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

shape smoothing using medial axis properties a shape smoothing algorithm is presented which uses properties of the medial axis to define segments of the shape border and their prominence relative to the local radius of the shape prominence values are used to classify axes as either major or minor and minor axes are then deleted an augmented medial axis transform is also defined using an approximate euclidean distance transform and medial axis interpolation and linking

transitions of the 3d medial axis under a one parameter family of deformations the instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications these instabilities or transitions occur when the structure of the medial axis graph changes abruptly under deformations of shape the recent classification of these transitions in 2d for the medial axis and for the shock graph was a key factor in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths the classification of generic transitions of the 3d medial axis could likewise potentially lead to a similar representation in 3d in this paper these transitions are classified by examining the order of contact of spheres with the surface leading to an enumeration of possible transitions which are then examined on a case by case basis some cases are ruled out as never occurring in any family of deformations while others are shown to be nongeneric in a one parameter family of deformations finally the remaining cases are shown to be viable by developing a specific example for each our work is inspired by that of bogaevsky who obtained the transitions as part of an investigation of viscosity solutions of hamilton jacobi equations our contribution is to give a more down to earth approach bringing this work to the attention of the computer vision community and to provide explicit constructions for the various transitions using simple surfaces we believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications

empirical bayesian analysis for high dimensional computer output this article proposes a two stage statistical method for the analysis of multivariate computer experiments when at least one of the output dimensions is large the stage one data are modeled by a multivariate extension of a widely used scalar statistical model for computer output conditioned on stage one data a simple statistical model is then proposed for the stage two data the method is demonstrated in a geophysical application involving an ocean model

symmetry maps of free form curve segments via wave propagation this paper presents an approach for computing the symmetries skeletons of an edge map consisting of a collection of curve segments this approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of pde s specifically waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions in addition to avoid error built up due to the discrete nature of propagation shockwaves are detected and explicitly propagated along a secondary dynamic grid the propagation of shockwaves integrated with the propagation of the wavefront along discrete directions leads to an exact simulation of propagation by the eikonal equation the resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally exactly and efficiently under local changes in an edge map gap completion removal of spurious elements etc the ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low level edge maps and high level object hypotheses

skeletonization based on error reduction the accuracy of a non pixel based skeletonization method is largely dependent on the contour information chosen as input when using a constrained delaunay triangulation to construct an object s skeleton a number of contour pixels must be chosen as a basis for triangulation this paper presents a new method of selecting these contour pixels a new method for measuring skeletonization error is proposed which quantifies the deviation of a skeleton segment from the true medial axis of a stroke in an image the goal of the proposed algorithm is to reduce this error to an acceptable level whilst retaining the superior efficiencies of previous non pixel based techniques experimental results show that the proposed method is adept at following the medial axis of an image and is capable of producing a skeleton that is confirmed by a human s perception of the image it is also computationally efficient and robust against noise

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

hierarchical decomposition and axial shape description a method for producing a segmented axial description of a given shape together with a hierarchical decomposition of the shape into its parts is presented the novelty of this approach lies in the combination of several competing approaches and tools into a unified scheme and an efficient implementation producing natural descriptions smooth local symmetries are used for the axial description of parts which are suggested by curvature sign changes parallel symmetries are used to provide information on global relationships within the shape this information is used for parsing shape into a hierarchy of parts this approach uses both region and contour information can handle shapes with corners and addresses the issue of local versus global information the issue of scale and the notion of part the method is computationally efficient robust and stable results that show that it provides an intuitive shape description are 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

medial axis transformation of a planar shape the medial axis transformation is a means first proposed by blum to describe a shape in this paper we present a 0 n log n algorithm for computing the medial axis of a planar shape represented by an n edge simple polygon the algorithm is an improvement over most previously known results interms of both efficiency and exactness and has been implemented in fortran some computer plotted output of the program are also shown in the paper

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

the delta medial axis in this paper we present the delta medial axis dma a quasi linear algorithmic solution addressing several of the main concerns of discrete medial axes ma computation first its sensitivity to small shape perturbations is counterbalanced by a single parameter delta used in a pruning strategy that implicitly takes into account the local topology of the boundary second the discrete nature of images is addressed by approximating the original ma definition in continuous space third to allow real time performances a set of algorithmic optimizations is proposed we compare our approach both qualitatively and quantitatively to recent state of the art solutions and show that the dma presents an excellent choice for a wide range of applications finally to allow fast and efficient use of our algorithm we propose the complete pseudo code highlightsthe delta medial axis a fast and robust skeletonization algorithm is presented it includes a pruning parameter which considers the local topology of the boundary to allow real time performances optimizations are introduced with complete pseudo code a complete comparative study with four recent state of the art algorithms is proposed

multiscale skeletons by image foresting transform and its application to neuromorphometry abstract the image foresting transform ift reduces optimal image partition problems based on seed pixels to a shortest path forest problem in a graph whose solution can be obtained in linear time such a strategy has allowed a unified and efficient approach to the design of image processing operators such as edge tracking region growing watershed transforms distance transforms and connected filters this paper presents a fast and simple method based on the ift to compute multiscale skeletons and shape reconstructions without border shifting the method also generates one pixel wide connected skeletons and the skeleton by influence zones simultaneously for objects of arbitrary topologies the results of the work are illustrated with respect to skeleton quality execution time and its application to neuromorphometry

on kinetic ising models in one dimension it is shown that glauber dynamics in 1d ising spin systems is not universal this is illustrated on a periodic model with a basic unit cell j1 jn containing an arbitrary set of n ferromagnetic coupling constants the dynamic critical exponent z is calculated exactly as z 1 max ji min ji the known value z 2 is recovered only for n 1 the extension of this result to other types of dynamics is briefly discussed

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

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

discrete voronoi diagrams and the skiz operator a dynamic algorithm the voronoi diagram vd is a popular tool for partitioning the support of an image an algorithm is presented for constructing vd when the seed set which determines the voronoi regions can be modified by adding and removing seeds the number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized a result on cocircular seeds is presented the adjacency or dual graph of the vd is readily obtained the use of the algorithm for constructing skeletons by influence zones skiz is demonstrated

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

object matching with hierarchical skeletons the skeleton of an object provides an intuitive and effective abstraction which facilitates object matching and recognition however without any human interaction traditional skeleton based descriptors and matching algorithms are not stable for deformable objects specifically some fine grained topological and geometrical features would be discarded if the skeleton was incomplete or only represented significant visual parts of an object moreover the performance of skeleton based matching highly depends on the quality and completeness of skeletons in this paper we propose a novel object representation and matching algorithm based on hierarchical skeletons which capture the shape topology and geometry through multiple levels of skeletons for object representation we reuse the pruned skeleton branches to represent the coarse and fine grained shape topological and geometrical features moreover this can improve the stability of skeleton pruning without human interaction we also propose an object matching method which considers both global shape properties and fine grained deformations by defining singleton and pairwise potentials for similarity computation between hierarchical skeletons our experiments attest our hierarchical skeleton based method a significantly better performance than most existing shape based object matching methods on six datasets achieving a 99 21 bulls eye score on the mpeg7 shape dataset highlightsit represents the coarse and fine grained shape topological and geometrical features it improves the stability of skeleton pruning without human interaction it considers both global and fine shape properties by different potential functions it achieves a better performance than most existing methods on six datasets experiments attest our method a better performance than most related approaches we achieve a 99 21 bulls eye score on the mpeg7 shape dataset

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

extracting 3d parametric curves from 2d images of helical objects helical objects occur in medicine biology cosmetics nanotechnology and engineering extracting a 3d parametric curve from a 2d image of a helical object has many practical applications in particular being able to extract metrics such as tortuosity frequency and pitch we present a method that is able to straighten the image object and derive a robust 3d helical curve from peaks in the object boundary the algorithm has a small number of stable parameters that require little tuning and the curve is validated against both synthetic and real world data the results show that the extracted 3d curve comes within close hausdorff distance to the ground truth and has near identical tortuosity for helical objects with a circular profile parameter insensitivity and robustness against high levels of image noise are demonstrated thoroughly and quantitatively

an unified multiscale framework for planar surface and curve skeletonization computing skeletons of 2d shapes and medial surface and curve skeletons of 3d shapes is a challenging task in particular there is no unified framework that detects all types of skeletons using a single model and also produces a multiscale representation which allows to progressively simplify or regularize all skeleton types in this paper we present such a framework we model skeleton detection and regularization by a conservative mass transport process from a shape s boundary to its surface skeleton next to its curve skeleton and finally to the shape center the resulting density field can be thresholded to obtain a multiscale representation of progressively simplified surface or curve skeletons we detail a numerical implementation of our framework which is demonstrably stable and has high computational efficiency we demonstrate our framework on several complex 2d and 3d shapes

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

continuous medial representation for anatomical structures the m rep approach pioneered by pizer 2003 is a powerful morphological tool that makes it possible to employ features derived from medial loci skeletons in shape analysis this paper extends the medial representation paradigm into the continuous realm modeling skeletons and boundaries of three dimensional objects as continuous parametric manifolds while also maintaining the proper geometric relationship between these manifolds the parametric representation of the boundary medial relationship makes it possible to fit shape based coordinate systems to the interiors of objects providing a framework for combined statistical analysis of shape and appearance our approach leverages the idea of inverse skeletonization where the skeleton of an object is defined first and the object s boundary is derived analytically from the skeleton this paper derives a set of sufficient conditions ensuring that inverse skeletonization is well posed for single manifold skeletons and formulates a partial differential equation whose solutions satisfy the sufficient conditions an efficient variational algorithm for deformable template modeling using the continuous medial representation is described and used to fit a template to the hippocampus in 87 subjects from a schizophrenia study with sub voxel accuracy and 95 mean overlap

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

non universal critical dynamics of the one dimensional potts models the one dimensional kinetic potts models are studied for both glauber and kawasaki dynamics with different transition rates rigorous lower bounds for the dynamical exponent z are obtained from the initial response rate and the scaling hypothesis for the relaxation time it is shown that the z exponent is non universal for several periodic models with a basic unit cell j1 j2 jm containing different coupling constants the critical dynamics of the alternating bond model is also studied by a simple argument about the movement of domain walls

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

a theory of kinematics for the potts model a master equation formulation of the kinetic q state potts model is presented it is shown that this formulation reduces to glauber s dynamics in the ising limit the single spin problem is considered and the properties of its q 1 temperature dependent relaxation times are studied the linear chain problem is much harder than in the glauber case a variational method is used to obtain a lower bound for the dynamical critical exponent z or 3 for q 2 a discussion of these and related problems is presented

enslaved finite difference approximations for quasigeostrophic shallow flows abstract we describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate the nonlinear pdes that govern barotropic two dimensional geophysical fluid dynamics our underlying strategy is to reduce the truncation error of a given difference scheme in such a way that the time step restrictions are not changed to accomplish this reduction we use information from the governing equations in the case of vanishing time derivatives our method is based on a change of variables from fine to coarse grids which allows us to order the various terms that appear and justify further approximations these approximations lead to algebraic closures for the new higher order variables and finally to a new enslaved scheme we demonstrate the utility of the procedure for the shallow water equations in both periodic and closed basins in the latter case we present results that demonstrate the ability of the enslaved scheme to capture dynamics on scales smaller than those resolved by the original scheme

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

finding shape axes using magnetic fields this paper presents a novel method based on magnetic field principles for obtaining the axes of shapes the method is based on directional information of the shape s boundary by simulating a parallel algorithm we are able to generate the inner as well as the outer axes axes of concavities of the shape the preprocessing phase for this algorithm involves obtaining the shape s gradient each point of the gradient is substituted by a minute magnetic dipole the cumulative magnetic field due to these dipoles is accumulated at all points in the image in a one pass algorithm the magnitude of the final magnetic vector field has valleys that are created from mutual and directionally balanced cancellations of opposing boundary segments these valleys signify the axes of the shape the axes are obtained by performing a valley search the magnetic field modeling mfm method has an advantage over previous approaches since it utilizes not only the location information of the boundary but also its directional information as demonstrated experimental results of the mfm method are much improved compared to other skeletonization algorithms which tend to generate spurious and noisy axes

non universality in the dynamics of the one dimensional potts model the authors discuss two recent papers on the dynamics of the one dimensional potts model which yield different values of the dynamical critical exponent they show that in fact there are choices of the transition rates which yield arbitrarily large non universal exponents

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

multiscale medial loci and their properties blum s medial axes have great strengths in principle in intuitively describing object shape in terms of a quasi hierarchy of figures but it is well known that derived from a boundary they are damagingly sensitive to detail in that boundary the development of notions of spatial scale has led to some definitions of multiscale medial axes different from the blum medial axis that considerably overcame the weakness three major multiscale medial axes have been proposed iteratively pruned trees of voronoi edges ogniewicz 1993 szekely 1996 naf 1996 shock loci of reaction diffusion equations kimia et al 1995 siddiqi and kimia 1996 and height ridges of medialness cores fritsch et al 1994 morse et al 1993 pizer et al 1998 these are different from the blum medial axis and each has different mathematical properties of generic branching and ending properties singular transitions and geometry of implied boundary and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries these mathematical properties and computational abilities are laid out and compared and contrasted in this paper

a skeleton family generator via physics based deformable models this paper presents a novel approach for object skeleton family extraction the introduced technique utilizes a 2 d physics based deformable model that parameterizes the objects shape deformation equations are solved exploiting modal analysis and proportional to model physical characteristics a different skeleton is produced every time generating in this way a family of skeletons the theoretical properties and the experiments presented demonstrate that obtained skeletons match to hand labeled skeletons provided by human subjects even in the presence of significant noise and shape variations cuts and tears and have the same topology as the original skeletons in particular the proposed approach produces no spurious branches without the need of any known skeleton pruning method

path similarity skeleton graph matching this paper proposes a novel graph matching algorithm and applies it to shape recognition based on object silhouettes the main idea is to match skeleton graphs by comparing the geodesic paths between skeleton endpoints in contrast to typical tree or graph matching methods we do not consider the topological graph structure our approach is motivated by the fact that visually similar skeleton graphs may have completely different topological structures the proposed comparison of geodesic paths between endpoints of skeleton graphs yields correct matching results in such cases the skeletons are pruned by contour partitioning with discrete curve evolution which implies that the endpoints of skeleton branches correspond to visual parts of the objects the experimental results demonstrate that our method is able to produce correct results in the presence of articulations stretching and contour deformations

skeletonization of ribbon like shapes based on a new wavelet function a wavelet based scheme to extract skeleton of ribbon like shape is proposed in this paper where a novel wavelet function plays a key role in this scheme which possesses three significant characteristics namely 1 the position of the local maximum moduli of the wavelet transform with respect to the ribbon like shape is independent of the gray levels of the image 2 when the appropriate scale of the wavelet transform is selected the local maximum moduli of the wavelet transform of the ribbon like shape produce two new parallel contours which are located symmetrically at two sides of the original one and have the same topological and geometric properties as that of the original shape 3 the distance between these two parallel contours equals to the scale of the wavelet transform which is independent of the width of the shape this new scheme consists of two phases 1 generation of wavelet skeleton based on the desirable properties of the new wavelet function symmetry analyses of the maximum moduli of the wavelet transform is described midpoints of all pairs of contour elements can be connected to generate a skeleton of the shape which is defined as wavelet skeleton 2 modification of the wavelet skeleton thereafter a set of techniques are utilized for modifying the artifacts of the primary wavelet skeleton the corresponding algorithm is also developed in this paper experimental results show that the proposed scheme is capable of extracting exactly the skeleton of the ribbon like shape with different width as well as different gray levels the skeleton representation is robust against noise and affine transformation

hierarchic voronoi skeletons abstract robust and time efficient skeletonization of a planar shape which is connectivity preserving and based on euclidean metrics can be achieved by first regularizing the voronoi diagram vd of a shape s boundary points i e by removal of noise sensitive parts of the tessellation and then by establishing a hierarchic organization of skeleton constituents each component of the vd is attributed with a measure of prominence which exhibits the expected invariance under geometric transformations and noise the second processing step a hierarchic clustering of skeleton branches leads to a multiresolution representation of the skeleton termed skeleton pyramid

a ridge following algorithm for finding the skeleton of a fuzzy image abstract a typical method for finding the skeleton of a grayscale image is to threshold the image then thin the result with one of several binary thinning algorithms this process loses information in the intensity dimension by reducing all intensities to either one of two values the loss of intensity information causes the skeleton of non uniform grayscale images to be centrally located within the thresholded image but may not lie along the maximum intensity ridge line in this paper we propose a method that avoids thresholding instead we transform the grayscale image into a fuzzy set and use the degree of membership in the underlying object to find the skeleton along the maximum intensity ridge line topographical labeling methods find a ridge line by topographically labeling pixels the ride line is the union of peak ridge and saddle point pixels these methods require preprocessing or postprocessing to find connected single pixel skeletons we show the results of our algorithm on images varying in uniformity from a gaussian smoothed binary rectangle to a scanned image of a grayscale object we compare results with the threshold and thin and topographical labeling methods

self organizing maps for the skeletonization of sparse shapes this paper presents a method for computing the skeleton of planar shapes and objects which exhibit sparseness lack of connectivity within their image regions such sparseness in images may occur due to poor lighting conditions incorrect thresholding or image sub sampling furthermore in document image analysis sparse shapes are characteristic of texts faded due to aging and or poor ink quality given the pixel distribution for a shape the proposed method involves an iterative evolution of a piecewise linear approximation of the shape skeleton by using a minimum spanning tree based self organizing map som by constraining the som to lie on the edges of the delaunay triangulation of the shape distribution the adjacency relationships between regions in the shape are detected and used in the evolution of the skeleton the som on convergence gives the final skeletal shape the skeletonization is invariant to euclidean transformations the potential of the method is demonstrated on a variety of sparse shapes from different application domains

continuous medial representation of brain structures using the biharmonic pde a new approach for constructing deformable continuous medial models for anatomical structures is presented medial models describe geometrical objects by first specifying the skeleton of the object and then deriving the boundary surface corresponding to the skeleton however an arbitrary specification of a skeleton will not be valid unless a certain set of sufficient conditions is satisfied the most challenging of these is the non linear equality constraint that must hold along the boundaries of the manifolds forming the skeleton the main contribution of this paper is to leverage the biharmonic partial differential equation as a mapping from a codimension 0 subset of euclidean space to the space of skeletons that satisfy the equality constraint the pde supports robust numerical solution on freeform triangular meshes providing additional flexibility for shape modeling the approach is evaluated by generating continuous medial models for a large dataset of hippocampus shapes generalizations to modeling more complex shapes and to representing branching skeletons are demonstrated

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

characterizing width uniformity by wave propagation this work describes a different image analysis approach to characterize the uniformity of objects in agglomerates by using the propagation of normal wave fronts the problem of width uniformity is discussed and its importance for the characterization of composite structures normally found in physics and biology highlighted the methodology involves identifying each cluster i e connected component of interest which can correspond to objects or voids and estimating the respective medial axes by using a recently proposed wave front propagation approach which is briefly reviewed the distance values along such axes are identified and their mean and standard deviation values obtained as illustrated with respect to synthetic and real objects in vitro cultures of neuronal cells the combined use of these two features provides a powerful description of the uniformity of the separation between the objects presenting potential for several applications in material sciences and biology

shape based normalization of the corpus callosum for dti connectivity analysis the continuous medial representation cm rep is an approach that makes it possible to model normalize and analyze anatomical structures on the basis of medial geometry having recently presented a partial differential equation pde based approach for 3 d cm rep modeling 1 here we present an equivalent 2 d approach that involves solving an ordinary differential equation this paper derives a closed form solution of this equation and shows how pythagorean hodograph curves can be used to express the solution as a piecewise polynomial function allowing efficient and robust medial modeling the utility of the approach in medical image analysis is demonstrated by applying it to the problem of shape based normalization of the midsagittal section of the corpus callosum using diffusion tensor tractography we show that shape based normalization aligns subregions of the corpus callosum defined by connectivity more accurately than normalization based on volumetric registration furthermore shape based normalization helps increase the statistical power of group analysis in an experiment where features derived from diffusion tensor tractography are compared between two cohorts these results suggest that cm rep is an appropriate tool for normalizing the corpus callosum in white matter studies

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

gaussian surrogates for computer models with time varying inputs and outputs computer models of dynamic systems produce outputs that are functions of time models that solve systems of differential equations often have this character in many cases time series output can be usefully reduced via principal components to simplify analysis time indexed inputs such as the functions that describe time varying boundary conditions are also common with such models however inputs that are functions of time often do not have one or a few characteristic shapes that are more common with output functions and so principal component representation has less potential for reducing the dimension of input functions in this article gaussian process surrogates are described for models with inputs and outputs that are both functions of time the focus is on construction of an appropriate covariance structure for such surrogates some experimental design issues and an application to a model of marrow cell dynamics

globally optimal grouping for symmetric closed boundaries by combining boundary and region information many natural and man made structures have a boundary that shows a certain level of bilateral symmetry a property that plays an important role in both human and computer vision in this paper we present a new grouping method for detecting closed boundaries with symmetry we first construct a new type of grouping token in the form of symmetric trapezoids by pairing line segments detected from the image a closed boundary can then be achieved by connecting some trapezoids with a sequence of gap filling quadrilaterals for such a closed boundary we define a unified grouping cost function in a ratio form the numerator reflects the boundary information of proximity and symmetry and the denominator reflects the region information of the enclosed area the introduction of the region area information in the denominator is able to avoid a bias toward shorter boundaries we then develop a new graph model to represent the grouping tokens in this new graph model the grouping cost function can be encoded by carefully designed edge weights and the desired optimal boundary corresponds to a special cycle with a minimum ratio form cost we finally show that such a cycle can be found in polynomial time using a previous graph algorithm we implement this symmetry grouping method and test it on a set of synthetic data and real images the performance is compared to two previous grouping methods that do not consider symmetry in their grouping cost functions

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

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

finding axes of symmetry from potential fields this paper addresses the problem of detecting axes of bilateral symmetry in images in order to achieve robustness to variation in illumination only edge gradient information is used to overcome the problem of edge breaks a potential field is developed from the edge map which spreads the information in the image plane pairs of points in the image plane are made to vote for their axes of symmetry with some confidence values to make the method robust to overlapping objects only local features in the form of taylor coefficients are used for quantifying symmetry we define an axis of symmetry histogram which is used to accumulate the weighted votes for all possible axes of symmetry to reduce the computational complexity of voting a hashing scheme is proposed wherein pairs of points whose potential fields are too asymmetric are pruned by not being counted for the vote experimental results indicate that the proposed method is fairly robust to edge breaks and is able to detect symmetries even when only 0 05 of the possible pairs are used for voting

extraction of the euclidean skeleton based on a connectivity criterion abstract the skeleton is essential for general shape representation the commonly required properties of a skeletonization algorithm are that the extracted skeleton should be accurate robust to noise position and rotation able to reconstruct the original object and able to produce a connected skeleton in order to preserve its topological and hierarchical properties however the use of a discrete image presents a lot of problems that may influence the extraction of the skeleton moreover most of the methods are memory intensive and computationally intensive and require a complex data structure in this paper we propose a fast efficient and accurate skeletonization method for the extraction of a well connected euclidean skeleton based on a signed sequential euclidean distance map a connectivity criterion is proposed which can be used to determine whether a given pixel is a skeleton point independently the criterion is based on a set of point pairs along the object boundary which are the nearest contour points to the pixel under consideration and its 8 neighbors our proposed method generates a connected euclidean skeleton with a single pixel width without requiring a linking algorithm or iteration process experiments show that the runtime of our algorithm is faster than the distance transformation and is linearly proportional to the number of pixels of an image

a width independent fast thinning algorithm the skeleton of a digital figure can often be regarded as a convenient alternative to the figure itself it is useful both to diminish drastically the amount of data to be handled and to simplify the computational procedures required for description and classification purposes thinning a digital figure down to its skeleton is a time consuming process when conventional sequential computers are employed the procedure we propose allows one to speed up the thinning transformation and to get a well shaped skeleton after cleaning of the input picture has been performed the pixels of the figure are labeled according to their distance from the background and a set whose pixels are symmetrically placed with respect to distinct contour parts of the figure is found this set is then given a linear structure by applying topology preserving removal operations finally a pruning step regarding branches not relevant in the framework of the problem domain completes the process the resulting skeleton is a labeled set of pixels which is shown to possess all the required properties particularly those concerning connectedness topology and shape moreover the original figure can almost completely be recovered by means of a reverse distance transformation only a fixed and small number of sequential passes through the picture is necessary to achieve the goal the computational effort is rather modest and the use of the proposed algorithm turns out to be more advantageous the greater the width of the figure to be thinned

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

simulating the grassfire transform using an active contour model a method for shape description of planar objects that integrates both region and boundary features is presented the method is an implementation of a 2d dynamic grassfire that relies on a distance surface on which elastic contours minimize an energy function the method is based on an active contour model numerous implementation aspects of the shape description method were optimized a euclidean metric was used for optimal accuracy and the active contour model permits bypassing some of the discretization limitations inherent in using a digital grid noise filtering was performed on the basis of both contour feature measures and region measures that is curvature extremum significance and ridge support respectively to obtain robust shape descriptors other improvements and variations of the algorithmic implementation are proposed

veinerization a new shape description for flexible skeletonization we introduce the new concept of veinerization which produces a graph that contains all the topological information needed to derive a wide variety of skeletons theoretically the main contribution is to provide a homogeneous framework for integration of the major concepts described in other related works on digital skeletonization in practice the new aspect of this approach is to provide the user with different criteria for selecting the most suitable skeleton for a given application e g the user can select a suitable threshold for obtaining the desirable balance between having a skeleton without noisy prunes and having a skeleton that reflects the initial shape this algorithm has been tested on numerous kinds of patterns including pathological ones like fractal sets well known for the complexity of their shapes

surface and curve skeletonization of large 3d models on the gpu we present a gpu based framework for extracting surface and curve skeletons of 3d shapes represented as large polygonal meshes we use an efficient parallel search strategy to compute point cloud skeletons and their distance and feature transforms fts with user defined precision we regularize skeletons by a new gpu based geodesic tracing technique which is orders of magnitude faster and more accurate than comparable techniques we reconstruct the input surface from skeleton clouds using a fast and accurate image based method we also show how to reconstruct the skeletal manifold structure as a polygon mesh and the curve skeleton as a polyline compared to recent skeletonization methods our approach offers two orders of magnitude speed up high precision and low memory footprints we demonstrate our framework on several complex 3d models

the groupwise medial axis transform for fuzzy skeletonization and pruning medial representations of shapes are useful due to their use of an object centered coordinate system that directly captures intuitive notions of shape such as thickness bending and elongation however it is well known that an object s medial axis transform mat is unstable with respect to small perturbations of its boundary this instability results in additional unwanted branches in the skeletons which must be pruned in order to recover the portions of the skeletons arising purely from the uncorrupted shape information almost all approaches to skeleton pruning compute a significance measure for each branch according to some heuristic criteria and then prune the least significant branches first current approaches to branch significance computation can be classified as either local solely using information from a neighborhood surrounding each branch or global using information about the shape as a whole in this paper we propose a third groupwise approach to branch significance computation we develop a groupwise skeletonization framework that yields a fuzzy significance measure for each branch derived from information provided by the group of shapes we call this framework the groupwise medial axis transform g mat we propose and evaluate four groupwise methods for computing branch significance and report superior performance compared to a recent leading method we measure the performance of each pruning algorithm using denoising classification and within class skeleton similarity measures this research has several applications including object retrieval and shape analysis

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

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

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

disconnected skeleton shape at its absolute scale we present a new skeletal representation along with a matching framework to address the deformable shape recognition problem the disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale our motivation is to rely on stable properties the shape instead of inaccurately measured secondary details the new representation does not suffer from the common instability problems of the traditional connected skeletons and the matching process gives quite successful results on a diverse database of 2d shapes an important difference of our approach from the conventional use of skeleton is that we replace the local coordinate frame with a global euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations as a result we can produce descriptions that are sensitive to any combination of changes in scale position orientation and articulation as well as invariant ones

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

a mathematical morphology approach to euclidean distance transformation a distance transformation technique for a binary digital image using a gray scale mathematical morphology approach is presented applying well developed decomposition properties of mathematical morphology one can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers first the distance transformation using mathematical morphology is developed then several approximations of the euclidean distance are discussed the decomposition of the euclidean distance structuring element is presented the decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator real valued distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image

automatic and robust computation of 3d medial models incorporating object variability this paper presents a novel processing scheme for the automatic and robust computation of a medial shape model which represents an object population with shape variability the sensitivity of medial descriptions to object variations and small boundary perturbations are fundamental problems of any skeletonization technique these problems are approached with the computation of a model with common medial branching topology and grid sampling this model is then used for a medial shape description of individual objects via a constrained model fit the process starts from parametric 3d boundary representations with existing point to point homology between objects the voronoi skeleton of each sampled object boundary is partitioned into non branching medial sheets and simplified by a novel pruning algorithm using a volumetric contribution criterion using the surface homology medial sheets are combined to form a common medial branching topology finally the medial sheets are sampled and represented as meshes of medial primitives results on populations of up to 184 biological objects clearly demonstrate that the common medial branching topology can be described by a small number of medial sheets and that even a coarse sampling leads to a close approximation of individual objects

hierarchical shape description via the multiresolution symmetric axis transform a method is proposed that produces a shape description in the form of a hierarchy by scale of simple symmetric axis sequences an axis segment that is a child of another has smaller scale and is seen as a branch of its parent the scale value and parent child relationship are induced by following the symmetric axis under successive reduction of resolution the result in two or three dimensions is a figure rather than boundary oriented shape description that has natural segments and is insensitive to noise in the object description the general approach of hierarchy production by following a feature through successive resolution reduction will be presented as will methods of resolution reduction and computer implementation also the relation of this figure based shape description to those based on boundary curvature will be briefly discussed

a skeletonization algorithm by maxima tracking on euclidean distance transform abstract a simple and efficient algorithm using the maxima tracking approach on euclidean distance transform to detect skeleton points is presented the advantages of the skeleton obtained are 1 connectivity preservation 2 single pixel in width and 3 its locations as close as to the most symmetrical axes besides the condition of the least slope change of skeleton is used to ensure the fairness of the digital medial axes with the least effort the algorithm can be modified to eliminate non significant short skeletal branches originating from the object contour while the critical shape informative medial axes are preserved

euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform a general algorithm for computing euclidean skeletons of 2d and 3d data sets in linear time is presented these skeletons are defined in terms of a new concept called the integer medial axis ima transform we prove a number of fundamental properties of the ima skeleton and compare these with properties of the cmd centers of maximal disks skeleton several pruning methods for ima skeletons are introduced constant linear and square root pruning and their properties studied the algorithm for computing the ima skeleton is based upon the feature transform using a modification of a linear time algorithm for euclidean distance transforms the skeletonization algorithm has a time complexity which is linear in the number of input points and can be easily parallelized we present experimental results for several data sets looking at skeleton quality memory usage and computation time both for 2d images and 3d volumes

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