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

parts of visual form computational aspects underlying recognition is an organization of objects and their parts into classes and hierarchies a representation of parts for recognition requires that they be invariant to rigid transformations robust in the presence of occlusions stable with changes in viewing geometry and be arranged in a hierarchy these constraints are captured in a general framework using notions of a part line and a partitioning scheme a proposed general principle of form from function motivates a particular partitioning scheme involving two types of parts neck based and limb based neck based parts arise from narrowings in shape or the local minima in distance between two points on the boundary while limb based parts arise from a pair of negative curvature minima which have co circular tangents in this paper we present computational support for the limb based and neck based parts by showing that they are invariant robust stable and yield a hierarchy of parts examples illustrate that the resulting decompositions are robust in the presence of occlusion and clutter for a range of man made and natural objects and lead to natural and intuitive parts which can be used for recognition

knots sutures and excision we develop monopole and instanton floer homology groups for balanced sutured manifolds in the spirit of 12 applications include a new proof of property p for knots

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

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

on computing structural changes in evolving surfaces and their appearance as a surface undergoes a one parameter family of deformations its shape and its appearance change smoothly except at certain critical parameter values where abrupt structural changes occur this paper considers the case of surfaces defined as the zero set of smooth density functions undergoing a gaussian diffusion process and addresses the problem of computing the critical parameter values corresponding to structural changes in the parabolic curves of a surface and in its aspect graph an algorithm based on homotopy continuation and curve tracing is proposed in the case of polynomial density functions whose zero set is an algebraic surface it has been implemented and examples are presented

shape based object detection via boundary structure segmentation we address the problem of object detection and segmentation using global holistic properties of object shape global shape representations are highly susceptible to clutter inevitably present in realistic images and thus can be applied robustly only using a precise segmentation of the object to this end we propose a figure ground segmentation method for extraction of image regions that resemble the global properties of a model boundary structure and are perceptually salient our shape representation called the chordiogram is based on geometric relationships of object boundary edges while the perceptual saliency cues we use favor coherent regions distinct from the background we formulate the segmentation problem as an integer quadratic program and use a semidefinite programming relaxation to solve it the obtained solutions provide a segmentation of the object as well as a detection score used for object recognition our single step approach achieves state of the art performance on several object detection and segmentation benchmarks

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

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

stochastic jump diffusion process for computing medial axes in markov random fields proposes a statistical framework for computing medial axes of 2d shapes in the paper the computation of medial axes is posed as a statistical inference problem not as a mathematical transform the paper contributes to three aspects in computing medial axes 1 prior knowledge is adopted for axes and junctions so that axes around junctions are regularized 2 multiple interpretations of axes are possible each being assigned a probability 3 a stochastic jump diffusion process is proposed for estimating both axes and junctions in markov random fields we argue that the stochastic algorithm for computing medial axes is compatible with existing algorithms for image segmentation such as region growing snake and region competition thus our method provides a new direction for computing medial axes from texture images experiments are demonstrated on both synthetic and real 2d shapes

a latent trait theory via a stochastic learning theory for a knowledge space to capture the cognitive organization of a set of questions or problems pertaining to a body of information doignon and falmagne have proposed and analyzed in a number of papers the concept of aknowledge space that is a distinguished collection of subsets of questions representing the possibleknowledge states this collection of sets is assumed to satisfy a number of conditions since this concept is a deterministic one the problem of empirical testing arises a stochastic version of a knowledge space is developed in this paper in which the knowledge states are considered as possible epochs in a subject s learning history the knowledge space is decomposed as a union of a number of possible learning paths calledgradations the model specifies how a subject is channelled through and progresses along a gradation a probabilistic axiom of the local indepencence type relates the knowledge states to the observable responses the predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions an application of the model to artificial data is described based on maximum likelihood methods the statistical analysis is shown to be capable of revealing the combinatoric core of the model

segmentation of external force field for automatic initialization and splitting of snakes active contours or snakes have been extensively utilized in handling image segmentation and classification problems in traditional active contour models snake initialization is performed manually by users and topological changes such as splitting of the snake cannot be automatically handled in this paper we introduce a new method to solve the snake initialization and splitting problem based on an area segmentation approach the external force field is segmented first and then the snake initialization and splitting can be automatically performed by using the segmented external force field such initialization and splitting produces multiple snakes each of which is within the capture range associated to an object and will be evolved to the object boundary the external force used in this paper is a gradient vector flow with an edge preserving property epgvf which can prevent the snakes from passing over weak boundaries to segment the external force field we represent it with a graph and a graph theory approach can be taken to determine the membership of each pixel experimental results establish the effectiveness of the proposed approach

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

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

geometric active deformable models in shape modeling this paper analyzes the problem of shape modeling using the principle of active geometric deformable models while the basic modeling technique already exists in the literature we highlight many of its drawbacks and discuss their source and steps to overcome them we propose a new stopping criterion to address the stopping problem we also propose to apply a level set algorithm to implement the active geometric deformable models thereby handling topology changes automatically to alleviate the numerical problems associated with the implementation of the level set algorithm we propose a new adaptive multigrid narrow band algorithm all the proposed new changes have been illustrated with experiments with synthetic images and medical images

prior learning and gibbs reaction diffusion this article addresses two important themes in early visual computation it presents a novel theory for learning the universal statistics of natural images and it proposes a general framework of designing reaction diffusion equations for image processing we studied the statistics of natural images including the scale invariant properties then generic prior models were learned to duplicate the observed statistics based on minimax entropy theory the resulting gibbs distributions have potentials of the form u i spl lambda s spl sigma sub spl alpha 1 sup k spl sigma sub x y spl lambda sup spl alpha f sup spl alpha i x y with s f sup 1 f sup 2 f sup k being a set of filters and spl lambda spl lambda sup 1 spl lambda sup 2 spl lambda sup k the potential functions the learned gibbs distributions confirm and improve the form of existing prior models such as line process but in contrast to all previous models inverted potentials were found to be necessary we find that the partial differential equations given by gradient descent on u i spl lambda s are essentially reaction diffusion equations where the usual energy terms produce anisotropic diffusion while the inverted energy terms produce reaction associated with pattern formation enhancing preferred image features we illustrate how these models can be used for texture pattern rendering denoising image enhancement and clutter removal by careful choice of both prior and data models of this type incorporating the appropriate features

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

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

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

shape classification using the inner distance part structure and articulation are of fundamental importance in computer and human vision we propose using the inner distance to build shape descriptors that are robust to articulation and capture part structure the inner distance is defined as the length of the shortest path between landmark points within the shape silhouette we show that it is articulation insensitive and more effective at capturing part structures than the euclidean distance this suggests that the inner distance can be used as a replacement for the euclidean distance to build more accurate descriptors for complex shapes especially for those with articulated parts in addition texture information along the shortest path can be used to further improve shape classification with this idea we propose three approaches to using the inner distance the first method combines the inner distance and multidimensional scaling mds to build articulation invariant signatures for articulated shapes the second method uses the inner distance to build a new shape descriptor based on shape contexts the third one extends the second one by considering the texture information along shortest paths the proposed approaches have been tested on a variety of shape databases including an articulated shape data set mpeg7 ce shape 1 kimia silhouettes the eth 80 data set two leaf data sets and a human motion silhouette data set in all the experiments our methods demonstrate effective performance compared with other algorithms

implementing continuous scale morphology via curve evolution abstract a new approach to digital implementation of continuous scale mathematical morphology is presented the approach is based on discretization of evolution equations associated with continuous multiscale morphological operations those equations and their corresponding numerical implementation can be derived either directly from mathematical morphology definitions or from curve evolution theory the advantages of the proposed approach over the classical discrete morphology are demonstrated

optical flow a curve evolution approach a novel approach for the computation of optical flow based on an l sup 1 type minimization is presented it is shown that the approach has inherent advantages since it does not smooth the flow velocity across the edges and hence preserves edge information a numerical approach based on computation of evolving curves is proposed for computing the optical flow field computations are carried out on a number of real image sequences in order to illustrate the theory as well as the numerical approach

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

shape instabilities and pattern formation in solidification a new method for numerical solution of the moving boundary problem abstract when a solidification front is advancing into a region of supercooled liquid its shape is subject to instabilities which can lead to complex modes of growth the evaluation of the growth pattern is determined by the interaction of the driving force of the instability due to heat diffusion with the restabilizing force due to surface tension and its full development is a highly nonlinear process a new method for solving the two phase heat diffusion problem under these conditions is described the method is based on a single domain treatment weak formulation of the moving boundary problem with the temperature field near the interface approximated by a piecewise linear function in the spirit of the finite element technique calculated results for a model problem are in excellent agreement with the predictions of an analytical approximation

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

enhancement of planar shape through optimization of functionals for curves we show how optimization of the nordstrom and mumford shah functionals can be used to develop a type of curve evolution that is able to preserve salient features of closed curves while simultaneously suppressing noise and irrelevant details the idea is to characterize a curve by means of its angle function and apply the appropriate dynamics to this representation upon convergence the resulting form of the contour is reconstructed from the representation

representation and self similarity of shapes representing shapes in a compact and informative form is a significant problem for vision systems that must recognize or classify objects we describe a compact representation model for two dimensional 2d shapes by investigating their self similarities and constructing their shape axis trees sa trees our approach can be formulated as a variational one or equivalently as map estimation of a markov random field we start with a 2d shape its boundary contour and two different parameterizations for the contour one parameterization is oriented counterclockwise and the other clockwise to measure its self similarity the two parameterizations are matched to derive the best set of one to one point to point correspondences along the contour the cost functional used in the matching may vary and is determined by the adopted self similarity criteria e g cocircularity distance variation parallelism and region homogeneity the loci of middle points of the pairing contour points yield the shape axis and they can be grouped into a unique free tree structure the sa tree by implicitly encoding the local and global shape information into an sa tree a variety of vision tasks e g shape recognition comparison and retrieval can be performed in a more robust and efficient way via various tree based algorithms a dynamic programming algorithm gives the optimal solution in o n sup 1 where n is the size of the contour

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

shape from shading level set propagation and viscosity solutions we present a new implementation of an algorithm aimed at recovering a 3d shape from its 2d gray level picture in order to reconstruct the shape of the object an almost arbitrarily initialized 3d function is propagated on a rectangular grid so that a level set of this function tracks the height contours of the shape the method imports techniques from differential geometry fluid dynamics and numerical analysis and provides an accurate shape from shading algorithm the method solves some topological problems and gracefully handles cases of non smooth surfaces that give rise to shocks in the propagating contours real and synthetic images of 3d profiles were submitted to the algorithm and the reconstructed surfaces are presented demonstrating the effectiveness of the proposed method

semi continuity of skeletons in two manifold and discrete voronoi approximation the skeleton of a 2d shape is an important geometric structure in pattern analysis and computer vision in this paper we study the skeleton of a 2d shape in a two manifold mathcal m based on a geodesic metric we present a formal definition of the skeleton s omega for a shape omega in mathcal m and show several properties that make s omega distinct from its euclidean counterpart in mathbb r 2 we further prove that for a shape sequence lbrace omega i rbrace that converge to a shape omega in mathcal m the mapping omega rightarrow overline s omega is lower semi continuous a direct application of this result is that we can use a set p of sample points to approximate the boundary of a 2d shape omega in mathcal m and the voronoi diagram of p inside omega subset mathcal m gives a good approximation to the skeleton s omega examples of skeleton computation in topography and brain morphometry are illustrated

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

pseudo linear scale space theory it has been observed that linear gaussian scale space and nonlinear morphological erosion and dilation scale spaces generated by a quadratic structuring function have a lot in common indeed far reaching analogies have been reported which seems to suggest the existence of an underlying isomorphism however an actual mapping appears to be missing in the present work a one parameter isomorphism is constructed in closed form which encompasses linear and both types of morphological scale spaces as non uniform limiting cases the unfolding of the one parameter family provides a means to transfer known results from one domain to the other moreover for any fixed and non degenerate parameter value one obtains a novel type of pseudo linear multiscale representation that is in a precise way in between the familiar ones this is of interest in its own right as it enables one to balance pros and cons of linear versus morphological scale space representations in any particular situation

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

discrete active models and applications abstract optimization processes based on active models play central roles in many areas of computational vision as well as computational geometry unfortunately current models usually require highly complex and sophisticated mathematical machinery and at the same time they suffer from a number of limitations which impose restrictions on their applicability in this paper a simple class of discrete active models called migration processes mps is presented the processes are based on iterated averaging over neighborhoods defined by constant geodesic distance it is demonstrated that the mp model a system of self organizing active particles has a number of advantages over previous models both parametric active models snakes and implicit contour evolution models due to the generality of the mp model the process can be applied to derive natural solutions to a variety of optimization problems including defining minimal surface patches given their boundary curves finding shortest paths joining sets of points and decomposing objects into primitive parts

flux maximizing geometric flows several geometric active contour models have been proposed for segmentation in computer vision and image analysis the essential idea is to evolve a curve in 2d or a surface in 3d under constraints from image forces so that it clings to features of interest in an intensity image recent variations on this theme take into account properties of enclosed regions and allow for multiple curves or surfaces to be simultaneously represented however it is still unclear how to apply these techniques to images of narrow elongated structures such as blood vessels where intensity contrast may be low and reliable region statistics cannot be computed to address this problem we derive the gradient flows which maximize the rate of increase of flux of an appropriate vector field through a curve in 2d or a surface in 3d the key idea is to exploit the direction of the vector field along with its magnitude the calculations lead to a simple and elegant interpretation which is essentially parameter free and has the same form in both dimensions we illustrate its advantages with several level set based segmentations of 2d and 3d angiography images of blood vessels

symmetry reductions and exact solutions of the affine heat equation lie symmetry group method is applied to study the affine heat equation for surface its symmetry groups and corresponding optimal systems are determined and group invariant solutions associated to the symmetries are obtained and classified

the vector distance functions we present a novel method for representing and evolving objects of arbitrary dimension the method called the vector distance function vdf method uses the vector that connects any point in space to its closest point on the object it can deal with smooth manifolds with and without boundaries and with shapes of different dimensions it can be used to evolve such objects according to a variety of motions including mean curvature if discontinuous velocity fields are allowed the dimension of the objects can change the evolution method that we propose guarantees that we stay in the class of vdf s and therefore that the intrinsic properties of the underlying shapes such as their dimension curvatures can be read off easily from the vdf and its spatial derivatives at each time instant the main disadvantage of the method is its redundancy the size of the representation is always that of the ambient space even though the object we are representing may be of a much lower dimension this disadvantage is also one of its strengths since it buys us flexibility

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

deformotion deforming motion shape average and the joint registration and approximation of structures in images what does it mean for a deforming object to be moving how can we separate the overall motion a finite dimensional group action from the more general deformation a diffeomorphism in this paper we propose a definition of motion for a deforming object and introduce a notion of shape average as the entity that separates the motion from the deformation our definition allows us to derive novel and efficient algorithms to register non identical shapes using region based methods and to simultaneously approximate and align structures in greyscale images we also extend the notion of shape average to that of a moving average in order to track moving and deforming objects through time the algorithms we propose extend prior work on landmark based matching to smooth curves and involve the numerical integration of partial differential equations which we address within the framework of level set methods

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

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

stochastic differential equations and geometric flows in previous years curve evolution applied to a single contour or to the level sets of an image via partial differential equations has emerged as an important tool in image processing and computer vision curve evolution techniques have been utilized in problems such as image smoothing segmentation and shape analysis we give a local stochastic interpretation of the basic curve smoothing equation the so called geometric heat equation and show that this evolution amounts to a tangential diffusion movement of the particles along the contour moreover assuming that a priori information about the shapes of objects in an image is known we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise we also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features in contrast to the original geometric heat flow which tends to circularize any closed curve

marginal maximum likelihood estimation for a psychometric model of discontinuous development item response theory models posit latent variables to account for regularities in students performances on test items wilson s saltus model extends the ideas of irt to development that occurs in stages where expected changes can be discontinuous show different patterns for different types of items or even exhibit reversals in probabilities of success on certain tasks examples include piagetian stages of psychological development and siegler s rule based learning this paper derives marginal maximum likelihood mml estimation equations for the structural parameters of the saltus model and suggests a computing approximation based on the em algorithm for individual examinees empirical bayes probabilities of learning stage are given along with proficiency parameter estimates conditional on stage membership the mml solution is illustrated with simulated data and an example from the domain of mixed number subtraction

graph characteristics from the heat kernel trace graph structures have been proved important in high level vision since they can be used to represent structural and relational arrangements of objects in a scene one of the problems that arises in the analysis of structural abstractions of objects is graph clustering in this paper we explore how permutation invariants computed from the trace of the heat kernel can be used to characterize graphs for the purposes of measuring similarity and clustering the heat kernel is the solution of the heat equation and is a compact representation of the path length distribution on a graph the trace of the heat kernel is given by the sum of the laplacian eigenvalues exponentiated with time we explore three different approaches to characterizing the heat kernel trace as a function of time our first characterization is based on the zeta function which from the mellin transform is the moment generating function of the heat kernel trace our second characterization is unary and is found by computing the derivative of the zeta function at the origin the third characterization is derived from the heat content i e the sum of the elements of the heat kernel we show how the heat content can be expanded as a power series in time and the coefficients of the series can be computed using the laplacian spectrum we explore the use of these characterizations as a means of representing graph structure for the purposes of clustering and compare them with the use of the laplacian spectrum experiments with the synthetic and real world databases reveal that each of the three proposed invariants is effective and outperforms the traditional laplacian spectrum moreover the heat content invariants appear to consistently give the best results in both synthetic sensitivity studies and on real world object recognition problems

the medial scaffold of 3d unorganized point clouds we introduce the notion of the medial scaffold a hierarchical organization of the medial axis of a 3d shape in the form of a graph constructed from special medial curves connecting special medial points a key advantage of the scaffold is that it captures the qualitative aspects of shape in a hierarchical and tightly condensed representation we propose an efficient and exact method for computing the medial scaffold based on a notion of propagation along the scaffold itself starting from initial sources of the flow and constructing the scaffold during the propagation we examine this method specifically in the context of an unorganized cloud of points in 3d e g as obtained from laser range finders which typically involve hundreds of thousands of points but the ideas are generalizable to data arising from geometrically described surface patches the computational bottleneck in the propagation based scheme is in finding the initial sources of the flow we thus present several ideas to avoid the unnecessary consideration of pairs of points which cannot possibly form a medial point source such as the visibility of a point from another given a third point and the interaction of clusters of points an application of using the medial scaffold for the representation of point samplings of real life objects is also illustrated

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

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

floer cohomology and pencils of quadrics there is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces we investigate symplectic aspects of this relationship with a view to applications in low dimensional topology

the attack of the psychometricians this paper analyzes the theoretical pragmatic and substantive factors that have hampered the integration between psychology and psychometrics theoretical factors include the operationalist mode of thinking which is common throughout psychology the dominance of classical test theory and the use of construct validity as a catch all category for a range of challenging psychometric problems pragmatic factors include the lack of interest in mathematically precise thinking in psychology inadequate representation of psychometric modeling in major statistics programs and insufficient mathematical training in the psychological curriculum substantive factors relate to the absence of psychological theories that are sufficiently strong to motivate the structure of psychometric models following the identification of these problems a number of promising recent developments are discussed and suggestions are made to further the integration of psychology and psychometrics

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

highly accurate schemes for pde based morphology with general convex structuring elements the two fundamental operations in morphological image processing are dilation and erosion these processes are defined via structuring elements it is of practical interest to consider a variety of structuring element shapes the realisation of dilation erosion for convex structuring elements by use of partial differential equations pdes allows for digital scalability and subpixel accuracy however numerical schemes suffer from blur by dissipative artifacts in our paper we present a family of so called flux corrected transport fct schemes that addresses this problem for arbitrary convex structuring elements the main characteristics of the fct schemes are i they keep edges very sharp during the morphological evolution process and ii they feature a high degree of rotational invariance we validate the fct scheme theoretically by proving consistency and stability numerical experiments with diamonds and ellipses as structuring elements show that fct schemes are superior to standard schemes in the field of pde based morphology

constructing simple stable descriptions for image partitioning a new formulation of the image partitioning problem is presented construct a complete and stable description of an image in terms of a specified descriptive language that is simplest in the sense of being shortest we show that a descriptive language limited to a low order polynomial description of the intensity variation within each region and a chain code like description of the region boundaries yields intuitively satisfying partitions for a wide class of images the advantage of this formulation is that it can be extended to deal with subsequent steps of the image understanding problem or to deal with other attributes such as texture in a natural way by augmenting the descriptive language experiments performed on a variety of both real and synthetic images demonstrate the superior performance of this approach over partitioning techniques based on clustering vectors of local image attributes and standard edge detection techniques

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

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

area and length minimizing flows for shape segmentation a number of active contour models have been proposed that unify the curve evolution framework with classical energy minimization techniques for segmentation such as snakes the essential idea is to evolve a curve in two dimensions or a surface in three dimensions under constraints from image forces so that it clings to features of interest in an intensity image the evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional tailored to features such as edges however because the flow may be slow to converge in practice a constant hyperbolic term is added to keep the curve surface moving in the desired direction we derive a modification of this term based on the gradient flow derived from a weighted area functional with image dependent weighting factor when combined with the earlier modified length gradient flow we obtain a partial differential equation pde that offers a number of advantages as illustrated by several examples of shape segmentation on medical images in many cases the weighted area flow may be used on its own with significant computational savings

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

recovery of parametric models from range images the case for superquadrics with global deformations a method for recovery of compact volumetric models for shape representation of single part objects in computer vision is introduced the models are superquadrics with parametric deformations bending tapering and cavity deformation the input for the model recovery is three dimensional range points model recovery is formulated as a least squares minimization of a cost function for all range points belonging to a single part during an iterative gradient descent minimization process all model parameters are adjusted simultaneously recovery position orientation size and shape of the model such that most of the given range points lie close to the model s surface a specific solution among several acceptable solutions where are all minima in the parameter space can be reached by constraining the search to a part of the parameter space the many shallow local minima in the parameter space are avoided as a solution by using a stochastic technique during minimization results using real range data show that the recovered models are stable and that the recovery procedure is fast

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

minimal surfaces a geometric three dimensional segmentation approach a novel geometric approach for three dimensional object segmentation is presented the scheme is based on geometric deformable surfaces moving towards the objects to be detected we show that this model is related to the computation of surfaces of minimal area local minimal surfaces the space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected the general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework the scheme is stable robust and automatically handles changes in the surface topology during the deformation results related to existence uniqueness stability and correctness of the solution to this geometric deformable model are presented as well based on an efficient numerical algorithm for surface evolution we present a number of examples of object detection in real and synthetic images

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

a moving superimposed finite element method for structural topology optimization level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures in this paper a dynamic implicit boundary based moving superimposed finite element method s version fem or s fem is developed for structural topology optimization using the level set methods in which the variational interior and exterior boundaries are represented by the zero level set both a global mesh and an overlaying local mesh are integrated into the moving s fem analysis model a relatively coarse fixed eulerian mesh consisting of bilinear rectangular elements is used as a global mesh the local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function in numerical integration using the gauss quadrature rule the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes a double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co ordinate systems an element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency a simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint a physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated pde solving procedure the proposed moving s fem is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems for the classical elasticity problems in the literature the present s fem can achieve numerical results in good agreement with those from the theoretical solutions and or numerical results from the standard fem for the minimum compliance topology optimization problems in structural optimization the present approach significantly outperforms the well recognized ersatz material approach as expected in the accuracy of the strain field numerical stability and representation fidelity at the expense of increased computational time it is also shown that the present approach is able to produce structures near the theoretical optimum it is suggested that the present s fem can be a promising tool for shape and topology optimization using the level set methods copyright 2005 john wiley sons ltd

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

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

prior knowledge level set representations visual grouping in this paper we propose a level set method for shape driven object extraction we introduce a voxel wise probabilistic level set formulation to account for prior knowledge to this end objects are represented in an implicit form constraints on the segmentation process are imposed by seeking a projection to the image plane of the prior model modulo a similarity transformation the optimization of a statistical metric between the evolving contour and the model leads to motion equations that evolve the contour toward the desired image properties while recovering the pose of the object in the new image upon convergence a solution that is similarity invariant with respect to the model and the corresponding transformation are recovered promising experimental results demonstrate the potential of such an approach

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

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

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

on the evolution of curves via a function of curvature i the classical case abstract the problem of curve evolution as a function of its local geometry arises naturally in many physical applications a special case of this problem is the curve shortening problem which has been extensively studied here we consider the general problem and prove an existence theorem for the classical solution the main theorem rests on lemmas that bound the evolution of length curvature and how far the curve can travel

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