an optimum solution for scale invariant object recognition based on the multiresolution approximation this paper presents a multiresolution approximation approach to obtaining boundary representations for object recognition our technique combines a multiresolution approximation and the curvature scale space representation for obtaining representations our research consists of two main parts in the first part of our research we introduce the continuous multiresolution approximation cma in terms of the continuous wavelet transform cwt then we implement a fast algorithm to compute the cma we apply the cma to a boundary to obtain approximations of the boundary at various resolutions the cma provides a consistent interpretation of objects with scale variations moreover we can quickly compute our representations by using the fast algorithm for the cma in the second part we propose three representations for object recognition which cover most boundary based object recognition problems all three representations use the approximations obtained by the cma each representation has different features and covers different types of matching problems but all representations are constructed by using curvature zero crossings of the approximations our representations provide a general but reliable solution to most boundary based object matching problems finally we investigate the properties of our representations such as validity efficiency and reliability we verified our results experimentally to demonstrate the feasibility of using our representations for object recognition

shape description and matching using integral invariants on eccentricity transformed images matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision to keep track of changes inside the breast for example it is important for a computer aided detection system to establish correspondences between regions of interest shape transformations computed both with integral invariants ii and with geodesic distance yield signatures that are invariant to isometric deformations such as bending and articulations integral invariants describe the boundaries of planar shapes however they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure conversely eccentricity transforms ecc can match shapes by signatures of geodesic distance histograms based on information from inside the shape but they ignore the boundary information we describe a method that combines the boundary signature of a shape obtained from ii and structural information from the ecc to yield results that improve on them separately

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

on critical point detection of digital shapes in this paper we present a nonlinear algorithm for critical point detection cpd of 2d digital shapes the algorithm eliminates the problems arising from curvature approximation and gaussian filtering in the existing algorithms based on the definition of critical level we establish a set of criteria for the design of an effective cpd algorithm for the first time by quantifying the critical level to the modified area confined by three consecutive pseudocritical points a simple but very effective algorithm is developed the comparison of our experimental results with those of many other cpd algorithms shows that the proposed algorithm is superior in that it provides a sequence of figures at every detail level and each has a smaller integral error than the others with the same number of critical points the experimental results on shapes with various complexities also show the algorithm is reliable and robust with regard to noise

tangent bundle elastica and computer vision visual curve completion an early visual process that completes the occluded parts between observed boundary fragments a k a inducers is a major problem in perceptual organization and a critical step toward higher level visual tasks in both biological and machine vision most computational contributions to solving this problem suggest desired perceptual properties that the completed contour should satisfy in the image plane and then seek the mathematical curves that provide them alternatively few studies including by the authors have suggested to frame the problem not in the image plane but rather in the unit tangent bundler 2 s 1 the space that abstracts the primary visual cortex where curve completion allegedly occurs combining both schools here we propose and develop a biologically plausible theory of elastica in the tangent bundle that provides not only perceptually superior completion results but also a rigorous computational prediction that inducer curvatures greatly affects the shape of the completed curve as indeed indicated by human perception

shape classification via image based multiscale description we introduce a new multiscale fourier based object description in 2 d space using a low pass gaussian filter lpgf and a high pass gaussian filter hpgf separately using the lpgf at different scales standard deviation represents the inner and central part of an object more than the boundary on the other hand using the hpgf at different scales represents the boundary and exterior parts of an object more than the central part our algorithms are also organized to achieve size translation and rotation invariance evaluation indicates that representing the boundary and exterior parts more than the central part using the hpgf performs better than the lpgf based multiscale representation and in comparison to zernike moments and elliptic fourier descriptors with respect to increasing noise multiscale description using hpgf in 2 d also outperforms wavelet transform based multiscale contour fourier descriptors and performs similar to the perimeter descriptors without any noise

sharp eigenvalue enclosures for the perturbed angular kerr newman dirac operator we examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients the strategy relies on computing the second order spectrum relative to subspaces of continuous piecewise linear functions for smooth perturbations of the angular kerr newman dirac operator explicit rates of convergence linked to regularity of the eigenfunctions are established numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included

filtering closed curves a closed curve in the plane can be described in several ways we show that a simple representation in terms of radius of curvature versus normal direction has certain advantages in particular convolutional filtering of the extended circular image leads to a closed curve similar filtering operations applied to some other representations of the curve do not guarantee that the result corresponds to a closed curve in one case where a closed curve is produced it is smaller than the original a description of a curve can be based on a sequence of smoothed versions of the curve this is one reason why smoothing of closed curves is of interest

implicit polynomials orthogonal distance regression and the closest point on a curve implicit polynomials i e multinomials have a number of properties that make them attractive for modeling curves and surfaces in computer vision the paper considers the problem of finding the best fitting implicit polynomial or algebraic curve to a collection of points in the plane using an orthogonal distance metric approximate methods for orthogonal distance regression have been shown by others to be prone to the problem of cusps in the solution and this is confirmed here consequently this work focuses on exact methods for orthogonal distance regression the most difficult and costly part of exact methods is computing the closest point on the algebraic curve to an arbitrary point in the plane the paper considers three methods for achieving this in detail the first is the standard newton s method the second is based on resultants which are making a resurgence in computer graphics and the third is a novel technique based on successive circular approximations to the curve it is shown that newton s method is the quickest but that it can fail sometimes even with a good initial guess the successive circular approximation algorithm is not as fast but is robust the resultant method is the slowest of the three but does not require an initial guess the driving application of this work was the fitting of implicit quartics in two variables to thinned oblique ionogram traces

an adaptive and stable method for fitting implicit polynomial curves and surfaces representing 2d and 3d data sets with implicit polynomials ips has been attractive because of its applicability to various computer vision issues therefore many ip fitting methods have already been proposed however the existing fitting methods can be and need to be improved with respect to computational cost for deciding on the appropriate degree of the ip representation and to fitting accuracy while still maintaining the stability of the fit we propose a stable method for accurate fitting that automatically determines the moderate degree required our method increases the degree of ip until a satisfactory fitting result is obtained the incrementability of qr decomposition with gram schmidt orthogonalization gives our method computational efficiency furthermore since the decomposition detects the instability element precisely our method can selectively apply ridge regression based constraints to that element only as a result our method achieves computational stability while maintaining fitting accuracy experimental results demonstrate the effectiveness of our method compared with prior methods

robustness of shape descriptors to incomplete contour representations with inspiration from psychophysical researches of the human visual system we propose a novel aspect and a method for performance evaluation of contour based shape recognition algorithms regarding their robustness to incompleteness of contours we use complete contour representations of objects as a reference training set incomplete contour representations of the same objects are used as a test set the performance of an algorithm is reported using the recognition rate as a function of the percentage of contour retained we call this evaluation procedure the icr test we consider three types of contour incompleteness viz segment wise contour deletion occlusion and random pixel depletion as an illustration the robustness of two shape recognition algorithms to contour incompleteness is evaluated these algorithms use a shape context and a distance multiset as local shape descriptors qualitatively both algorithms mimic human visual perception in the sense that recognition performance monotonously increases with the degree of completeness and that they perform best in the case of random depletion and worst in the case of occluded contours the distance multiset method performs better than the shape context method in this test framework

multiscale signal enhancement beyond the normality and independence assumption current approaches to denoising or signal enhancement in a wavelet based framework have generally relied on the assumption of normally distributed perturbations in practice this assumption is often violated and sometimes prior information of the probability distribution of a noise process is not even available to relax this assumption we propose a novel nonlinear filtering technique in this paper the key idea is to project a noisy signal onto a wavelet domain and to suppress wavelet coefficients by a mask derived from curvature extrema in its scale space representation for a piecewise smooth signal it can be shown that filtering by this curvature mask is equivalent to preserving the signal pointwise holder exponents at the singular points and lifting its smoothness elsewhere

hierarchical string cuts a translation rotation scale and mirror invariant descriptor for fast shape retrieval this paper presents a novel approach for both fast and accurately retrieving similar shapes a hierarchical string cuts hscs method is proposed to partition a shape into multiple level curve segments of different lengths from a point moving around the contour to describe the shape gradually and completely from the global information to the finest details at each hierarchical level the curve segments are cut by strings to extract features that characterize the geometric and distribution properties in that particular level of details the translation rotation scale and mirror invariant hsc descriptor enables a fast metric based matching to achieve the desired high accuracy encouraging experimental results on four databases demonstrated that the proposed method can consistently achieve higher or similar retrieval accuracies than the state of the art benchmarks with a more than 120 times faster speed this may suggest a new way of developing shape retrieval techniques in which a high accuracy can be achieved by a fast metric matching algorithm without using the time consuming correspondence optimization strategy

shape error concealment based on a shape preserving boundary approximation in object based video representation video scenes are composed of several arbitrarily shaped video objects vos defined by their texture shape and motion in error prone communications packet loss results in missing information at the decoder the impact of transmission errors is minimized through error concealment in this paper we propose a spatial error concealment technique for recovering lost shape data we consider a geometric shape representation consisting of the object boundary which can be extracted from the plane missing macroblocks result in a broken boundary a b spline curve is constructed to replace a missing boundary segment based on a t spline representation of the received boundary we use t splines because they produce shape preserving approximations and do not change the characteristics of the original boundary the representation ensures a good estimation of the first derivatives at the points touching the missing segment applying smoothing conditions we manage to construct a new spline that joins smoothly with the received boundary leading to successful concealment results experimental results on object shapes with different concealment difficulty demonstrate the performance of the proposed method comparisons with prior proposed methods are also presented

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

a new approach to spectral approximation a new technique for approximating eigenvalues and eigenvectors of a self adjoint operator is presented the method does not incur spectral pollution uses trial spaces from the form domain has a self adjoint algorithm and exhibits superconvergence

viewpoint independent matching of planar curves in 3d space we present a new approach to match planar curves using the weak perspective projection model this is based on a set of shape parameters that can be extracted from a closed or open contour derived from the original image as a s boundary code in order to reduce the complexity and increase the robustness ofthe matching process the original parameters are reduced to a set ofthree intermediate variables each ofwhich can be calculated independently these variables are contained within a system oflinear equations which de4ne the angles and the ratio ofthe heights ofcorresponding point pairs on the two contours with respect to a 5oating coordinate system the shape matching process is scale and orientation independent and the original parameters that describe the relative pose ofthe two contours in 3d space can be recovered subsequently the approach can be applied to featured and featureless contours to whole and partial contours and is demonstrated on images ofcontours and mechanical parts and tools to recover identity and pose

analysis of singularities from modulus maxima of complex wavelets complex valued wavelets are normally used to measure instantaneous frequencies while real wavelets are normally used to detect singularities we prove that the wavelet modulus maxima with a complex valued wavelet can detect and characterize singularities this is an extension of the previous wavelet work of mallat and hwang on modulus maxima using a real wavelet with this extension we can simultaneously detect instantaneous frequencies and singularities from the wavelet modulus maxima of a complex valued wavelet some results of singularity detection with the modulus maxima from a real wavelet and an analytic complex valued wavelet are compared we also demonstrate that singularity detection methods can be employed to detect the corners of a planar object

a robust method for scale independent detection of curvature based criticalities and intersections in line drawings a novel two phase iterative method is proposed to identify curvature based criticalities and intersections in line drawings the method is based on evaluating a field obtained from the image via diffusion using adaptively changing ellipse shaped analysis windows the deviation of the average field strength values within the analysis windows from that of an external reference field is used as a metric to evaluate the criticality of a region the external reference field is computed from an image of a straight line the experimental results depict that the method is effective in detecting curvature based criticalities and intersections even for noisy and disconnected drawings as well as drawings drawn with a variety of brush characteristics such as glass ocean ripple scatter stamp strokes effects our method can be employed as a part of a design grammar interpreter highlightsthe method detects curvature based criticalities and intersections efficiently the method is effective for the noisy and disconnected line drawings the method is effective for line drawings with a variety of brush characteristics the method is utilized to represent shapes in a design grammar interpreter features in varying scales can be detected in digital drawings

estimating derivatives and curvature of open curves this article presents an effective spectral approach to estimate derivatives and curvature of open parametric curves as the method is based on the discrete fourier transform the discontinuities of the curve as well as of its derivatives must be controlled to minimize the gibbs phenomenon we address this problem by obtaining a smooth extension of the curve in such a way as to suitably close it which is done through a variational approach taking into account the spectral energy of differentiated versions of the extended curves this novel method presents potential for applications in a broad class of problems ranging from applied and experimental physics to image analysis

remark on algorithms to find roots of polynomials the problem of finding the roots of a polynomial is equivalent to finding the eigenvalues of an upper hessenberg matrix which can be done with the qr algorithm it is shown that the qr algorithm has considerable advantages over other standard algorithms to find the roots of a polynomial

learning affine transformations under the assumption of weak perspective two views of the same planar object are related through an affine transformation in this paper we consider the problem of training a simple neural network to learn to predict the parameters of the affine transformation although the proposed scheme has similarities with other neural network schemes its practical advantages are more profound first of all the views used to train the neural network are not obtained by taking pictures of the object from different viewpoints instead the training views are obtained by sampling the space of affine transformed views of the object this space is constructed using a single view of the object fundamental to this procedure is a methodology based on singular value decomposition svd and interval arithmetic ia for estimating the ranges of values that the parameters of affine transformation can assume second the accuracy of the proposed scheme is very close to that of a traditional least squares approach with slightly better space and time requirements a front end stage to the neural network based on principal components analysis pca shows to increase its noise tolerance dramatically and also to guides us in deciding how many training views are necessary in order for the network to learn a good noise tolerant mapping the proposed approach has been tested using both artificial and real data

towards effective planar shape representation with multiscale digital curvature analysis based on signal processing techniques abstract this paper presents and discusses a new approach to multiscale curvature analysis of digital contours that is based on digital signal processing techniques the shape contour is expressed in terms of two one dimensional signals linearized x and y coordinates and the derivative theorem is applied as a means of obtaining an interesting expression relating contour curvature and the spectra of its parametrized x and y signals multiscale curvature analysis is achieved through gaussian lowpass filtering and the shrinkage of the original signals implied by such a process is effectively circumvented by an energy based compensation scheme which has allowed accurate quantitative identification of the curvature value the concept of curvegram is introduced and exemplified and the overall performance of the proposed technique for curvature estimation is formally assessed with respect to a contour defined in terms of b splines application examples to synthetic and real images have been included

wavelet based approach to character skeleton character skeleton plays a significant role in character recognition the strokes of a character may consist of two regions i e singular and regular regions the intersections and junctions of the strokes belong to singular region while the straight and smooth parts of the strokes are categorized to regular region therefore a skeletonization method requires two different processes to treat the skeletons in theses two different regions all traditional skeletonization algorithms are based on the symmetry analysis technique the major problems of these methods are as follows 1 the computation of the primary skeleton in the regular region is indirect so that its implementation is sophisticated and costly 2 the extracted skeleton cannot be exactly located on the central line of the stroke 3 the captured skeleton in the singular region may be distorted by artifacts and branches to overcome these problems a novel scheme of extracting the skeleton of character based on wavelet transform is presented in this paper this scheme consists of two main steps namely a extraction of primary skeleton in the regular region and b amendment processing of the primary skeletons and connection of them in the singular region a direct technique is used in the first step where a new wavelet based symmetry analysis is developed for finding the central line of the stroke directly a novel method called smooth interpolation is designed in the second step where a smooth operation is applied to the primary skeleton and thereafter the interpolation compensation technique is proposed to link the primary skeleton so that the skeleton in the singular region can be produced experiments are conducted and positive results are achieved which show that the proposed skeletonization scheme is applicable to not only binary image but also gray level image and the skeleton is robust against noise and affine transform

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

shape matching using multiscale integral invariants we present a shape descriptor based on integral kernels shape is represented in an implicit form and it is characterized by a series of isotropic kernels that provide desirable invariance properties the shape features are characterized at multiple scales which form a signature that is a compact description of shape over a range of scales the shape signature is designed to be invariant with respect to group transformations which include translation rotation scaling and reflection in addition the integral kernels that characterize local shape geometry enable the shape signature to be robust with respect to undesirable perturbations while retaining discriminative power use of our shape signature is demonstrated for shape matching based on a number of synthetic and real examples

an approach to the parameterization of structure for fast categorization a decomposition is described which parameterizes the geometry and appearance of contours and regions of gray scale images with the goal of fast categorization to express the contour geometry a contour is transformed into a local global space from which parameters are derived classifying its global geometry arc inflexion or alternating and describing its local aspects degree of curvature edginess symmetry regions are parameterized based on their symmetric axes which are evolved with a wave propagation process enabling to generate the distance map for fragmented contour images the methodology is evaluated on three image sets the caltech 101 set and two sets drawn from the corel collection the performance nearly reaches the one of other categorization systems for unsupervised learning

the second order spectrum and optimal convergence the method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self adjoint operator we extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces the convergence to eigenspaces is new while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude

logarithmic hodge witt sheaves on normal crossing varieties in this paper we define two kinds homological and cohomological of etale logarithmic hodge witt sheaves on normal crossing varieties over a perfect field of positive characteristic and discuss some fundamental properties in particular puity and duality

multiscale curvature based shape representation using b spline wavelets this paper presents a new multiscale curvature based shape representation technique with application to curve data compression using b spline wavelets the evolution of the curve is implemented in the b spline scale space which enjoys a number of advantages over the classical gaussian scale space for instance the availability of fast algorithms the b spline wavelet transforms are used to efficiently estimate the multiscale curvature functions based on the curvature scale space image we introduce a coarse to fine matching algorithm which automatically detects the dominant points and uses them as knots for curve interpolation

a novel rotation scale invariant template matching algorithm using weighted adaptive lifting scheme transform this paper presents a novel algorithm for detecting user selected objects in given test images based on a new adaptive lifting scheme transform given an object as a template we first select a set of coefficients as object features in the wavelet transform domain and then build an adaptive transform based on the selected features the goal of the new adaptive transform is to vanish the selected features in the transform domain after applying both non adaptive and adaptive transforms to a given test image the corresponding transform domain coefficients are compared for detecting the object of interest in addition the proposed detection algorithm is combined with the proper log polar mapping model in the parametric template space to attain rotation scale invariance property finally we have verified the properties of our proposed algorithm with experimental results

statistical bias of conic fitting and renormalization introducing a statistical model of noise in terms of the covariance matrix of the n vector we point out that the least squares conic fitting is statistically biased we present a new fitting scheme called renormalization for computing an unbiased estimate by automatically adjusting to noise relationships to existing methods are discussed and our method is tested using real and synthetic data

using geometric distance fits for 3 d object modeling and recognition addresses the problems of automatically constructing algebraic surface models from sets of 2d and 3d images and using these models in pose computation motion and deformation estimation and object recognition we propose using a combination of constrained optimization and nonlinear least squares estimation techniques to minimize the mean squared geometric distance between a set of points or rays and a parameterized surface in modeling tasks the unknown parameters are the surface coefficients while in pose and deformation estimation tasks they represent the transformation which maps the observer s coordinate system onto the modeled surface s own coordinate system we have applied this approach to a variety of real range computerized tomography and video images

repeatedly smooting discrete scale space evolution and dominant point detection in this paper the fourier analysis is used to derive the properties of an evolving curve an arbitrary kernel repeatedly smoothing akrs evolution of a curve is then introduced it is shown that when the repeated number is large the akrs evolution of a curve is an approximately discrete implementation of the scale based evolution of this curve in the euclidean space as a special case an exponential repeatedly smoothing is proposed to implement the scale based evolution it is shown that in addition to its simple implementation and its desired approximation to a gaussian kernel an exponential function is a function that when it is selected as a repeatedly smoothing kernel the motion both magnitude and direction of a point on a curve from the i 1 th to ith instant is equal to the curvature of the ith smoothed curve at this point finally a perimeter controlled evolution method is proposed to extract dominant points it is shown experimentally that the proposed method is robust to noise object rotation and object changes in sizes

silhouette based isolated object recognition through curvature scale space a complete fast and practical isolated object recognition system has been developed which is very robust with respect to scale position and orientation changes of the objects as well as noise and local deformations of shape due to perspective projection segmentation errors and non rigid material used in some objects the system has been tested on a wide variety of three dimensional objects with different shapes and material and surface properties a light box setup is used to obtain silhouette images which are segmented to obtain the physical boundaries of the objects which are classified as either convex or concave convex curves are recognized using their four high scale curvature extrema points curvature scale space css representations are computed for concave curves the css representation is a multi scale organization of the natural invariant features of a curve curvature zero crossings or extrema and useful for very reliable recognition of the correct model since it places no constraints on the shape of objects a three stage coarse to fine matching algorithm prunes the search space in stage one by applying the css aspect ratio test the maxima of contours in css representations of the surviving models are used for fast css matching in stage two finally stage three verifies the best match and resolves any ambiguities by determining the distance between the image and model curves transformation parameter optimization is then used to find the best fit of the input object to the correct model

wavelet approximation based affine invariant shape representation functions in this paper new wavelet based affine invariant functions for shape representation are presented unlike the previous representation functions only the approximation coefficients are used to obtain the proposed functions one of the derived functions is computed by applying a single wavelet transform the other function is calculated by applying two different wavelet transforms with two different wavelet families one drawback of the previously derived detail based invariant representation functions is that they are sensitive to noise at the finer scale levels which limits the number of scale levels that can be used the experimental results in this paper demonstrate that the proposed functions are more stable and less sensitive to noise than the detail based functions

faithful polygonal representation of the convex and concave parts of a digital curve from results about digital convexity we define a reversible polygon that faithfully represents the maximal convex and concave parts of a digital curve such a polygon always exists and is unique in the general case it is computed from a given digital curve in linear time using well known routines adding a point at the front of a digital straight segment and removing a point from the back of a digital straight segment it may helps to extract perceptually meaningful parts of shape outlines or lines

polygonal approximation of digital planar curves through break point suppression this paper presents a new algorithm that detects a set of dominant points on the boundary of an eight connected shape to obtain a polygonal approximation of the shape itself the set of dominant points is obtained from the original break points of the initial boundary where the integral square is zero for this goal most of the original break points are deleted by suppressing those whose perpendicular distance to an approximating straight line is lower than a variable threshold value the proposed algorithm iteratively deletes redundant break points until the required approximation which relies on a decrease in the length of the contour and the highest error is achieved a comparative experiment with another commonly used algorithm showed that the proposed method produced efficient and effective polygonal approximations for digital planar curves with features of several sizes

integral invariants for space motion trajectory matching and recognition motion trajectories provide a key and informative clue in motion characterization of humans robots and moving objects in this paper we propose some new integral invariants for space motion trajectories which benefit effective motion trajectory matching and recognition integral invariants are defined as the line integrals of a class of kernel functions along a motion trajectory a robust estimation of the integral invariants is formulated based on the blurred segment of noisy discrete curve then a non linear distance of the integral invariants is defined to measure the similarity for trajectory matching and recognition such integral invariants in addition to being invariant to transformation groups have some desirable properties such as noise insensitivity computational locality and uniqueness of representation experimental results on trajectory matching and sign recognition show the effectiveness and robustness of the proposed integral invariants in motion trajectory matching and recognition a meaningful definition of integral invariants for space trajectories is proposed we derive a novel estimation method of integral invariants for discrete trajectories a non linear distance is defined to measure the similarity between trajectories integral invariants show robustness to noise occlusions and transformations experiments are carried out to examine their effectiveness and robustness in applications

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

matching shapes with self intersections application to leaf classification we address the problem of two dimensional 2 d shape representation and matching in presence of self intersection for large image databases this may occur when part of an object is hidden behind another part and results in a darker section in the gray level image of the object the boundary contour of the object must include the boundary of this part which is entirely inside the outline of the object the curvature scale space css image of a shape is a multiscale organization of its inflection points as it is smoothed the css based shape representation method has been selected for mpeg 7 standardization we study the effects of contour self intersection on the curvature scale space image when there is no self intersection the css image contains several arch shape contours each related to a concavity or a convexity of the shape self intersections create contours with minima as well as maxima in the css image an efficient shape representation method has been introduced in this paper which describes a shape using the maxima as well as the minima of its css contours this is a natural generalization of the conventional method which only includes the maxima of the css image contours the conventional matching algorithm has also been modified to accommodate the new information about the minima the method has been successfully used in a real world application to find for an unknown leaf similar classes from a database of classified leaf images representing different varieties of chrysanthemum for many classes of leaves self intersection is inevitable during the scanning of the image therefore the original contributions of this paper is the generalization of the curvature scale space representation to the class of 2 d contours with self intersection and its application to the classification of chrysanthemum leaves

c algebras and numerical linear algebra given a self adjoint operator a on a hilbert space suppose that one wishes to compute the spectrum of a numerically in practice these problems often arise in such a way that the matrix of a relative to a natural basis is sparse for example discretized second order differential operators can be represented by doubly infinite tridiagonal matrices in these cases it is easy and natural to compute the eigenvalues of large n n submatrices of the infinite operator matrix and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of a numerical analysts call this the galerkin method while this hope is often realized in practice it often fails as well and it can fail in spectacular ways the sequence of eigenvalue distributions may not converge as n or they may converge to something that has little to do with the original operator a at another level even the meaning of convergence has not been made precise in general in this paper we determine the proper general setting in which one can expect convergence and we describe the asymptotic behavior of the n n eigenvalue distributions in all but the most pathological cases under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of a is recovered from the sequence of eigenvalues of the n n compressions in broader terms our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms ofc algebras indeed it is only when the single operator a is viewed as an element of an appropriate c algebra a that one can see the precise nature of the limit of the n n eigenvalue distributions the limit is associated with a tracial state on a normally a is highly noncommutative and in our main applications it is a simple c algebra having a unique tracial state we obtain precise asymptotic results for operators which represent discretized hamiltonians of one dimensional quantum systems with arbitrary continuous potentials

bayesian multiscale modeling of closed curves in point clouds modeling object boundaries based on image or point cloud data is frequently necessary in medical and scientific applications ranging from detecting tumor contours for targeted radiation therapy to the classification of organisms based on their structural information in low contrast images or sparse and noisy point clouds there is often insufficient data to recover local segments of the boundary in isolation thus it becomes critical to model the entire boundary in the form of a closed curve to achieve this we develop a bayesian hierarchical model that expresses highly diverse 2d objects in the form of closed curves the model is based on a novel multiscale deformation process by relating multiple objects through a hierarchical formulation we can successfully recover missing boundaries by borrowing structural information from similar objects at the appropriate scale furthermore the model s latent parameters help interpret the population indicating dimensions of significant structural variability

a statistical model for signature verification a bayesian model for off line signature verification involving the representation of a signature through its curvature is developed the prior model makes use of a spatial point process for specifying the knots in an approximation restricted to a buffer region close to a template curvature along with an independent time warping mechanism in this way prior shape information about the signature can be built into the analysis the observation model is based on additive white noise superimposed on the underlying curvature the approach is implemented using markov chain monte carlo and applied to a collection of documented instances of william shakespeare s signature

the multiresolution gradient vector field skeleton many algorithms suppress skeleton associated with boundary perturbation by preventing their formation or by costly branch pruning this work proposes a novel concept of structural and textural skeletons the former is associated with the general shape structure and the latter with boundary perturbations these skeletons remain disconnected to facilitate gross shape matching without the need for branch pruning they are extracted from a multiresolution gradient vector field that is efficiently generated within a pyramidal framework experimental results show that these skeletons are scale and rotation invariant they are less affected by boundary noise compared to skeletons extracted by popular iterative and non iterative techniques

a unified curvature definition for regular polygonal and digital planar curves in this paper we propose a new definition of curvature called visual curvature it is based on statistics of the extreme points of the height functions computed over all directions by gradually ignoring relatively small heights a multi scale curvature is obtained the theoretical properties and the experiments presented demonstrate that multi scale visual curvature is stable even in the presence of significant noise to our best knowledge the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves moreover it yields stable curvature estimates of curves in digital images even under sever distortions we also show a relation between multi scale visual curvature and convexity of simple closed curves

shape representation and recognition through morphological curvature scale spaces a multiscale morphological method for the purpose of shape based object recognition is presented a connected operator similar to the morphological hat transform is defined and two scale space representations are built using the curvature function as the underlying one dimensional signal each peak and valley of the curvature is extracted and described by its maximum and average heights and by its extent and represents an entry in the top or bottom hat transform scale spaces we demonstrate object recognition based on hat transform scale spaces for three large data sets a set of diatom contours the set of silhouettes from the mpeg 7 database and the set of two dimensional views of three dimensional objects from the coil 20 database our approach outperforms other methods for which comparative results exist

shape similarity retrieval under affine transforms the maxima of curvature scale space css image have already been used to represent 2 d shapes in different applications the representation has shown robustness under the similarity transformations scaling orientation changes translation and even noise can be easily handled by the representation and its associated matching algorithm in this paper we examine the robustness of the representation under general affine transforms we have a database of 1100 images of marine creatures the contours in this database demonstrate a great range of shape variation a database of 5000 contours has been constructed using 500 real object boundaries and 4500 contours which are the affine transformed versions of real objects the css representation is then used to find similar shapes from this prototype database the results provide substantial evidence of stability of the css image and its contour maxima under affine transformation the method is also evaluated objectively through a large classified database and its performance is compared with the performance of two well known methods namely fourier descriptors and moment invariants the css shape descriptor has been selected for mpeg 7 standardization

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

on the convergence of planar curves under smoothing curve smoothing has two important applications in computer vision and image processing 1 the curvature scale space css technique for shape analysis and 2 the gaussian filter for noise suppression in this paper we study how planar curves converge as they are smoothed with increasing scales first two types of convergence behavior are clarified the coined term shrinkage refers to the reduction of arc length of a smoothed planar curve which describes the convergence of the curve latitudinally and another coined term collapse refers to the movement of each point to its limiting position which describes the convergence of the curve longitudinally a systematic study on the shrinkage and collapse of three categories of curve models is then presented the corner models helps to reveal how the local structures of planar curves collapse and what the smoothed curves may converge to the sawtooth models allows us to gain insights regarding how noise is suppressed from noisy planar curves by the gaussian filter our investigation on the closed curves shows that each curve collapses to a point at its center of mass however different curves may yield different limiting shapes at the infinity scale finally based upon the derived results the performance of the css technique in corner detection and shape representation is analyzed and a fast implementation method of the gaussian filter for noise suppression is proposed

parameterized families of polynomials for bounded algebraic curve and surface fitting interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for different model based computer vision tasks has increased in recent years and although their properties make them a natural choice for object recognition and positioning applications algebraic curve and surface fitting algorithms often suffer from instability problems one of the main reasons for these problems is that while the data sets are always bounded the resulting algebraic curves or surfaces are in most cases unbounded in this paper the authors propose to constrain the polynomials to a family with bounded zero sets and use only members of this family in the fitting process for every even number d the authors introduce a new parameterized family of polynomials of degree d whose level sets are always bounded in particular its zero sets this family has the same number of degrees of freedom as a general polynomial of the same degree three methods for fitting members of this polynomial family to measured data points are introduced experimental results of fitting curves to sets of points in r sup 2 and surfaces to sets of points in r sup 3 are presented

spectral pollution and how to avoid it this paper devoted to the study of spectral pollution contains both abstract results and applications to some self adjoint operators with a gap in their essential spectrum occuring in quantum mechanics first we consider galerkin basis which respect the decomposition of the ambient hilbert space into a direct sum h ph oplus 1 p h given by a fixed orthogonal projector p and we localize the polluted spectrum exactly this is followed by applications to periodic schrodinger operators pollution is absent in a wannier type basis and to dirac operator several natural decompositions are considered in the second part we add the constraint that within the galerkin basis there is a certain relation between vectors in ph and vectors in 1 p h abstract results are proved and applied to several practical methods like the famous kinetic balance of relativistic quantum mechanics

using dominant points and variants for profile extraction from chromosomes abstract the most accurate methods for automatic analysis of chromosomes under a light microscope today extract numerical features from band pattern profiles along their longitudinal axes the construction of a reliable axis is a crucial step in this process we propose a new way based on the dominant points of the contour and cubic splines the dominant points serve as candidates for the tips of the chromosome ambiguities are dissolved by the recently developed method of variants for object identification a voronoi diagram decomposes the chromosome in slices for profile extraction the method improves the currently best classification results significantly yielding a test set error rate of 0 61

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

optimal estimation of contour properties by cross validated regularization the problem of estimating the properties of smooth continuous contours from discrete noisy samples is used as vehicle to demonstrate the robustness of cross validated regularization applied to a vision problem a method for estimation of contour properties based on smoothing spline approximations is presented generalized cross validation is to devise an automatic algorithm for finding the optimal value of the smoothing regularization parameter from the data the cross validated smoothing splines are then used to obtain optimal estimates of the derivatives of quantized contours experimental results are presented which demonstrate the robustness of the method applied to the estimation of curvature of quantized contours under variable scale rotation and partial occlusion these results suggest the application of generalized cross validation to other computer vision algorithms involving regularization

limiting set of second order spectra let m be a self adjoint operator acting on a hilbert space h a complex number z is in the second order spectrum of m relative to a finite dimensional subspace l c dom m 2 iff the truncation to l of m z 2 is not invertible this definition was first introduced in davies 1998 and according to the results of levin and shargorodsky in 2004 these sets provide a method for estimating eigenvalues free from the problems of spectral pollution in this paper we investigate various aspects related to the issue of approximation using second order spectra our main result shows that under fairly mild hypothesis on m the uniform limit of these sets as l increases towards h contain the isolated eigenvalues of m of finite multiplicity therefore unlike the majority of the standard methods second order spectra combine nonpollution and approximation at a very high level of generality

on pseudospectra of matrix polynomials and their boundaries in the first part of this paper sections 2 4 the main concern is with the boundary of the pseudospectrum of a matrix polynomial and particularly with smoothness properties of the boundary in the second part sections 5 6 results are obtained concerning the number of connected components of pseudospectra as well as results concerning matrix polynomials with multiple eigenvalues or the proximity to such polynomials

estimation of planar curves surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation the author addresses the problem of parametric representation and estimation of complex planar curves in 2 d surfaces in 3 d and nonplanar space curves in 3 d curves and surfaces can be defined either parametrically or implicitly with the latter representation used here a planar curve is the set of zeros of a smooth function of two variables x y a surface is the set of zeros of a smooth function of three variables x y z and a space curve is the intersection of two surfaces which are the set of zeros of two linearly independent smooth functions of three variables x y z for example the surface of a complex object in 3 d can be represented as a subset of a single implicit surface with similar results for planar and space curves it is shown how this unified representation can be used for object recognition object position estimation and segmentation of objects into meaningful subobjects that is the detection of interest regions that are more complex than high curvature regions and hence more useful as features for object recognition

on the convergence of second order spectra and multiplicity the notion of second order relative spectrum of a self adjoint operator acting on a hilbert space has been studied recently in connection with the phenomenon of spectral pollution in the galerkin method in this paper we examine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of the underlying operator our theoretical findings are supported by various numerical experiments on the computation of guaranteed eigenvalue inclusions via finite element bases

symmetry as a continuous feature symmetry is treated as a continuous feature and a continuous measure of distance from symmetry in shapes is defined the symmetry distance sd of a shape is defined to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape this general definition of a symmetry measure enables a comparison of the amount of symmetry of different shapes and the amount of different symmetries of a single shape this measure is applicable to any type of symmetry in any dimension the symmetry distance gives rise to a method of reconstructing symmetry of occluded shapes the authors extend the method to deal with symmetries of noisy and fuzzy data finally the authors consider grayscale images as 3d shapes and use the symmetry distance to find the orientation of symmetric objects from their images and to find locally symmetric regions in images

skeleton pruning by contour partitioning with discrete curve evolution in this paper we introduce a new skeleton pruning method based on contour partitioning any contour partition can be used but the partitions obtained by discrete curve evolution dce yield excellent results the theoretical properties and the experiments presented demonstrate that obtained skeletons are in accord with human visual perception and stable even in the presence of significant noise and shape variations and have the same topology as the original skeletons in particular we have proven that the proposed approach never produces spurious branches which are common when using the known skeleton pruning methods moreover the proposed pruning method does not displace the skeleton points consequently all skeleton points are centers of maximal disks again many existing methods displace skeleton points in order to produces pruned skeletons

on chow and brauer groups of a product of mumford curves let c1 cd be mumford curves defined over a finite extension of open image in new window and let x c1 cd we shall show the following 1 the cycle map ch0 x n h2d x n d is injective for any non zero integer n 2 the kernel of the canonical map ch0 x hom br x open image in new window defined by the brauer manin pairing coincides with the maximal divisible subgroup in ch0 x

the generalized uniqueness wavelet descriptor for planar closed curves in the problem of specifying a well defined wavelet description of a planar closed curve defining a unique start point on the curve is crucial for wavelet representation in this paper a generalized uniqueness property inherent in the one dimensional 1 d discrete periodized wavelet transformation dpwt is derived the uniqueness property facilitates a quantitative analysis of the one to one mapping between the variation of 1 d dpwt coefficients and the starting point shift of the originally sampled curve data by employing the uniqueness property a new shape descriptor called the uniqueness wavelet descriptor uwd by which the starting point is fixed entirely within the context of the wavelet representation is proposed the robustness of the uwd against input noise is analyzed on the basis of local shape characteristic enhancement several experiments were conducted to illustrate the adaptability property of the uwd for desirable starting point determination our experiments of pattern recognition show that the uwd can provide a supervised pattern classifier with optimal features to obtain the best matching performance in the presence of heavy noise in addition the generalized uniqueness property can be used for the shape regularity measurement the uwd does not have local support and therefore it can not be applied to contour segments

2 d object recognition using invariant contour descriptor and projective refinement abstract this paper presents an efficient model based recognition method to recognize 2 d objects and to obtain correspondences between models and scene boundaries with a subpixel positioning error as a shape signature for a contour we propose a descriptor consisting of five point invariants which are used to index a hash table also we propose a projective refinement as a verification method to compute exact correspondences between models and scene contour points this method repeatedly computes projective transformation using a weighted pseudo inverse we present an error model for five point invariants which are used to define a similarity between two descriptors to determine a searching bound in indexing and to obtain the weights in the projective refinement in experiments using seriously distorted images of forty models this method led to the recognition of planar curved objects a transformation using the correspondence between the model and scene points on contours was also obtained

on the approximation of spectra of linear operators on hilbert spaces abstract we present several new techniques for approximating spectra of linear operators not necessarily bounded on an infinite dimensional separable hilbert space our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized to an infinite dimensional setting to provide approximations of spectra of elements in a large class of operators we conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n pseudospectrum and argue how that can be used as an approximation to the spectrum

on spectral pollution in the finite element approximation of thin elastic membrane shells the bending terms in a shell are small with respect to membrane ones as the thickness tends to zero consequently the membrane approximation gives a good description of vibration properties of a thin shell this vibration problem is associated with a non compact resolvent operator and spectral pollution could appear when computing galerkin approximations that is to say there could exist sequences of eigenvalues of the approximate problems that converge to points of the resolvent set of the exact problem we give an account of the state of the art of this problem in shell theory a description of the phenomenon and its interpretation in terms of spectral families are given a theorem of localization of the region where pollution may appear is stated and its complete proof is published for the first time recipes are given for avoiding the pollution as well as indications on the possibility of a posteriori elimination

a theory of multiscale curvature based shape representation for planar curves a shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented the method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve three different ways of computing the representation are described they result in three different representations the curvature scale space image the renormalized curvature scale space image and the resampled curvature scale space image the process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve several evolution and arc length evolution properties of planar curves are discussed

scale based description and recognition of planar curves and two dimensional shapes the problem of finding a description at varying levels of detail for planar curves and matching two such descriptions is posed and solved in this paper a number of necessary criteria are imposed on any candidate solution method path based gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail the result is the generalized scale space image of a planar curve which is invariant under rotation uniform scaling and translation of the curve these properties make the scale space image suitable for matching the matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images it is argued that this is preferable to matching in a so called stable scale of the curve because no such scale may exist for a given curve this technique is applied to register a landsat satellite image of the strait of georgia b c manually corrected for skew to a map containing the shorelines of an overlapping area