complete sets of complex zernike moment invariants and the role of the pseudoinvariants moment invariants are important shape descriptors in computer vision in this paper rules are presented to form complete sets of rotational invariants of arbitrary order given complex zernike moments the relevance of the pseudoinvariants is shown by inverting the rotational invariants to reconstruct the normalized image pattern

generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

invariant character recognition with zernike and orthogonal fourier mellin moments abstract in this paper we consider the use of orthogonal moments for invariant classification of alphanumeric characters of different size in addition to the zernike and pseudo zernike moments zms and pzms which have been previously proposed for invariant character recognition a new method of combining orthogonal fourier mellin moments ofmms with centroid bounding circle scaling is introduced which is shown to be useful in characterizing images with large variability through extensive experimentation using zms and ofmms as features different scaling methodologies and classifiers it is shown that ofmms give the best overall performance in terms of both image reconstruction and classification accuracy

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

finite time entropy a probabilistic approach for measuring nonlinear stretching abstract transport and mixing processes in dynamical systems are often difficult to study analytically and therefore a variety of numerical methods have been developed finite time lyapunov exponents ftles or related stretching indicators are frequently used as a means to estimate transport barriers alternatively eigenvectors singular vectors or oseledets vectors of numerical transfer operators find almost invariant sets finite time coherent sets or time asymptotic coherent sets respectively which are minimally dispersed under the dynamics while these families of approaches geometric ftles and the probabilistic transfer operator often give compatible results a formal link is still missing here we present a small step towards providing a mathematical link we propose a new entropy based methodology for estimating finite time expansive behaviour along trajectories in autonomous and nonautonomous dynamical systems we introduce the finite time entropy fte field as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localised density evolved by the transfer operator the fte construction elucidates in a straightforward way the connection between the evolution of probability densities and the local stretching experienced we develop an extremely simple and numerically efficient method of constructing an estimate of the fte field the fte field is instantaneously calculable from a numerical transfer operator a transition matrix of conditional probabilities that describes a discretised version of the dynamical system once one has such a transition matrix the fte field may be computed for free we also show i how to avoid long time integrations in autonomous and time periodic systems ii how to perform backward time computations by a fast matrix manipulation rather than backward time integration and iii how to easily employ adaptive methods to focus on high value fte regions

a manifold independent approach to understanding transport in stochastic dynamical systems abstract we develop a new collection of tools aimed at studying stochastically perturbed dynamical systems specifically in the setting of bi stability that is a two attractor system it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero noise case barriers in the phase space pseudo barriers thus creating a pre heteroclinic tangency chaos like behavior the stochastic dynamical system has a corresponding frobenius perron operator with a stochastic kernel which describes how densities of initial conditions move under the noisy map thus in studying the action of the frobenius perron operator we learn about the transport of the map we have employed a galerkin ulam like method to project the frobenius perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space graph theoretic methods allow us to re order the resulting finite dimensional markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo barriers of the stochastic dynamics our toolbox allows us to find 1 regions of high activity of transport 2 flux across pseudo barriers and also 3 expected time of escape from pseudo basins some of these quantities are also possible via the manifold dependent stochastic melnikov method but melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available our methods are unique in that they can essentially be considered as a black box of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures we use here a model of childhood diseases to showcase our methods our tools will allow us to make specific observations of 1 loss of reducibility between basins with increasing noise 2 identification in the phase space of active regions of stochastic transport 3 stochastic flux which essentially completes the heteroclinic tangle

skew estimation of document images using bagging this paper proposes a general purpose method for estimating the skew angles of document images rather than to derive a skew angle merely from text lines the proposed method exploits various types of visual cues of image skew available in local image regions the visual cues are extracted by radon transform and then outliers of them are iteratively rejected through a floating cascade a bagging bootstrap aggregating estimator is finally employed to combine the estimations on the local image blocks our experimental results show significant improvements against the state of the art methods in terms of execution speed and estimation accuracy as well as the robustness to short and sparse text lines multiple different skews and the presence of nontextual objects of various types and quantities

a robust and fast skew detection algorithm for generic documents abstract a robust and fast skew detection algorithm based on hierarchical hough transform is proposed it is capable of detecting the skew angle for various document images including technical articles postal labels handwritten text forms drawings and bar codes the algorithm is robust even when black margins introduced by photocopying are present in the image and when the document is scanned at a low resolution of 50 dpi the algorithm consists of two steps in the first step we quickly extract the centroids of connected components using a graph data structure then a hierarchical hough transform at two different angular resolutions is applied to the selected centroids the skew angle corresponds to the location of the highest peak in the hough space the performance of the algorithm is shown on a number of document images collected from various application domains the algorithm is not very sensitive to algorithmic parameters for an a4 size document image scanned at 50 dpi typically 413 575 pixels our algorithm is able to detect the skew angle with an accuracy of 0 1 in 0 4s of cpu time on a sunsparc 20 workstation

a variational theory of hyperbolic lagrangian coherent structures abstract we develop a mathematical theory that clarifies the relationship between observable lagrangian coherent structures lcss and invariants of the cauchy green strain tensor field motivated by physical observations of trajectory patterns we define hyperbolic lcss as material surfaces i e codimension one invariant manifolds in the extended phase space that extremize an appropriate finite time normal repulsion or attraction measure over all nearby material surfaces we also define weak lcss wlcss as stationary solutions of the above variational problem solving these variational problems we obtain computable sufficient and necessary criteria for wlcss and lcss that link them rigorously to the cauchy green strain tensor field we also prove a condition for the robustness of an lcs under perturbations such as numerical errors or data imperfection on several examples we show how these results resolve earlier inconsistencies in the theory of lcs finally we introduce the notion of a constrained lcs clcs that extremizes normal repulsion or attraction under constraints this construct allows for the extraction of a unique observed lcs from linear systems and for the identification of the most influential weak unstable manifold of an unstable node

distinguished material surfaces and coherent structures in three dimensional fluid flows we prove analytic criteria for the existence of finite time attracting and repelling material surfaces and lines in three dimensional unsteady flows the longest lived such structures define coherent structures in a lagrangian sense our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories an alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite time lyapunov exponent field computed directly from particle paths both approaches provide effective tools for extracting distinguished lagrangian structures from three dimensional velocity data we illustrate the results on steady and unsteady abc type flows

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

multiple phases in stochastic dynamics geometry and probabilities stochastic dynamics is generated by a matrix of transition probabilities certain eigenvectors of this matrix provide observables and when these are plotted in the appropriate multidimensional space the phases in the sense of phase transitions of the underlying system become manifest as extremal points this geometrical construction which we call an observable representation of state space can allow hierarchical structure to be observed it also provides a method for the calculation of the probability that an initial points ends in one or another asymptotic state

partial orderings preorderings and the polar decomposition of matrices abstract when a unique decomposition of a complex rectangular matrix into two components is given then a partial ordering on the domain of one of the components induces a preordering on the whole set of matrices and partial orderings on the domains of both components induce a partial ordering on the whole set in this not we consider the three main partial orderings lo wner star and rank subtractivity on the respective domains of the two components of matrices subjected to polar decomposition a ben israel t n e greville generalized inverses theory and applications wiley new york 1974 p 255 and investigate the resulting pre and partial orderings

discriminative zernike and pseudo zernike moments for face recognition usually magnitude coefficients of some selected orders of zms and pzms have been used as invariant image features the careful selection of the set of features with higher discrimination competence may increase the recognition performance in this paper the authors have used a statistical method to estimate the discrimination strength of all the extracted coefficients of zms and pzms whereas for classification only the coefficients with estimated higher discrimination strength are used in the feature vector the performance of these selected discriminative zms dzms and discriminative pzms dpzms features have been compared to that of their corresponding conventional approaches on yale orl and feret databases against illumination expression scale and pose variations an extension to these dzms and dpzms have been proposed by combining them with pca and fld it has been observed from the exhaustive experimentation that the recognition rate is improved by 2 6 at reduced dimensions and with less computational complexity than that of using the successive zms and pzms features

innovation based on gaussian elimination to compute generalized inverse at s 2 abstract in this paper we execute elementary row and column operations on the partitioned matrix g a g g g 0 into i s 0 0 0 0 0 a t s 2 to compute generalized inverse a t s 2 of a given complex matrix a where g is a matrix such that r g t and n g s the total number of multiplications and divisions operations is t m n s 2 m n 2 4 m s 1 2 n s m s n s m n s and the upper bound of t m n s is less than 6 m n 2 3 2 n 3 1 2 n 2 when n m a numerical example is shown to illustrate that this method is correct

characterizations and representations of the p q outer generalized inverse we present a new explicit expression for the p q outer generalized inverse based on the idempotent operators p and q we derive properties of the p q outer generalized inverse and analyse relations among the various p q outer generalized inverses

hyperbolic and elliptic transport barriers in three dimensional unsteady flows abstract we develop a general theory of transport barriers for three dimensional unsteady flows with arbitrary time dependence the barriers are obtained as two dimensional lagrangian coherent structures lcss that create locally maximal deformation along hyperbolic lcss this deformation is induced by locally maximal normal repulsion or attraction along shear lcss the deformation is created by locally maximal tangential shear hyperbolic lcss therefore play the role of generalized stable and unstable manifolds while closed shear lcss elliptic lcss act as generalized kam tori or kam type cylinders all these barriers can be computed from our theory as explicitly parametrized surfaces we illustrate our results by visualizing two dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the abc flow

partial orders based on core nilpotent decomposition abstract in this paper we study partial orders in terms of the core nilpotent decomposition we derive some characterizations of the c n partial ordering create a new partial ordering the g drazin partial ordering which is a generalization of c n partial ordering and give some properties and characterizations of the g drazin partial ordering

image analysis by circularly semi orthogonal moments various types of circularly orthogonal moments have been widely used for image reconstruction and rotation invariant classification however they suffer from two errors namely numerical integration error and geometric error which affect their reconstruction capability and pattern recognition accuracy in this paper a novel category of circular moments named circularly semi orthogonal moments is proposed in the proposed moment a set of orthogonal basis functions modulated by a negative power exponential envelope is utilized as the radial basis function for a given degree n the radial basis function possesses more compact bandwidth less cutoff frequency and more zeros compared with the frequently used circularly orthogonal moments including zernike and orthogonal fourier mellin moments and so the circularly semi orthogonal moment calculated with the zeroth order approximation is more robust to numerical error than the frequently used circularly orthogonal moments furthermore the capability of the semi orthogonal moment to describe high spatial frequency components of images is relatively higher than that of the frequently used circularly orthogonal moments experimental results demonstrate that the semi orthogonal moments calculated with the zeroth order approximation perform better than the frequently used circularly orthogonal moments in terms of image reconstruction capability and invariant recognition accuracy in noise free noisy and smooth distortion conditions it is also shown that the proposed high order moments are more numerically stable than the circularly orthogonal moments a kind of circular moments named circularly semi orthogonal moment is proposed it outperforms in image reconstruction over the circularly orthogonal moments it performs better in invariant image recognition under noisy and smooth distortions

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

an analytic framework for identifying finite time coherent sets in time dependent dynamical systems abstract the study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems barriers to transport which mitigate mixing are currently the subject of intense study in the autonomous setting the use of transfer operators perron frobenius operators to identify invariant and almost invariant sets has been particularly successful in the nonautonomous time dependent setting coherent sets a time parameterised family of minimally dispersive sets are a natural extension of almost invariant sets the present work introduces a new analytic transfer operator construction that enables the calculation of finite time coherent sets sets are that minimally dispersive over a finite time interval this new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude the construction can also be applied to general markov processes on continuous state space

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

nonautonomous control of stable and unstable manifolds in two dimensional flows abstract we outline a method for controlling the location of stable and unstable manifolds in the following sense from a known location of the stable and unstable manifolds in a steady two dimensional flow the primary segments of the manifolds are to be moved to a user specified time varying location which is near the steady location we determine the nonautonomous perturbation to the vector field required to achieve this control and give a theoretical bound for the error in the manifolds resulting from applying this control the efficacy of the control strategy is illustrated via a numerical example

improved quality of reconstructed images using floating point arithmetic for moment calculation zernike moments which are superior to geometric moments because of their special properties of image reconstruction and immunity to noise suffer from several discretization errors these errors lead to poor quality of reconstructed image and wide variations in the numerical values of the moments the predominant factor as observed in this paper is due to the discrete integer implementation of the steps involved in moment calculation it is shown in this paper that by modifying the algorithms to include discrete float implementation the quality of the reconstructed image improves significantly and the first order moment becomes zero low order zernike moments have been found to be stable under linear transformations while the high order moments have large variations the large variations in high order moments however do not greatly affect the quality of the reconstructed image implying that they should be ignored when numerical values of moments are used as features the 11 functions based on geometric moments have also been found to be stable under linear transformations and thus these can be used as features pixel level analysis of the images has been carried out to strengthen the results

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

e pcp a robust skew detection method for scanned document images we present here an enhanced algorithm e pcp for skew detection in scanned documents based on the work on piecewise covering by parallelogram pcp for robust determination of skew angles c h chou s y chu f chang estimation of skew angles for scanned documents based on piecewise covering by parallelograms pattern recognition 40 2007 443 455 our algorithm achieves even better robustness for detection of skew angle than the original pcp algorithm we have shown accurate determination of skew angles in document images where the original pcp algorithm fails further the increased robustness of performance is achieved with reduced number of computation compared to the originally proposed pcp algorithm the e pcp algorithm also outputs a confidence measure which is important in automated systems to filter cases where the estimated skew angle may not be very accurate and thus can be handled by manual intervention the proposed algorithm was tested extensively on all categories of real time documents and comparisons with pcp method is also provided useful details regarding faster execution of the proposed algorithm is provided in appendix

stochastic sensitivity a computable lagrangian uncertainty measure for unsteady flows uncertainties in velocity data are often ignored when computing lagrangian particle trajectories of fluids modeling these as noise in the velocity field leads to a random deviation from each traje

accurate calculation of zernike moments zernike moments zms are very effective global image descriptors which are used in many digital image processing applications the digitization process compromises the accuracy of the moments and therefore several of its properties are affected there are two major discretization errors namely the geometric error and numerical integration error in this paper we propose two new algorithms which eliminate these errors the first algorithm performs the exact computation of geometric moments gms over a unit disk and then uses gms to zms relationship to compute the latter this algorithm is computationally more expensive and it becomes numerically instable for higher order moments therefore we develop a second algorithm based on gaussian quadrature numerical integration the second algorithm reduces both the errors simultaneously and its accuracy increases as the degree of gaussian quadrature numerical integration increases the proposed algorithms are observed to provide very accurate zms which result in improved image reconstruction reduction in reconstruction error and improvement in rotation and scale invariance exhaustive experiments are provided to support improved accuracy of zms and time complexity analysis is performed for the existing and the proposed methods

a note on the rank subtractivity ordering abstract some well known matrix partial orderings can be created by supplementing the so called rank subtractivity partial ordering by an appropriate preordering in this note a new partial ordering with the same property is introduced this new ordering is seen to be weaker than the other orderings but still preserving some inheritance properties

fast and accurate method for high order zernike moments computation abstract zernike moments zms which belong to a class of circularly orthogonal rotation invariant moments are very useful image descriptors for various image processing applications they however suffer from various errors such as discretization error geometric error and numerical integration error in this paper we present an approach for their computation which reduces geometric error and numerical integration error geometric error is reduced by performing exact mapping of a square grid domain into a circular grid by using both the square and arc grids numerical integration error is reduced by resorting to gaussian numerical integration over the square and arc grids it is observed that the contribution of numerical integration error to the total error is much more than the geometric error the effect of these two errors is more significant in small images as compared to large images the use of numerical integration however enhances the time complexity this problem is resolved by using recursive algorithms for the computation of radial polynomials 8 way symmetry of radial polynomials and 8 way symmetry anti symmetry of angular part of the kernel function exhaustive experiments on synthetic and real images are performed to demonstrate the effectiveness of the proposed approach

estimation of skew angles for scanned documents based on piecewise covering by parallelograms we propose a fast and robust skew estimation method for scanned documents that estimates skew angles based on piecewise covering of objects such as textlines figures forms or tables the method first divides a document image into a number of non overlapping slabs in which each object is covered by parallelograms it then estimates the skew angle based on these parallelograms or equivalently their complementary regions putting our method to a systematic test and comparing it with some alternatives we find that it yields favorable results in terms of accuracy sensitivity to non textual objects effectiveness in dealing with documents of unspecified reading order and computational efficiency some work is also conducted to find an effective way to further shorten its computation time at the expense of an extremely small loss of accuracy

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

geometric invariance in image watermarking surviving geometric attacks in image watermarking is considered to be of great importance in this paper the watermark is used in an authentication context two solutions are being proposed for such a problem both geometric and invariant moments are used in the proposed techniques an invariant watermark is designed and tested against attacks performed by stirmark using the invariant moments on the other hand an image normalization technique is also proposed which creates a normalized environment for watermark embedding and detection the proposed algorithms have the advantage of being robust computationally efficient and no overhead needs to be transmitted to the decoder side the proposed techniques have proven to be highly robust to all geometric manipulations filtering compression and slight cropping which are performed as part of stirmark attacks as well as noise addition both gaussian and salt pepper

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

partial orders based on outer inverses classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders these classes are then compared with those obtained by using g inverses inner inverses

on the accuracy of zernike moments for image analysis we give a detailed analysis of the accuracy of zernike moments in terms of their discretization errors and the reconstruction power it is found that there is an inherent limitation in the precision of computing the zernike moments due to the geometric nature of a circular domain this is explained by relating the accuracy issue to a celebrated problem in analytic number theory of evaluating the lattice points within a circle

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

block decomposition and segmentation for fast hough transform evaluation the decomposition of binary images using rectangular blocks of foreground pixels as primitives is considered based on this type of decomposition a fast method for evaluating the hough transform is introduced a complexity analysis of the proposed block hough transform algorithm sets constraints on the complexity of algorithms used for block decomposition so that the total decomposition and hough transform application time is much less than the time consumed by the usual point hough transform using this analysis we propose two algorithms for the decomposition and segmentation of binary images into rectangular blocks a combination of these methods leads to significant acceleration in the identification of linear features which is demonstrated in various image processing experiments 1999 pattern recognition society published by elsevier science ltd all rights reserved

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

hough transform based fast skew detection and accurate skew correction methods the hough transform provides a robust technique for skew detection in document images but suffers from high time complexity which becomes prohibitive for detecting skew in large documents analysis of time complexity on various stages of skew detection process is carried out in this paper a complete skew detection and correction process is divided into three parts a preprocessing stage using a simplified form of block adjacency graph bag voting process using the hough transform and de skewing the image using rotation skew correction phase which is hitherto a neglected area is analysed for the quality of de skewed images with respect to the type of rotation fast algorithms for all the three stages are presented and exhaustive analysis on time complexity is conducted it is shown that the overall time taken for the whole process is less than one second even for very large documents it is also observed that time taken in rotation is as significant as in skew detection which is reduced with the help of fast algorithms using integer operations while the bag algorithm is found to be effective for documents with roman script it does not provide satisfactory results for indian scripts where headline is a part of a script

translation invariants of zernike moments moment functions defined using a polar coordinate representation of the image space such as radial moments and zernike moments are used in several recognition tasks requiring rotation invariance however this coordinate representation does not easily yield translation invariant functions which are also widely sought after in pattern recognition applications this paper presents a mathematical framework for the derivation of translation invariants of radial moments defined in polar form using a direct application of this framework translation invariant functions of zernike moments are derived algebraically from the corresponding central moments both derived functions are developed for non symmetrical as well as symmetrical images they mitigate the zero value obtained for odd order moments of the symmetrical images vision applications generally resort to image normalization to achieve translation invariance the proposed method eliminates this requirement by providing a translation invariance property in a zernike feature set the performance of the derived invariant sets is experimentally confirmed using a set of binary latin and english characters

stochastic stability of lyapunov exponents and oseledets splittings for semi invertible matrix cocycles we establish i stability of lyapunov exponents and ii convergence in probability of oseledets spaces for semi invertible matrix cocycles subjected to small random perturbations the first part extends results of ledrappier and young to the semi invertible setting the second part relies on the study of evolution of subspaces in the grassmannian the analysis developed here is based on higher dimensional mobius transformations and is likely to be of wider interest

fast and numerically stable methods for the computation of zernike moments zernike moments zms are used in many image processing applications due to their superior performance over other moments however they suffer from high computation cost and numerical instability at high order of moments in the past many recursive methods have been developed to improve their speed performance and considerable success has been achieved the analysis of numerical stability has also gained momentum as it affects the accuracy of moments and their invariance property there are three recursive methods which are normally used in zms calculation prata s kintner s and q recursive methods the earlier studies have found the q recursive method outperforming the two other methods in this paper we modify prata s method and present a recursive relation which is proved to be faster than the q recursive method numerical instability is observed at high orders of moments with the q recursive method suffering from the underflow problem while the modified prata s method suffering from finite precision error the modified kintner s method is the least susceptible to these errors keeping in view the better numerical stability we further make the modified kintner s method marginally faster than the q recursive method we recommend the modified prata s method for low orders 90 and kintner s fast method for high orders 90 of zms

robust perron cluster analysis in conformation dynamics the key to molecular conformation dynamics is the direct identification of metastable conformations which are almost invariant sets of molecular dynamical systems once some reversible markov operator has been discretized a generalized symmetric stochastic matrix arises this matrix can be treated by perron cluster analysis a rather recent method involving a perron cluster eigenproblem the paper presents an improved perron cluster analysis algorithm which is more robust than earlier suggestions numerical examples are included

automatic image orientation detection we present an algorithm for automatic image orientation estimation using a bayesian learning framework we demonstrate that a small codebook the optimal size of codebook is selected using a modified mdl criterion extracted from a learning vector quantizer lvq can be used to estimate the class conditional densities of the observed features needed for the bayesian methodology we further show how principal component analysis pca and linear discriminant analysis lda can be used as a feature extraction mechanism to remove redundancies in the high dimensional feature vectors used for classification the proposed method is compared with four different commonly used classifiers namely k nearest neighbor support vector machine svm a mixture of gaussians and hierarchical discriminating regression hdr tree experiments on a database of 16 344 images have shown that our proposed algorithm achieves an accuracy of approximately 98 on the training set and over 97 on an independent test set a slight improvement in classification accuracy is achieved by employing classifier combination techniques

statistically optimal almost invariant sets abstract chaotic dynamical systems are often transitive although this transitivity is sometimes very weak it is of interest to divide the phase space into large regions between which there is relatively little communication of trajectories we present fast simple algorithms to find such divisions the present work builds on the results of froyland and dellnitz g froyland m dellnitz detecting and locating near optimal almost invariant sets and cycles siam j sci comput 24 6 2003 1839 1863 focussing on a statistical description of transitivity that takes into account the fact that trajectories tend to visit different regions of phase space with different frequencies the new work takes advantage of theoretical results from the theory of reversible markov chains a new adaptive algorithm is put forward to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated this algorithm is illustrated with the standard map relevant convergence results are proven

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

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

polar rotation angle identifies elliptic islands in unsteady dynamical systems abstract we propose rotation inferred from the polar decomposition of the flow gradient as a diagnostic for elliptic or vortex type invariant regions in non autonomous dynamical systems we consider here two and three dimensional systems in which polar rotation can be characterized by a single angle for this polar rotation angle pra we derive explicit formulas using the singular values and vectors of the flow gradient we find that closed level sets of the pra reveal elliptic islands in great detail and singular level sets of the pra uncover centers of such islands both features turn out to be objective frame invariant for two dimensional systems we illustrate the diagnostic power of pra for elliptic structures on several examples

detecting and locating near optimal almost invariant sets and cycles the behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict rather than characterizing dynamical behavior at the level of trajectories we consider following the evolution of sets there are often collections of sets that behave in a very predictable way in spite of the fact that individual trajectories are entirely unpredictable such special collections of sets are invisible to studies of long trajectories we describe a global set oriented method to detect and locate these large dynamical structures our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms

definition and properties of lagrangian coherent structures from finite time lyapunov exponents in two dimensional aperiodic flows this paper develops the theory and computation of lagrangian coherent structures lcs which are defined as ridges of finite time lyapunov exponent ftle fields these ridges can be seen as finite time mixing templates such a framework is common in dynamical systems theory for autonomous and time periodic systems in which examples of lcs are stable and unstable manifolds of fixed points and periodic orbits the concepts defined in this paper remain applicable to flows with arbitrary time dependence and in particular to flows that are only defined computed or measured over a finite interval of time previous work has demonstrated the usefulness of ftle fields and the associated lcss for revealing the lagrangian behavior of systems with general time dependence however ridges of the ftle field need not be exactly advected with the flow the main result of this paper is an estimate for the flux across an lcs which shows that the flux is small and in most cases negligible for well defined lcss or those that rotate at a speed comparable to the local eulerian velocity field and are computed from ftle fields with a sufficiently long integration time under these hypotheses the structures represent nearly invariant manifolds even in systems with arbitrary time dependence the results are illustrated on three examples the first is a simplified dynamical model of a double gyre flow the second is surface current data collected by high frequency radar stations along the coast of florida and the third is unsteady separation over an airfoil in all cases the existence of lcss governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

further results on iterative methods for computing generalized inverses in this paper we reconsider the iterative method x k x k email protected i ax k 1 k 1 2 email protected email protected 0 for computing the generalized inverse a t s 2 over banach spaces or the generalized drazin inverse a d of a banach algebra element a reveal the intrinsic relationship between the convergence of such iterations and the existence of a t s 2 or a d and present the error bounds of the iterative methods for approximating a t s 2 or a d moreover we deduce some necessary and sufficient conditions for iterative convergence to a t s 2 or a d

some properties of matrix partial orderings abstract the matrix partial orderings considered are 1 the star ordering and 2 the minus ordering or rank subtractivity both in the set of m n complex matrices and 3 the lowner ordering in the set of m m matrices the problems discussed are 1 inheriting certain properties under a given ordering 2 preserving an ordering under some matrix multiplications 3 relationships between an ordering among direct or kronecker and hadamard products and the corresponding orderings between the factors involved 4 orderings between generalized inverses of a given matrix and 5 preserving or reversing a given ordering under generalized inversions several generalizations of results known in the literature and a number of new results are derived

orthogonal moment features for use with parametric and non parametric classifiers this research examines a variety of approaches for using two dimensional orthogonal polynomials for the recognition of handwritten arabic numerals it also makes use of parametric and non parametric statistical and neural network classifiers polynomials including legendre zernike and pseudo zernike are used to generate moment based features which are invariant to location size and optionally rotation an efficient method for computing the moments via geometric moments is presented a side effect of this method also yields scale invariance a new approach to location invariance using a minimum bounding circle is presented and a detailed analysis of the rotational properties of the moments is given data partitioning tests are performed to evaluate the various feature types and classifiers for rotational invariant character recognition the highest percentage of correctly classified characters was 91 7 and for non rotational invariant recognition it was 97 6 this compares with a previous effort using the same data and test conditions of 94 8 the techniques developed here should also be applicable to other areas of shape recognition

core ep decomposition and its applications abstract in this paper we introduce a new decomposition called the core ep decomposition in the present paper for square matrices and give some of its applications by applying the decomposition we derive several characterizations of the core ep inverse introduce two orders the core ep order and the core minus partial order and characterize their properties we also give some properties of the decomposition

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses