zak transform matrix method is developed for such super gabor systems we characterize complete super gabor systems and super gabor frames given a super gabor frame we obtain an explicit expression of its canonical dual and a parametrization of all its super gabor duals and prove that the canonical dual is the norm minimal one among all super gabor duals we also prove that a super gabor frame has a unique super gabor dual if and only if it is a riesz basis

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

system theory for system identification abstract prediction and control problems require for their solution mathematical models in the form of dynamic systems in system theory the approach to this problem is to investigate realization problems for system identification novel in this paper is the emphasis on the realization problem for observation based realizations and on the realization problem for control a major aspect of these realization problems is computation and complexity algebraic approaches to realization problems may lead to the distinction of subclasses of systems with required finiteness properties to decompositions and to abstractions for hierarchical systems

true bargmann transforms for rank one automorphic functions associated with landau levels for a given discrete subgroup of c and given real number 0 we study the spectral properties of the magnetic laplacian operator acting on the hilbert space l 2 c of l2 automorphic functions see below for notations we show that its spectrum is reduced to the eigenvalues m m 0 1 we also give a concrete description of each eigenspace in terms of the hermite polynomials this description will be used to characterize the range of true bargmann transform of l2 periodic functions on the real line r

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

classification of symmetric positive definite matrices based on bilinear isometric riemannian embedding abstract because covariance features with the form of symmetric positive definite matrices lie on riemannian manifold classification on riemannian manifold could possess high performance in many applications unfortunately the applicability of classification methods developed on riemannian manifold is limited by their huge computational complexities particularly with the feature data on high dimensional riemannian manifold to alleviate the problem of computational cost in this paper a simple yet efficient dimensionality reduction algorithm bilinear isometric riemannian embedding is derived to construct a low dimensional embedding from high dimensional riemannian manifold to this end we model the bilinear isometric mapping to identify a low dimensional embedding that maximizes the preservation of riemannian geodesic distance a supervised classification method embedding discriminant analysis is then proposed based on the low dimensional embedding experimental results on image and electroencephalogram reveal that the proposed algorithms can efficiently extract the distance preserving embedding and obtain higher classification performance

reprint of a bilinear differential forms approach to parametric structured state space modelling abstract we use one variable loewner techniques to compute polynomial parametric models for mimo systems from vector exponential data gathered at various points in the parameter space instrumental in our approach are the connections between vector exponential modelling via bilinear differential forms and the loewner framework

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

vector valued gabor frames associated with periodic subsets of the real line the notion of vector valued frame also called superframe was first introduced by balan in the context of multiplexing it has significant applications in mobile communication satellite communication and computer area network for vector valued gabor analysis existent literatures mostly focus on l 2 r c l instead of its subspace let a 0 and s be an a z periodic measurable set in r i e s a z s this paper addresses gabor frames in l 2 s c l with rational time frequency product they can model vector valued signals to appear periodically but intermittently and the projections of gabor frames in l 2 r c l onto l 2 s c l cannot cover all gabor frames in l 2 s c l if s r by introducing a suitable zak transform matrix we characterize completeness and frame condition of gabor systems obtain a necessary and sufficient condition on gabor duals of type i resp ii for a general gabor frame and establish a parametrization expression of gabor duals of type i resp ii all our conclusions are closely related to corresponding zak transform matrices this allows us to easily realize these conclusions by designing the corresponding matrix valued functions an example theorem is also presented to illustrate the efficiency of our method

jensen bregman logdet divergence with application to efficient similarity search for covariance matrices covariance matrices have found success in several computer vision applications including activity recognition visual surveillance and diffusion tensor imaging this is because they provide an easy platform for fusing multiple features compactly an important task in all of these applications is to compare two covariance matrices using a dis similarity function for which the common choice is the riemannian metric on the manifold inhabited by these matrices as this riemannian manifold is not flat the dissimilarities should take into account the curvature of the manifold as a result such distance computations tend to slow down especially when the matrix dimensions are large or gradients are required further suitability of the metric to enable efficient nearest neighbor retrieval is an important requirement in the contemporary times of big data analytics to alleviate these difficulties this paper proposes a novel dissimilarity measure for covariances the jensen bregman logdet divergence jbld this divergence enjoys several desirable theoretical properties and at the same time is computationally less demanding compared to standard measures utilizing the fact that the square root of jbld is a metric we address the problem of efficient nearest neighbor retrieval on large covariance datasets via a metric tree data structure to this end we propose a k means clustering algorithm on jbld we demonstrate the superior performance of jbld on covariance datasets from several computer vision applications

a nonparametric riemannian framework on tensor field with application to foreground segmentation background modeling on tensor field has recently been proposed for foreground detection tasks taking into account the riemannian structure of the tensor manifold recent research has focused on developing parametric methods on the tensor domain e g mixture of gaussians gmm however in some scenarios simple parametric models do not accurately explain the physical processes kernel density estimators kdes have been successful to model on euclidean sample spaces the nonparametric nature of complex time varying and non static backgrounds founded on a mathematically rigorous kde paradigm on general riemannian manifolds recently proposed in the literature we define a kde specifically to operate on the tensor manifold in order to nonparametrically reformulate the existing tensor based algorithms we present a mathematically sound framework for nonparametric modeling on tensor field to foreground detection we endow the tensor manifold with two well founded riemannian metrics i e affine invariant and log euclidean theoretical aspects are presented and the metrics are compared experimentally by inducing a space with a null curvature the log euclidean metric considerably simplifies the scheme from a practical point of view while maintaining the mathematical soundness and the excellent segmentation performance theoretic analysis and experimental results demonstrate the promise and effectiveness of this framework

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

diffusion maps and coarse graining a unified framework for dimensionality reduction graph partitioning and data set parameterization we provide evidence that nonlinear dimensionality reduction clustering and data set parameterization can be solved within one and the same framework the main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise our construction which is based on a markov random walk on the data offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry we show that clustering in embedding spaces is equivalent to compressing operators the objective of data partitioning and clustering is to coarse grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy we show that the quantization distortion in diffusion space bounds the error of compression of the operator thus giving a rigorous justification for k means clustering in diffusion space and a precise measure of the performance of general clustering algorithms

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

the duals of gabor frames on discrete periodic sets due to its good potential for digital signal processing analysis the discrete version of gabor analysis has interested many mathematicians in this paper we focus on gabor analysis on discrete periodic sets which can model a signal to appear periodically but intermittently for a gabor frame on a discrete periodic set three types of dual frames with gabor structure are introduced and investigated in terms of discrete zak transform given a gabor frame on a discrete periodic set a characterization of a gabor system being separately its dual of types i iii is obtained a method to construct these three types of duals is presented an explicit expression of its canonical dual is established the uniqueness of these three types of duals is characterized some examples are also provided to illustrate the generality of the theory

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

a linear cost algorithm to compute the discrete gabor transform in this paper we propose an alternative efficient method to calculate the gabor coefficients of a signal given a synthesis window with a support of size much lesser than the length of the signal the algorithm uses the canonical dual of the window which does not need to be calculated beforehand and achieves a computational cost that is linear with the signal length in both analysis and synthesis this is done by exploiting the block structure of the matrices and using an ad hoc cholesky decomposition of the gabor frame matrix

multiwindow real valued discrete gabor transform and its fast algorithms based on the biorthogonal analysis approach a multiwindow real valued discrete gabor transform m rdgt for periodic sequences is presented to efficiently analyze the dynamic time frequency content of a signal containing components with multiple and or time varying frequencies the m rdgt offers a computationally efficient implementation as well as a real valued formulation of the multiwindow complex valued discrete gabor transform m cdgt the completeness condition of the m rdgt is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows the m rdgt can utilize the fast discrete hartley transform algorithms for fast computation and has a simple relationship with the m cdgt such that its coefficients can be directly computed from the m rdgt coefficients therefore the m rdgt offers an efficient method to compute the m cdgt since the analyzed sequence analysis and synthesis windows in the existing m cdgt must have an equal period if the period of a sequence is very long solving its windows requires a huge amount of computation and memory and could lead to numerical instability to overcome this problem a modified m rdgt for long periodic or even infinite sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long periodic or even in finite sequence finally the multirate based parallel implementation of the m rdgt is presented which has shown to be effective and fast for time frequency analysis

on constructing a shortest linear recurrence relation it has been shown in the literature that a formulation of the minimal partial realization problem in terms of exact modeling of a behavior lends itself to an iterative polynomial solution for the scalar case we explicitly present such a solution in full detail unlike classical solution methods based on hankel matrices the algorithm is constructive it iteratively constructs a partial realization of minimal mcmillian degree the algorithm is known in information theory as the berlekamp massey algorithm and is used for constructing a shortest linear recurrence relation for a finite sequence of numbers

on the takagi interpolation problem in this paper we approach the interpolation problem of takagi from the perspective of time series modeling the framework of exact identification and the notion of mpum are used in order to generalize some results on the nevanlinna interpolation problem these results are of independent interest and concern the relation between the roots of the determinant of the denominator of an unfalsified model and the signature of the pick matrix of the data

convergence of an iterative time variant filtering based on discrete gabor transform an iterative time variant filtering based on the discrete gabor transform dgt has been previously proposed by the authors in this article we present a proof of the convergence of the iterative algorithm under a sufficient condition on the analysis and synthesis window functions of the dgt in the meantime we show that the iterative algorithm refines the least squares solution

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

discrete multi gabor expansions a discrete multi gabor expansion dmge is developed to meet the requirements of localized and refined time frequency tf representation of signals the dmge uses multiple windows and their translations and complex modulations as synthesis or analysis waveforms it includes and generalizes the metaplectic translation modulation and dilation representations which are useful in signal analysis uniform nonuniform and proportional time sampling schemes are analyzed the fundamental features and the importance of the dmge are discussed we focus on the construction of dmge and deriving fast algorithms for the computation of related multi analysis sequences with matrix algebra the algorithms derived apply to both multi gabor expansions and uni window gabor expansions another useful feature of the dmge lies in the fact that the multi gabor transform can be realized in a parallel fft based implementation structure examples of dmge and their applications to tf analysis are also discussed

on the problem of stable rational interpolation abstract aspects of the stability issue in connection with rational interpolation are investigated in particular it is shown that unconstrained interpolation of a given set of points together with an associated mirror image set of points yields a one parameter family of stable interpolating functions as an application it is also shown that if a certain number of markov parameters are given by appropriate choice of the moments stable realizations are obtained