wavelet domain blur invariants for image analysis radiometric degradation is a common problem in the image acquisition part of many applications there is much research carried out in an effort to deblur such images however it has been proven that it is not always necessary to go through a burdensome process of deblurring to tackle this problem different blur invariant descriptors have been proposed so far which are either in the spatial domain or based on the properties available in the fourier domain in this paper wavelet domain blur invariants are proposed for the first time for discrete 2 d signals these descriptors which are invariant to centrally symmetric blurs inherit the advantages that this domain provides it is also proven that the spatial domain blur invariants are a special version of the proposed invariants the performance of these invariants will be demonstrated through experiments

central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

hypothesis testing for automated community detection in networks summary community detection in networks is a key exploratory tool with applications in a diverse set of areas ranging from finding communities in social and biological networks to identifying link farms in the world wide web the problem of finding communities or clusters in a network has received much attention from statistics physics and computer science however most clustering algorithms assume knowledge of the number of clusters k we propose to determine k automatically in a graph generated from a stochastic block model by using a hypothesis test of independent interest our main contribution is twofold first we theoretically establish the limiting distribution of the principal eigenvalue of the suitably centred and scaled adjacency matrix and use that distribution for our test of the hypothesis that a random graph is of erd s renyi noise type secondly we use this test to design a recursive bipartitioning algorithm which naturally uncovers nested community structure using simulations and quantifiable classification tasks on real world networks with ground truth we show that our algorithm outperforms state of the art methods

registration of images with n fold dihedral blur in this paper we extend our recent registration method designed specifically for registering blurred images the original method works for unknown blurs assuming the blurring point spread function psf exhibits an n fold rotational symmetry here we also generalize the theory to the case of dihedrally symmetric blurs which are produced by the psfs having both rotational and axial symmetries such kind of blurs are often found in unfocused images acquired by digital cameras as in out of focus shots the psf typically mimics the shape of the shutter aperture this makes our registration algorithm particularly well suited in applications where blurred image registration must be used as a preprocess step of an image fusion algorithm and where common registration methods fail due to the amount of blur we demonstrate that the proposed method leads to an improvement of the registration performance and we show its applicability to real images by providing successful examples of blurred image registration followed by depth of field extension and multichannel blind deconvolution

a blur robust descriptor with applications to face recognition understanding the effect of blur is an important problem in unconstrained visual analysis we address this problem in the context of image based recognition by a fusion of image formation models and differential geometric tools first we discuss the space spanned by blurred versions of an image and then under certain assumptions provide a differential geometric analysis of that space more specifically we create a subspace resulting from convolution of an image with a complete set of orthonormal basis functions of a prespecified maximum size that can represent an arbitrary blur kernel within that size and show that the corresponding subspaces created from a clean image and its blurred versions are equal under the ideal case of zero noise and some assumptions on the properties of blur kernels we then study the practical utility of this subspace representation for the problem of direct recognition of blurred faces by viewing the subspaces as points on the grassmann manifold and present methods to perform recognition for cases where the blur is both homogenous and spatially varying we empirically analyze the effect of noise as well as the presence of other facial variations between the gallery and probe images and provide comparisons with existing approaches on standard data sets

criteria for generalized invex monotonicities abstract in this paper under appropriate conditions we establish that i if the gradient of a function is strictly pseudo monotone then the function is strictly pseudo invex ii if the gradient of a function is quasi monotone then the function is quasi invex and iii if the gradient of a function is strong pseudo monotone then the function is strong pseudo invex

projection operators and moment invariants to image blurring in this paper we introduce a new theory of blur invariants blur invariants are image features which preserve their values if the image is convolved by a point spread function psf of a certain class we present the invariants to convolution with an arbitrary n fold symmetric psf both in fourier and image domain we introduce a notion of a primordial image as a canonical form of all blur equivalent images it is defined in spectral domain by means of projection operators we prove that the moments of the primordial image are invariant to blur and we derive recursive formulae for their direct computation without actually constructing the primordial image we further prove they form a complete set of invariants and show how to extent their invariance also to translation rotation and scaling we illustrate by simulated and real data experiments their invariance and recognition power potential applications of this method are wherever one wants to recognize objects on blurred images

moment invariants under similarity transformation this paper presents a method of deriving moment invariants under similarity transformation the invariants sets are algebraically independent and complete the method can be adopted for arbitrary dimension and order this paper also presents some explicit formulas for computing invariant sets dimension which reveals how the number of invariants increases with moments order and dimension elevating we present a method for constructing moment invariant under similarity transformation our method can be used for deriving moment invariants of arbitrary dimension and order our invariants are algebraically independent and complete we present some explicit formulas for computing invariant sets dimensions

criteria for quasi convexity and pseudo convexity relationships and comparisons a first order criterion for pseudo convexity and second order criteria for quasi convexity and pseudo convexity are given for twice differentiable functions on open convex sets the relationships between these second order criteria and other known criteria are also analysed finally the numbers of operations required to verify these criteria are calculated and compared

projective invariants of co moments of 2d images functions of moments of 2d images that are invariant under some changes are important in image analysis and pattern recognition one of the most basic changes to a 2d image is geometric change two images of the same plane taken from different viewpoints are related by a projective transformation unfortunately it is well known that geometric moment invariants for projective transformations do not exist in general yet if we generalize the standard definition of the geometric moments and utilize some additional information from the images certain type of projective invariants of 2d images can be derived this paper first defines co moment as a moment like function of image that contains two reference points then a set of functions of co moments that is invariant under general projective transformations is derived the invariants are simple and in explicit form experimental results validated the mathematical derivations

invex sets and preinvex functions on riemannian manifolds the concept of a geodesic invex subset of a riemannian manifold is introduced geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined the relation between geodesic invexity and preinvexity of functions on manifolds is studied using proximal subdifferential certain results concerning extremum points of a non smooth geodesic preinvex function on a geodesic invex set are obtained the main value inequality and the mean value theorem in invexity analysis are extended to cartan hadamard manifolds

eigenvalues outside the bulk of inhomogeneous erdős renyi random graphs in this article an inhomogeneous erd s renyi random graph on 1 ldots n is considered where an edge is placed between vertices i and j with probability varepsilon n f i n j n for i le j the choice being made independently for each pair the integral operator i f associated with the bounded function f is assumed to be symmetric non negative definite and of finite rank k we study the edge of the spectrum of the adjacency matrix of such an inhomogeneous erd s renyi random graph under the assumption that n varepsilon n rightarrow infty sufficiently fast although the bulk of the spectrum of the adjacency matrix scaled by sqrt n varepsilon n is compactly supported the kth largest eigenvalue goes to infinity it turns out that the largest eigenvalue after appropriate scaling and centering converges to a gaussian law if the largest eigenvalue of i f has multiplicity 1 if i f has k distinct non zero eigenvalues then the joint distribution of the k largest eigenvalues converge jointly to a multivariate gaussian law the first order behaviour of the eigenvectors is derived as a byproduct of the above results the results complement the homogeneous case derived by 18

the revised fundamental theorem of moment invariants a revised fundamental theorem of moment invariants for pattern recognition which corrects the fundamental theorem proposed by m k hu 1962 is presented the correction affects neither similitude scale nor rotation invariants derived using the original theorem but it does affect features invariant to general linear transformations four of the latter invariants were presented originally by hu these are revised to take the correction to the fundamental theorem into account furthermore these four invariants are combined to yield three new invariants which are additionally invariant to changes in the illumination of an image

subordination for the sum of two random matrices this paper is about the relation of random matrix theory and the subordination phenomenon in complex analysis we find that the resolvent of the sum of two random matrices is approximately subordinated to the resolvents of the original matrices we estimate the error terms in this relation and in the subordination relation for the traces of the resolvents this allows us to prove a local limit law for eigenvalues and a delocalization result for eigenvectors of the sum of two random matrices in addition we use subordination to determine the limit of the largest eigenvalue for the rank one deformations of unitary invariant random matrices

pattern recognition by affine moment invariants abstract the paper deals with moment invariants which are invariant under general affine transformation and may be used for recognition of affine deformed objects our approach is based on the theory of algebraic invariants the invariants from second and third order moments are derived and shown to be complete the paper is a significant extension and generalization of recent works several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described

determination of blur and affine combined invariants by normalization the determination of invariant characteristics is an important problem in pattern recognition in many situations images to be processed are usually subjected to geometric distortion and or blur degradation in this paper we introduce an approach to derive blur and affine combined invariants bai firstly we normalize the image to a standard form by using blur invariant moments as normalization constraints then we construct the blur and affine combined invariants at the standard form using the method proposed in this paper a set of blur and affine combined invariant features can be obtained easily and effectively several experimental results are presented to illustrate the performance of the invariants for simultaneously affine deformed and blur degraded images

a robust tracking system for low frame rate video tracking in low frame rate lfr videos is one of the most important problems in the tracking literature most existing approaches treat lfr video tracking as an abrupt motion tracking problem however in lfr video tracking applications lfr not only causes abrupt motions but also large appearance changes of objects because the objects poses and the illumination may undergo large changes from one frame to the next this adds extra difficulties to lfr video tracking in this paper we propose a robust and general tracking system for lfr videos the tracking system consists of four major parts dominant color spatial based object representation bin ratio based similarity measure annealed particle swarm optimization pso based searching and an integral image based parameter calculation the first two parts are combined to provide a good solution to the appearance changes and the abrupt motion is effectively captured by the annealed pso based searching moreover an integral image of model parameters is constructed which provides a look up table for parameters calculation this greatly reduces the computational load experimental results demonstrate that the proposed tracking system can effectively tackle the difficulties caused by lfr

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

normalization and shape recognition of three dimensional objects by 3d moments abstract in this paper we are primarily concerned with the recognition of three dimensional 3d objects on the basis of the geometry of their physical surface the 3d moments are defined on the object s surface with two main aims 1 to derive a normalized version irrespective of position size and orientation for each imaged object and 2 to establish global descriptors extracted from normalized shapes in order to make up a representative feature vector to be used in recognition tasks the source data are multiple orthographic views of 3d objects with known viewpoint specifications a volume intersection procedure is used in the recovery of 3d object surface from two dimensional 2d data the problem of normalization to achieve recognition of 3d objects is dealt with and a heuristic solution is proposed to lessen the inherent ambiguity of the principal axes method for object orientation experimental results are described with ten object categories showing excellent percentage classification success rates by using only a small number of normalized moments as elements of the feature vector

true multi image alignment and its application to mosaicing and lens distortion correction multiple images of a scene are related through 2d 3d view transformations and linear and nonlinear camera transformations we present an algorithm for true multi image alignment that does not rely on the measurements of a reference image being distortion free the algorithm is developed to specifically align and mosaic images using parametric transformations in the presence of lens distortion when lens distortion is present none of the images can be assumed to be ideal in our formulation all the images are modeled as intensity measurements represented in their respective coordinate systems each of which is related to an ideal coordinate system through an interior camera transformation and an exterior view transformation the goal of the accompanying algorithm is to compute an image in the ideal coordinate system while solving for the transformations that relate the ideal system with each of the data images key advantages of the technique presented in this paper are i no reliance on one distortion free image ii ability to register images and compute coordinate transformations even when the multiple images are of an extended scene with no overlap between the first and last frame of the sequence and iii ability to handle linear and nonlinear transformations within the same framework results of applying the algorithm are presented for the correction of lens distortion and creation of video mosaics

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs diluted regime we study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity we prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a gaussian random variable with zero mean and variance which coincides with non gaussian part of the wigner ensemble variance

invariant image recognition by zernike moments the problem of rotation scale and translation invariant recognition of images is discussed a set of rotation invariant features are introduced they are the magnitudes of a set of orthogonal complex moments of the image known as zernike moments scale and translation invariance are obtained by first normalizing the image with respect to these parameters using its regular geometrical moments a systematic reconstruction based method for deciding the highest order zernike moments required in a classification problem is developed the quality of the reconstructed image is examined through its comparison to the original one the orthogonality property of the zernike moments which simplifies the process of image reconstruction make the suggest feature selection approach practical features of each order can also be weighted according to their contribution to the reconstruction process the superiority of zernike moment features over regular moments and moment invariants was experimentally verified

n dimensional moment invariants and conceptual mathematical theory of recognition n dimensional solids the proof of the generalized fundamental theorem of moment invariants gftmi is presented for n dimensional pattern recognition on the basis of gftmi the moment invariants of affine transformation and subgroups of affine transformation are constructed using these invariants the conceptual mathematical theory of recognition of geometric figures solids and their n dimensional generalizations is worked out by means of this theory it is possible for the first time to analyze scenes consisting not only of polygons and polyhedra but also scenes consisting of geometric figures and solids with curved contours and surfaces respectively in general it is the author s opinion that this theory is a useful step toward the essential development of robot vision and toward creating machine intelligence to make machines able to think by means of geometric concepts of different generalities and dimensions and by associations of these concepts

spectral rigidity for addition of random matrices at the regular edge abstract we consider the sum of two large hermitian matrices a and b with a haar unitary conjugation bringing them into a general relative position we prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of a and b as the dimension of the matrix increases this implies optimal rigidity of the eigenvalues and optimal rate of convergence in voiculescu s theorem our previous works 4 5 established these results in the bulk spectrum the current paper completely settles the problem at the spectral edges provided they have the typical square root behavior the key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge our results also hold if the haar unitary matrix is replaced by the haar orthogonal matrix

combined blur and affine moment invariants and their use in pattern recognition abstract the paper is devoted to the recognition of objects and patterns deformed by imaging geometry as well as by unknown blurring we introduce a new class of features invariant simultaneously to blurring with a centrosymmetric psf and to affine transformation as we prove in the paper they can be constructed by combining affine moment invariants and blur invariants derived earlier combined invariants allow to recognize objects in the degraded scene without any restoration

invariant monotone vector fields on riemannian manifolds abstract various concepts of invariant monotone vector fields on riemannian manifolds are introduced some examples of invariant monotone vector fields are given several notions of invexities for functions on riemannian manifolds are defined and their relations with invariant monotone vector fields are studied

recognitive aspects of moment invariants moment invariants are evaluated as a feature space for pattern recognition in terms of discrimination power and noise tolerance the notion of complex moments is introduced as a simple and straightforward way to derive moment invariants through this relation properties of complex moments are used to characterize moment invariants aspects of information loss suppression and redundancy encountered in moment invariants are investigated and significant results are derived the behavior of moment invariants in the presence of additive noise is also described

pattern recognition with moment invariants a comparative study and new results abstract moment invariants have been proposed as pattern sensitive features in classification and recognition applications in this paper the authors present a comprehensive study of the effectiveness of different moment invariants in pattern recognition applications by considering two sets of data handwritten numerals and aircrafts the authors also present a detailed study of zernike and pseudo zernike moment invariants including a new procedure for deriving the moment invariants in addition the authors introduce a new normalization scheme that reduces the large dynamic range of these invariants as well as implicit redundancies in these invariants based on a comprehensive study with both handwritten numerals and aircraft data the authors show that the new method of deriving zernike moment invariants along with the new normalization scheme yield the best overall performance even when the data are degraded by additive noise

3d rotation invariants by complex moments a generalization of the complex moments from 2d to 3d is described group representation theory is used to construct 3d rotation invariants from them the algorithm for automatic generating of the invariants of higher orders is proposed an algorithm for automatic generation of higher order invariants is proposed the linearly dependent invariants are eliminated the invariants are experimentally tested on 3d graphical models and also on real volumetric data

generalized monotonicity and convexity of non differentiable functions the relationships between strict strong convexity of non differentiable functions and strict strong monotonicity of set valued mappings and strict strong sharp pseudo convexity of non differentiable functions and strict strong pseudo monotonicity of set valued mappings as well as quasi convexity of non differentiable functions and quasi monotonicity of set valued mappings are studied in this paper in addition the relations between generalized convexity of non differentiable functions and generalized co coerciveness of set valued mappings are also analyzed

a numerical recipe for accurate image reconstruction from discrete orthogonal moments recursive procedures used for sequential calculations of polynomial basis coefficients in discrete orthogonal moments produce unreliable results for high moment orders as a result of error accumulation this paper demonstrates accurate reconstruction of arbitrary size images using full order orders as large as the image size tchebichef and krawtchouk moments by calculating polynomial coefficients directly from their definition formulas in hypergeometric functions and by creating lookup tables of these coefficients off line an arbitrary precision calculator is used to achieve greater numerical range and precision than is possible with software using standard 64 bit ieee floating point arithmetic this reconstruction scheme is content and noise independent

combined invariants to similarity transformation and to blur using orthogonal zernike moments the derivation of moment invariants has been extensively investigated in the past decades in this paper we construct a set of invariants derived from zernike moments which is simultaneously invariant to similarity transformation and to convolution with circularly symmetric point spread function psf two main contributions are provided the theoretical framework for deriving the zernike moments of a blurred image and the way to construct the combined geometric blur invariants the performance of the proposed descriptors is evaluated with various psfs and similarity transformations the comparison of the proposed method with the existing ones is also provided in terms of pattern recognition accuracy template matching and robustness to noise experimental results show that the proposed descriptors perform on the overall better

recognition of blurred images by the method of moments the article is devoted to the feature based recognition of blurred images acquired by a linear shift invariant imaging system against an image database the proposed approach consists of describing images by features that are invariant with respect to blur and recognizing images in the feature space the psf identification and image restoration are not required a set of symmetric blur invariants based on image moments is introduced a numerical experiment is presented to illustrate the utilization of the invariants for blurred image recognition robustness of the features is also briefly discussed

blur invariants a novel representation in the wavelet domain blur invariants in the wavelet domain are proposed for the first time in this paper wavelet domain blur invariants take advantage of several benefits that this domain provides i e different alternatives for wavelet function and analysis in different scales it is not required to model the blur system in order to extract the invariants it will be shown how the space domain blur invariants are a special case of the proposed invariants it will also be explained how the proposed invariants would not have the null space that their special case in the spatial domain have which limits their discriminative power the performance of these invariants will be demonstrated through experiments and compared to its counterpart which is defined in the spatial domain

wigner random matrices with non symmetrically distributed entries we show that the spectral radius of an n n random symmetric matrix with i i d bounded centered but non symmetrically distributed entries is bounded from above by 2 o n 6 11 e where 2 is the variance of the matrix entries and e is an arbitrary small positive number our bound improves the earlier results by z furedi and j komlos 1981 and van vu 2005

degraded image analysis an invariant approach analysis and interpretation of an image which was acquired by a nonideal imaging system is the key problem in many application areas the observed image is usually corrupted by blurring spatial degradations and random noise classical methods like blind deconvolution try to estimate the blur parameters and to restore the image we propose an alternative approach we derive the features for image representation which are invariant with respect to blur regardless of the degradation psf provided that it is centrally symmetric as we prove in the paper there exist two classes of such features the first one in the spatial domain and the second one in the frequency domain we also derive so called combined invariants which are invariant to composite geometric and blur degradations knowing these features we can recognize objects in the degraded scene without any restoration

local law and tracy widom limit for sparse random matrices we consider spectral properties and the edge universality of sparse random matrices the class of random matrices that includes the adjacency matrices of the erd s renyi graph model g n p we prove a local law for the eigenvalue density up to the spectral edges under a suitable condition on the sparsity we also prove that the rescaled extremal eigenvalues exhibit goe tracy widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included for the adjacency matrix of the erd s renyi graph this establishes the tracy widom fluctuations of the second largest eigenvalue when p is much larger than n 2 3 with a deterministic shift of order np 1

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

bulk universality of sparse random matrices we consider the adjacency matrix of the ensemble of erd s renyi random graphs which consists of graphs on n vertices in which each edge occurs independently with probability p we prove that in the regime pn 1 these matrices exhibit bulk universality in the sense that both the averaged n point correlation functions and distribution of a single eigenvalue gap coincide with those of the goe our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

contraction theory on riemannian manifolds abstract contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another in this work we present the fundamental results of contraction theory in an intrinsic coordinate free setting with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties we provide coordinate free proofs of the main results for autonomous vector fields and clarify the assumptions under which the results hold we state and prove several interesting corollaries to the main result study cascade and feedback interconnections of contracting systems study some simple examples and highlight how contraction theory has arisen independently in other scientific disciplines we conclude by illustrating the developed theory for the case of gradient dynamics

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

depth from motion and optical blur with an unscented kalman filter space variantly blurred images of a scene contain valuable depth information in this paper our objective is to recover the 3 d structure of a scene from motion blur optical defocus in the proposed approach the difference of blur between two observations is used as a cue for recovering depth within a recursive state estimation framework for motion blur we use an unblurred blurred image pair since the relationship between the observation and the scale factor of the point spread function associated with the depth at a point is nonlinear we propose and develop a formulation of unscented kalman filter for depth estimation there are no restrictions on the shape of the blur kernel furthermore within the same formulation we address a special and challenging scenario of depth from defocus with translational jitter the effectiveness of our approach is evaluated on synthetic as well as real data and its performance is also compared with contemporary techniques

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

generalized gradients and characterization of epi lipschitz sets in riemannian manifolds abstract in this paper a notion of generalized gradient on riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented a characterization of the tangent cone to a nonempty subset s of a riemannian manifold m at a point x is obtained then these results are applied to characterize epi lipschitz subsets of complete riemannian manifolds

3 d moment forms their construction and application to object identification and positioning the 3 d moment method is applied to object identification and positioning a general theory of deriving 3 d moments invariants is proposed the notion of complex moments is introduced complex moments are defined as linear combinations of moments with complex coefficients and are collected into multiplets such that each multiplet transforms irreducibly under 3 d rotations the application of the 3 d moment method to motion estimation is also discussed using group theoretic techniques various invariant scalars are extracted from compounds of complex moments via clebsch gordon expansion twelve moment invariants consisting of the second order and third order moments are explicitly derived based on a perturbation formula it is shown that the second order moment invariants can be used to predict whether the estimation using noisy data is reliable or not the new derivation of vector forms also facilities the calculation of motion estimation in a tensor approach vectors consisting of the third order moments can be derived in a similar manner

geometric moment invariants this paper derives moment invariants in an intuitive way by multiple integrals of invariant geometric primitives like distance area and volume many existing 2 d and 3 d moment invariants are re expressed to prove the correctness of this method furthermore this construction method can be easily extended to higher dimensional space and get higher order moment invariants explicit expressions of 3 d moment invariants are generated automatically with the help of symbolic computation software and can be regarded as a kind of shape descriptor for the representation of solid objects

on the independence of rotation moment invariants abstract the problem of the independence and completeness of rotation moment invariants is addressed in this paper first a general method for constructing invariants of arbitrary orders by means of complex moments is described as a major contribution of the paper it is shown that for any set of invariants there exists a relatively small basis by means of which all other invariants can be generated the method how to construct such a basis and how to prove its independence and completeness is presented some practical impacts of the new results are mentioned at the end of the paper

local convergence of the proximal point method for a special class of nonconvex functions on hadamard manifolds abstract local convergence analysis of the proximal point method for a special class of nonconvex functions on hadamard manifold is presented in this paper the well definedness of the sequence generated by the proximal point method is guaranteed moreover it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and under additional assumptions its convergence for a minimizer is obtained

local stability of the free additive convolution abstract we prove that the system of subordination equations defining the free additive convolution of two probability measures is stable away from the edges of the support and blow up singularities by showing that the recent smoothness condition of kargin is always satisfied as an application we consider the local spectral statistics of the random matrix ensemble a u b u where u is a haar distributed random unitary or orthogonal matrix and a and b are deterministic matrices in the bulk regime we prove that the empirical spectral distribution of a u b u concentrates around the free additive convolution of the spectral distributions of a and b on scales down to n 2 3

convergence rate for spectral distribution of addition of random matrices abstract let a and b be two n by n deterministic hermitian matrices and let u be an n by n haar distributed unitary matrix it is well known that the spectral distribution of the sum h a u b u converges weakly to the free additive convolution of the spectral distributions of a and b as n tends to infinity we establish the optimal convergence rate 1 n in the bulk of the spectrum

blurred image recognition by legendre moment invariants processing blurred images is a key problem in many image applications existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments in this paper we propose a new method to construct a set of blur invariants using the orthogonal legendre moments some important properties of legendre moments for the blurred image are presented and proved the performance of the proposed descriptors is evaluated with various point spread functions and different image noises the comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided the experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments

multichannel blind deconvolution of spatially misaligned images existing multichannel blind restoration techniques assume perfect spatial alignment of channels correct estimation of blur size and are prone to noise we developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral this stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise we observe that the exact knowledge of the blur size is not necessary and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process

local law of addition of random matrices on optimal scale the eigenvalue distribution of the sum of two large hermitian matrices when one of them is conjugated by a haar distributed unitary matrix is asymptotically given by the free convolution of their spectral distributions we prove that this convergence also holds locally in the bulk of the spectrum down to the optimal scales larger than the eigenvalue spacing the corresponding eigenvectors are fully delocalized similar results hold for the sum of two real symmetric matrices when one is conjugated by haar orthogonal matrix

on image analysis by the methods of moments various types of moments have been used to recognize image patterns in a number of applications a number of moments are evaluated and some fundamental questions are addressed such as image representation ability noise sensitivity and information redundancy moments considered include regular moments legendre moments zernike moments pseudo zernike moments rotational moments and complex moments properties of these moments are examined in detail and the interrelationships among them are discussed both theoretical and experimental results are presented

a goodness of fit test for stochastic block models the stochastic block model is a popular tool for studying community structures in network data we develop a goodness of fit test for the stochastic block model the test statistic is based on the largest singular value of a residual matrix obtained by subtracting the estimated block mean effect from the adjacency matrix asymptotic null distribution is obtained using recent advances in random matrix theory the test is proved to have full power against alternative models with finer structures these results naturally lead to a consistent sequential testing estimate of the number of communities

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

central limit theorems for linear statistics of heavy tailed random matrices we show central limit theorems clt for the stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries including entries in the domain of attraction of alpha stable laws and entries with moments exploding with the dimension as in the adjacency matrices of erdos renyi graphs for the second model we also prove a central limit theorem of the moments of its empirical eigenvalues distribution the limit laws are gaussian but unlike to the case of standard wigner matrices the normalization is the one of the classical clt for independent random variables

blur invariant translational image registration for n fold symmetric blurs in this paper we propose a new registration method designed particularly for registering differently blurred images such a task cannot be successfully resolved by traditional approaches our method is inspired by traditional phase correlation which is now applied to certain blur invariant descriptors instead of the original images this method works for unknown blurs assuming the blurring psf exhibits an n fold rotational symmetry it does not require any landmarks we have experimentally proven its good performance which is not dependent on the amount of blur in this paper we explicitly address only registration with respect to translation but the method can be readily generalized to rotation and scaling

morphological characterization of intracranial aneurysms using 3 d moment invariants rupture of intracranial saccular aneurysms is the most common cause of spontaneous subarachnoid hemorrhage which has significant morbidity and mortality although there is still controversy regarding the decision on which unruptured aneurysms should be treated this is based primarily on their size nonetheless many large lesions do not rupture whereas some small ones do it is commonly accepted that hemodynamical factors are important to better understand the natural history of cerebral aneurysms however it might not always be practical to carry out a detailed computational analysis of such factors if a prompt assessment is required since shape is likely to be dependent on the balance between hemodynamic forces and the aneurysmal surrounding environment an appropriate morphological 3 d characterization is likely to provide a practical surrogate to quickly evaluate the risk of rupture in this paper an efficient and novel methodology for 3 d shape characterization of cerebral aneurysms is described the aneurysms are isolated by taking into account a portion of their adjacent vessels two methods to characterize the morphology of the aneurysms models using moment invariants have been considered geometrical moment invariants gmi and zernike moment invariants zmi the results have been validated in a database containing 53 patients with a total of 31 ruptured aneurysms and 24 unruptured aneurysms it has been found that zmi indices are more robust than gmi and seem to provide a reliable way to discriminate between ruptured and unruptured aneurysms correct rupture prediction rates of sime80 were achieved in contrast to 66 that is found when the aspect ratio index is considered

bulk universality for deformed wigner matrices we consider n n random matrices of the form h w v where w is a real symmetric or complex hermitian wigner matrix and v is a random or deterministic real diagonal matrix whose entries are independent of w we assume subexponential decay for the matrix entries of w and we choose v so that the eigenvalues of w and v are typically of the same order for a large class of diagonal matrices v we show that the local statistics in the bulk of the spectrum are universal in the limit of large n

new classes of radiometric and combined radiometric geometric invariant descriptors real images can contain geometric distortions as well as photometric degradations analysis and characterization of those images without recourse to either restoration or geometric standardization is of great importance for the computer vision community as those two processes are often ill posed problems to this end it is necessary to implement image descriptors that make it possible to identify the original image in a simple way independently of the imaging system and imaging conditions ideally descriptors that capture image characteristics must be invariant to the whole range of geometric distortions and photometric degradations such as blur that may affect the image in this paper we introduce two new classes of radiometric and or geometric invariant descriptors the first class contains two types of radiometric invariant descriptors the first of these type is based on the mellin transform and the second one is based on central moments both descriptors are invariant to contrast changes and to convolution with any kernel having a symmetric form with respect to the diagonals the second class contains two subclasses of combined invariant descriptors the first subclass includes central moment based descriptors invariant simultaneously to horizontal and vertical translations to uniform and anisotropic scaling to stretching to convolution and to contrast changes the second subclass contains central complex moment based descriptors that are simultaneously invariant to similarity transformation and to contrast changes we apply these invariant descriptors to the matching of geometric transformed and or blurred images experimental results confirm both the robustness and the effectiveness of the proposed invariants

anisotropic local laws for random matrices we develop a new method for deriving local laws for a large class of random matrices it is applicable to many matrix models built from sums and products of deterministic or independent random matrices in particular it may be used to obtain local laws for matrix ensembles that are anisotropic in the sense that their resolvents are well approximated by deterministic matrices that are not multiples of the identity for definiteness we present the method for sample covariance matrices of the form open image in new window where t is deterministic and x is random with independent entries we prove that with high probability the resolvent of q is close to a deterministic matrix with an optimal error bound and down to optimal spectral scales as an application we prove the edge universality of q by establishing the tracy widom airy statistics of the eigenvalues of q near the soft edges this result applies in the single cut and multi cut cases further applications include the distribution of the eigenvectors and an analysis of the outliers and bbp type phase transitions in finite rank deformations they will appear elsewhere we also apply our method to wigner matrices whose entries have arbitrary expectation i e we consider w a where w is a wigner matrix and a a hermitian deterministic matrix we prove the anisotropic local law for w a and use it to establish edge universality

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

bounds for the stieltjes transform and the density of states of wigner matrices we consider ensembles of wigner matrices whose entries are up to the symmetry constraints independent and identically distributed random variables we show the convergence of the stieltjes transform towards the stieltjes transform of the semicircle law on optimal scales and with the optimal rate our bounds improve previous results in particular from erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 by removing the logarithmic corrections as applications we establish the convergence of the eigenvalue counting functions with the rate log n n and the rigidity of the eigenvalues of wigner matrices on the same scale these bounds improve the results of erd s et al adv math 229 3 1435 1515 2012 electron j probab 18 59 1 58 2013 gotze and tikhomirov 2013

moment forms invariant to rotation and blur in arbitrary number of dimensions we present the construction of combined blur and rotation moment invariants in arbitrary number of dimensions moment invariants to convolution with an arbitrary centrosymmetric filter are derived first and then their rotationally invariant forms are found by means of group representation theory to achieve the desired combined invariance several examples of the invariants are calculated explicitly to illustrate the proposed procedure their invariance robustness and capability of using in template matching and in image registration are demonstrated on 3d mri data and 2d indoor images

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles