skeleton search category specific object recognition and segmentation using a skeletal shape model we describe a top down object detection and segmentation approach that uses a skeleton based shape model and that works directly on real images the approach is based on three components first we propose a fragment based generative model for shape that is based on the shock graph and has minimal dependency among its shape fragments the model is capable of generating a wide variation of shapes as instances of a given object category second we develop a progressive selection mechanism to search among the generated shapes for the category instances that are present in the image the search begins with a large pool of candidates identified by a dynamic programming dp algorithm and progressively reduces it in size by applying series of criteria namely local minimum criterion extent of shape overlap and thresholding of the objective function to select the final object candidates third we propose the partitioned chamfer matching pcm measure to capture the support of image edges for a hypothesized shape this measure overcomes the shortcomings of the oriented chamfer matching and is robust against spurious edges missing edges and accidental alignment between the image edges and the shape boundary contour we have evaluated our approach on the ethz dataset and found it to perform well in both object detection and object segmentation tasks

correcting curvature density effects in the hamilton jacobi skeleton the hamilton jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two dimensional 2 d shapes the approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero while the flux is negative at the locations of skeletal points nonetheless the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and hence unbounded near endpoints this makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter this problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision however here we develop a method for 2 d skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error this is done by taking into account variations of density due to boundary curvature this yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the hamilton jacobi method

discovering shape classes using tree edit distance and pairwise clustering this paper describes work aimed at the unsupervised learning of shape classes from shock trees we commence by considering how to compute the edit distance between weighted trees we show how to transform the tree edit distance problem into a series of maximum weight clique problems and show how to use relaxation labeling to find an approximate solution this allows us to compute a set of pairwise distances between graph structures we show how the edit distances can be used to compute a matrix of pairwise affinities using 2 statistics we present a maximum likelihood method for clustering the graphs by iteratively updating the elements of the affinity matrix this involves interleaved steps for updating the affinity matrix using an eigendecomposition method and updating the cluster membership indicators we illustrate the new tree clustering framework on shock graphs extracted from the silhouettes of 2d shapes

approximate vibration reanalysis of structures thecombinedapproximationsmethodisdevelopedforvibrationreanalysisofstructures inthesolution process the terms of local series approximations are used as basis vectors in global reduced basis approximations the solution steps are straightforward and the method can be readily used with a general e nite element system the computational effort needed by the method is usually much smaller than the effort needed for complete vibration analysis the efe ciency of the calculations and the accuracy of the results can be controlled by the amount of information considered efe cient solutions are obtained by low order approximations and accurate results can be achievedbyconsideringhigher orderterms numericalexamplesshowthataccurateresultsareachievedefe ciently for very large changes in the design nomenclature b matrix dee ned by eq 21 c matrix dee ned by eq 16 e modulus of elasticity i moment of inertia i identity matrix k stiffness matrix kr reduced stiffness matrix l length m mass matrix mr reduced mass matrix r right hand side vector r displacement vector rb matrix of basis vectors s number of basis vectors u upper triangular matrix y vector of coefe cients coefe cient dee ned by eq 35 angle between vectors 1k change in the stiffness matrix 1m change in the mass matrix kronecker delta eigenvalue circular frequency

graph embedding using tree edit union in this paper we address the problem of how to learn a structural prototype that can be used to represent the variations present in a set of trees the prototype serves as a pattern space representation for the set of trees to do this we construct a super tree to span the union of the set of trees this is a chicken and egg problem since before the structure can be estimated correspondences between the nodes of the super tree and the nodes of the sample tree must be to hand we demonstrate how to simultaneously estimate the structure of the super tree and recover the required correspondences by minimizing the sum of the tree edit distances over pairs of trees subject to edge consistency constraints each node of the super tree corresponds to a dimension of the pattern space and for each tree we construct a pattern vector in which the elements of the weights corresponding to each of the dimensions of the super tree we perform pattern analysis on the set of trees by performing principal components analysis on the vectors the method is illustrated on a shape analysis problem involving shock trees extracted from the skeletons of 2d objects

shape matching using a binary search tree structure of weak classifiers in this paper a novel algorithm for shape matching based on the hausdorff distance and a binary search tree data structure is proposed the shapes are stored in a binary search tree that can be traversed according to a hausdorff like similarity measure that allows us to make routing decisions at any given internal node each node functions as a classifier that can be trained using supervised learning these node classifiers are very similar to perceptrons and can be trained by formulating a probabilistic criterion for the expected performance of the classifier then maximizing that criterion methods for node insertion and deletion are also available so that a tree can be dynamically updated while offline training is time consuming all online training and both online and offline testing operations can be performed in o logn time experimental results on pedestrian detection indicate the efficiency of the proposed method in shape matching

efficient accurate reanalysis for structural optimization the combined approximations method developed recently is an efficient reanalysis method providing high quality results through the use of this approach the computed terms of a series expansion are used as basis vectors in a reduced basis expression by solving a reduced system of equations first and second order approximations were demonstrated in previous studies for small structures the efficiency and the accuracy of the method are improved and results are illustrated for larger structures by the utilization of a gram schmidt orthogonalization procedure new set of basis vectors is generated and normalized such that the reduced system of equations becomes uncoupled the advantage in using the latter vectors is that all expressions for evaluating the displacements are explicit functions of the design variables consequently additional vectors can be considered without modifying the calculations that have already been carried out in addition the uncoupled system is more well conditioned some considerations related to the efficiency of the solution process and the accuracy of the results are discussed and the effect of various parameters on the accuracy is studied numerical results are demonstrated for several medium and large scale structures it is shown that accurate and efficient approximations are achieved for very large changes in the design

bag of contour fragments for robust shape classification shape representation is a fundamental problem in computer vision current approaches to shape representation mainly focus on designing low level shape descriptors which are robust to rotation scaling and deformation of shapes in this paper we focus on mid level modeling of shape representation we develop a new shape representation called bag of contour fragments bcf inspired by classical bag of words bow model in bcf a shape is decomposed into contour fragments each of which is then individually described using a shape descriptor e g the shape context descriptor and encoded into a shape code finally a compact shape representation is built by pooling shape codes in the shape shape classification with bcf only requires an efficient linear svm classifier in our experiments we fully study the characteristics of bcf show that bcf achieves the state of the art performance on several well known shape benchmarks and can be applied to real image classification problem highlightsa new shape representation is proposed by encoding contour fragments in shape the proposed shape representation is compact yet informative the proposed shape representation is robust to shape deformation and conclusion we obtain the state of the art shape classification performance on several bench mark datasets

skeletal shape abstraction from examples learning a class prototype from a set of exemplars is an important challenge facing researchers in object categorization although the problem is receiving growing interest most approaches assume a one to one correspondence among local features restricting their ability to learn true abstractions of a shape in this paper we present a new technique for learning an abstract shape prototype from a set of exemplars whose features are in many to many correspondence focusing on the domain of 2d shape we represent a silhouette as a medial axis graph whose nodes correspond to partsrdquo defined by medial branches and whose edges connect adjacent parts given a pair of medial axis graphs we establish a many to many correspondence between their nodes to find correspondences among articulating parts based on these correspondences we recover the abstracted medial axis graph along with the positional and radial attributes associated with its nodes we evaluate the abstracted prototypes in the context of a recognition task

multilevel analysis of structural equation models summary this paper considers the multilevel analysis of structural equation models with unbalanced sampling designs the analysis is based on the maximum likelihood and the generalized least squares approaches basic statistical results for inference such as the asymptotic distribution of the estimators goodness of fit test statistics for the validity of the model and functional constraints are developed computationally the applications of the scoring algorithm and the gauss newton algorithm are discussed

calculating the statistics of forced response of a mistuned bladed disk assembly an analytical technique has been developed to calculate the statistics of the forced response of a mistuned bladed disk assembly the modal mass and stiffness of each blade have been assumed to be random variables with gaussian distribution the comparison with results from numerical simulations indicate that the statistics of distributions of the blades amplitudes predicted by the analytical technique are acceptable

unsupervised learning of finite mixture models this paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data the adjective unsupervised is justified by two properties of the algorithm 1 it is capable of selecting the number of components and 2 unlike the standard expectation maximization em algorithm it does not require careful initialization the proposed method also avoids another drawback of em for mixture fitting the possibility of convergence toward a singular estimate at the boundary of the parameter space the novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models instead we seamlessly integrate estimation and model selection in a single algorithm our technique can be applied to any type of parametric mixture model for which it is possible to write an em algorithm in this paper we illustrate it with experiments involving gaussian mixtures these experiments testify for the good performance of our approach

combined approximations for efficient probalistic analysis of structures real life analysis and design problems involve uncertainties quantification of the uncertainties in a system s response is important and requires a probabilistic analysis of the system a main challenge in probabilistic analysis of large structural systems is the high computational effort due to the multiple repeated analyses involved the combined approximations ca method which combines the strengths of both local and global approximations can be used for efficient probabilistic analysis of structures the ca method is a combination of binomial series local approximations also called neumann expansion approximations and reduced basis global approximations an efficient method is presented for probabilistic analysis of structural systems using the ca method the effectiveness of this method is demonstrated on analysis of mistuned bladed disk assemblies and systems with progressive collapse using monte carlo simulation it is shown that the method can predict accurately the probability distribution function of the responses of these systems at a considerably lower cost than a method using finite element analysis in each cycle of monte carlo simulation

accurate reanalysis of structures by a preconditioned conjugate gradient method a preconditioned conjugate gradient pcg method that is most suitable for reanalysis of structures is developed the method presented provides accurate results efficiently it is easy to implement and can be used in a wide range of applications including non linear analysis and eigenvalue problems it is shown that the pcg method presented and the combined approximations ca method developed recently provide theoretically identical results consequently available results from one method can be applied to the other method effective solution procedures developed for the ca method can be used for the pcg method and various criteria and error bounds developed for conjugate gradient methods can be used for the ca method numerical examples show that the condition number of the selected preconditioned matrix is much smaller than the condition number of the original matrix this property explains the fast convergence and accurate results achieved by the method copyright 2002 john wiley sons ltd

exact and accurate solutions in the approximate reanalysis of structures combined approximations ca is an efe cient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations in previous studies high quality approximations have been achieved for large changes in the design but the reasons for the high accuracy were not fully understood in this work some typical cases where exact and accurate solutions are achieved by the method are presented and discussed exact solutions are obtained when a basis vector is a linear combination of the previous vectors such solutions are obtained also for low rank modie cations to structures or scaling of the initial stiffness matrix in general the ca method provides approximate solutions but the results presented explain the high accuracy achieved with only a small number of basis vectors accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent such solutions are achieved also for changes in a small number of elements or when the angle between the two vectors representing the initial design and modie ed design is small numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges i introduction m ultiplerepeatedanalysesareneeded invariousdesign and optimization problems in general the structural response cannot be expressed explicitly in terms of the structure properties and structural analysis involves the solution of a set ofsimultaneous equations reanalysis methods are intended to efe ciently analyze structures modie ed due to changes in the design approximate reanalysis methods have been used extensively in structural optimization to reduce the number of exact analyses and the overall computational cost during the solution process the combined approximations ca method developed recently is considered in this paper the method combines several concepts and methods such as reduced basis series approximations matrix factorization and gram schmidt orthonormalization these and other methods are used to achieve effective solution procedures the effectivenessofthemethodinvariousoptimizationproblemshasbeen demonstrated in previous studies 1i5 initially the ca method was used only for linear reanalysis models recently the method has been used successfully also in eigenvalue 6 and nonlinear analysis 7 problems applications of the method in a large variety ofproblems are discussed elsewhere 8i11 high quality approximations of the structural response for large changes in the design have been achieved in previous studies but the reasons for the high accuracy were not fully understood in this paper some typical cases where exact and accurate solutions are achieved by the camethod are presented anddiscussed in general the ca method provides exact solutions but the results presented in the paper explain the high accuracy achieved with only a small number of basis vectors the solution procedure isbriee y described in sec ii three typical cases where exact solutions are achieved by the ca method are introduced and discussed in sec iii exact solutions are obtained when a basis vector is a linear combination of the previous vectors such solutions are obtained also for lowrank modie cations to structures or scaling of the initial stiffness matrix various cases of accurate solutions are discussed in sec iv convergence properties of the series of basis vectors and the series of the ca terms are presented and criteria intended to evaluate

an image understanding system using attributed symbolic representation and inexact graph matching this paper presents a powerful image understanding system that utilizes a semantic syntactic or attributed synibolic representation scheme in the form of attributed relational graphs arg s for comprehending the global information contents of images nodes in the arg represent the global image features while the relations between those features are represented by attributed branches between their corresponding nodes the extraction of arg representation from images is achieved by a multilayer graph transducer scheme this scheme is basically a rule based system that uses a combination of model driven and data driven concepts in performing a hierarchical symbolic mapping of the image information content from the spatial domain representation into a global representation further analysis and inter pretation of the imagery data is performed on the extracted arg representation a distance measure between images is defined in terms of the distance between their respective arg representations the distance between two arg s and the inexact matching of their respective components are calculated by an efficient dynamic programming technique the system handles noise distortion and ambiguity in real world images by two means namely through modeling and embedding them into the transducer s mapping rules as well as through the appropriate cost of error transformation for the inexact matching of the arg image representation two illustrative experiments are presented to demonstrate some capabilities of the proposed system experiment i deals with locating objects in multiobject scenes while experiment ii is concerned with target detection in sar images

shape clustering this paper aims to address the problem of shape clustering by discovering the common structure which captures the intrinsic structural information of shapes belonging to the same cluster it is based on a skeleton graph named common structure skeleton graph cssg which expresses possible correspondences between nodes of the individual skeletons of the cluster to construct the cssg we derive the correspondences by the optimal subsequence bijection osb to cluster the shape data we apply an agglomerative clustering scheme in each iteration the cssgs are formed from each cluster and the two closest clusters are merged into one the proposed agglomerative clustering algorithm has been evaluated on several shape data sets including three articulated shape data sets torsello s data set and a gesture data set in all experiments our method demonstrates effective performance compared to other algorithms highlights a method is proposed to discover the common structure of a cluster of shapes a cluster of shapes are represented by their cssg assigning weights to nodes and edges in cssgs to help measure their distances an agglomerative strategy is proposed for shape clustering based on cssgs

learning shape classes using a mixture of tree unions this paper poses the problem of tree clustering as that of fitting a mixture of tree unions to a set of sample trees the tree unions are structures from which the individual data samples belonging to a cluster can be obtained by edit operations the distribution of observed tree nodes in each cluster sample is assumed to be governed by a bernoulli distribution the clustering method is designed to operate when the correspondences between nodes are unknown and must be inferred as part of the learning process we adopt a minimum description length approach to the problem of fitting the mixture model to data we make maximum likelihood estimates of the bernoulli parameters the tree unions and the mixing proportions are sought so as to minimize the description length criterion this is the sum of the negative logarithm of the bernoulli distribution and a message length criterion that encodes both the complexity of the union trees and the number of mixture components we locate node correspondences by minimizing the edit distance with the current tree unions and show that the edit distance is linked to the description length criterion the method can be applied to both unweighted and weighted trees we illustrate the utility of the resulting algorithm on the problem of classifying 20 shapes using a shock graph representation

robustness of statistical inference in factor analysis and related models summary a class of latent variable models which includes the unrestricted factor analysis model is considered it is shown that minimum discrepancy test statistics and estimators derived under normality assumptions retain their asymptotic properties when the common factors are not normally distributed but the unique factors do have a multivariate normal distribution the minimum discrepancy test statistics and estimators considered include the usual likelihood ratio test statistic and maximum likelihood estimators

maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients the method of finding the maximum likelihood estimates of the parameters in a multivariate normal model with some of the component variables observable only in polytomous form is developed the main stratagem used is a reparameterization which converts the corresponding log likelihood function to an easily handled one the maximum likelihood estimates are found by a fletcher powell algorithm and their standard error estimates are obtained from the information matrix when the dimension of the random vector observable only in polytomous form is large obtaining the maximum likelihood estimates is computationally rather labor expensive therefore a more efficient method the partition maximum likelihood method is proposed these estimation methods are demonstrated by real and simulated data and are compared by means of a simulation study

reduced basis approximations of structural displacements for optimaldesign a new solution procedure for improved approximate reanalyses of structures using results of a single precise analysis is presented the proposed procedure is based on combining the computed terms of a series expansion used as high quality basis vectors with coefficients of a reduced basis expression the latter coefficients can readily be determined for each trial design by various criteria the results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations the proposed approach is suitable for various types of design variables and can be used with a general finite element model a general reanalysis procedure is introduced and the physical significance of first order approximations is demonstrated several numerical examples illustrate the effectiveness of the solution process it is shown that high quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables

polynomial time metrics for attributed trees we address the problem of comparing attributed trees and propose four novel distance measures centered around the notion of a maximal similarity common subtree the proposed measures are general and defined on trees endowed with either symbolic or continuous valued attributes and can be applied to rooted as well as unrooted trees we prove that our measures satisfy the metric constraints and provide a polynomial time algorithm to compute them this is a remarkable and attractive property since the computation of traditional edit distance based metrics is in general np complete at least in the unordered case we experimentally validate the usefulness of our metrics on shape matching tasks and compare them with an approximation of edit distance

a probabilistic spectral framework for grouping and segmentation this paper presents an iterative spectral framework for pairwise clustering and perceptual grouping our model is expressed in terms of two sets of parameters firstly there are cluster memberships which represent the affinity of objects to clusters secondly there is a matrix of link weights for pairs of tokens we adopt a model in which these two sets of variables are governed by a bernoulli model we show how the likelihood function resulting from this model may be maximised with respect to both the elements of link weight matrix and the cluster membership variables we establish the link between the maximisation of the log likelihood function and the eigenvectors of the link weight matrix this leads us to an algorithm in which we iteratively update the link weight matrix by repeatedly refining its modal structure each iteration of the algorithm is a three step process first we compute a link weight matrix for each cluster by taking the outer product of the vectors of current cluster membership indicators for that cluster second we extract the leading eigenvector from each modal link weight matrix third we compute a revised link weight matrix by taking the sum of the outer products of the leading eigenvectors of the modal link weight matrices

a general model for the analysis of multilevel data a general model is developed for the analysis of multivariate multilevel data structures special cases of the model include repeated measures designs multiple matrix samples multilevel latent variable models multiple time series and variance and covariance component models

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

region based hierarchical image matching this paper presents an approach to region based hierarchical image matching where given two images the goal is to identify the largest part in image 1 and its match in image 2 having the maximum similarity measure defined in terms of geometric and photometric properties of regions e g area boundary shape and color as well as region topology e g recursive embedding of regions to this end each image is represented by a tree of recursively embedded regions obtained by a multiscale segmentation algorithm this allows us to pose image matching as the tree matching problem to overcome imaging noise one to one many to one and many to many node correspondences are allowed the trees are first augmented with new nodes generated by merging adjacent sibling nodes which produces directed acyclic graphs dags then transitive closures of the dags are constructed and the tree matching problem reformulated as finding a bijection between the two transitive closures on dags while preserving the connectivity and ancestor descendant relationships of the original trees the proposed approach is validated on real images showing similar objects captured under different types of noise including differences in lighting conditions scales or viewpoints amidst limited occlusion and clutter

improving skeletal shape abstraction using multiple optimal solutions shape abstraction is an important problem faced by researchers in many fields such as pattern recognition computer vision and industrial design a recently developed previous shape abstraction framework demirci et al 20 generates an abstracted shape based on the correspondences between the features of the input shapes where the correspondences are obtained using the first optimal solution of a well known transportation problem as the size of the feature space grows the possibility of having more than one optimal solution for the same problem increases considering the case where multiple optimal solutions exist for the same transportation problem we first rank all optimal solutions based on how much they preserve the local neighborhood relations in this paper instead of creating the abstracted shape using the first optimal solution as done by the previous work we create the abstracted shape using the highest ranked optimal solution with this new property more effective abstracted shapes are generated experimental evaluation of the framework demonstrates that the proposed approach compares favorably with the previous technique in a set of shape retrieval experiments for different datasets highlightsskeletal shape abstraction is reformulated as the transportation problem a distortion free graph embedding is used for a better representation all optimal solutions for the transportation problem are considered the optimal solution which most resembles the neighborhood relations is used experiments demonstrate the improved performance

probabilistic vibration analysis of nearly periodic structures a disordered structure with weakly coupled subsystems has localized modes with the vibratory energy concentrated in one part of the mode shape the localization of modes may significantly affect the forced vibration response increasing the maximum vibration amplitude dramatically in some cases it is the scope of this paper to study the forced vibration of nearly periodic systems consisting of almost identical substructures and to analyze their free and forced dynamic responses probabilistically it is shown that the sensitivity of the forced response to the degree of localization depends on a combination of the symmetry of the mode which is excited and the phase difference between the forces acting on each substructure these results might explain the range of contrasting conclusions of earlier publications on the effects of forced response on mistuned structures a probabilistic analysis of the free and forced responses of a nearly periodic structure is shown to be useful in the design of structures that are sensitive to the degree of localization the results of the probabilistic analysis are verified using monte carlo simulation

unsupervised category modeling recognition and segmentation in images suppose a set of arbitrary unlabeled images contains frequent occurrences of 2d objects from an unknown category this paper is aimed at simultaneously solving the following related problems 1 unsupervised identification of photometric geometric and topological properties of multiscale regions comprising instances of the 2d category 2 learning a region based structural model of the category in terms of these properties and 3 detection recognition and segmentation of objects from the category in new images to this end each image is represented by a tree that captures a multiscale image segmentation the trees are matched to extract the maximally matching subtrees across the set which are taken as instances of the target category the extracted subtrees are then fused into a tree union that represents the canonical category model detection recognition and segmentation of objects from the learned category are achieved simultaneously by finding matches of the category model with the segmentation tree of a new image experimental validation on benchmark data sets demonstrates the robustness and high accuracy of the learned category models when only a few training examples are used for learning without any human supervision

a study of graph spectra for comparing graphs and trees the spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure it has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph there are however a number of potential drawbacks in using the spectrum as a representation of a graph firstly more than one graph may share the same spectrum it is well known for example that very few trees can be uniquely specified by their spectrum secondly the spectrum may change dramatically with a small change structure there are a wide variety of graph matrix representations from which the spectrum can be extracted among these are the adjacency matrix combinatorial laplacian normalised laplacian and unsigned laplacian spectra can also be derived from the heat kernel matrix and path length distribution matrix the choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks in this paper we investigate the performance of the spectra as a graph representation in a variety of situations firstly we investigate the cospectrality of the various matrix representations over large graph and tree sets extending the work of previous authors we then show that the euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs and we analyse the accuracy of this relationship we then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates these results are produced using both synthetic graphs and trees and graphs derived from shape and image data