on the generation of skeletons from discrete euclidean distance maps the skeleton is an important representation for shape analysis a common approach for generating discrete skeletons takes three steps 1 computing the distance map 2 detecting maximal disks from the distance map and 3 linking the centers of maximal disks cmds into a connected skeleton algorithms using approximate distance metrics are abundant and their theory has been well established however the resulting skeletons may be inaccurate and sensitive to rotation in this paper we study methods for generating skeletons based on the exact euclidean metric we first show that no previous algorithms identify the exact set of discrete maximal disks under the euclidean metric we then propose new algorithms and show that they are correct to link cmds into connected skeletons we examine two prevalent approaches connected thinning and steepest ascent we point out that the connected thinning approach does not work properly for euclidean distance maps only the steepest ascent algorithm produces skeletons that are truly medially placed the resulting skeletons have all the desirable properties they have the same simple connectivity as the figure they are well centered they are insensitive to rotation and they allow exact reconstruction the effectiveness of our algorithms is demonstrated with numerous examples

polynomial identities on superalgebras and exponential growth abstract let a be a finitely generated superalgebra over a field f of characteristic 0 to the graded polynomial identities of a one associates a numerical sequence cnsup a n 1 called the sequence of graded codimensions of a in case a satisfies an ordinary polynomial identity such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n c n sup a n exists and is a non negative integer we denote such integer by supexp a and we give an effective way for computing it as an application we construct eight superalgebras ai i 1 8 characterizing the identities of any finitely generated superalgebra a with supexp a 2 in the following way supexp a 2 if and only if idsup a idsup ai for some i 1 8 where idsup b is the ideal of graded identities of the algebra b we also compare the superexponent and the exponent see a giambruno m zaicev adv math 140 1998 145 155 of any finitely generated superalgebra

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

variational curve skeletons using gradient vector flow representing a 3d shape by a set of 1d curves that are locally symmetric with respect to its boundary i e curve skeletons is of importance in several machine intelligence tasks this paper presents a fast automatic and robust variational framework for computing continuous subvoxel accurate curve skeletons from volumetric objects a reference point inside the object is considered a point source that transmits two wave fronts of different energies the first front beta front converts the object into a graph from which the object salient topological nodes are determined curve skeletons are tracked from these nodes along the cost field constructed by the second front alpha front until the point source is reached the accuracy and robustness of the proposed work are validated against competing techniques as well as a database of 3d objects unlike other state of the art techniques the proposed framework is highly robust because it avoids locating and classifying skeletal junction nodes employs a new energy that does not form medial surfaces and finally extracts curve skeletons that correspond to the most prominent parts of the shape and hence are less sensitive to noise

shape representation using a generalized potential field model this paper is concerned with efficient derivation of the medial axis transform of a 2d polygonal region instead of using the shortest distance to the region border a potential field model is used for computational efficiency the region border is assumed to be charged and the valleys of the resulting potential field are used to estimate the axes for the medial axis transform the potential valleys are found by following the force field thus avoiding 2d search the potential field is computed in closed form using equations of the border segments the simple newtonian potential is shown to be inadequate for this purpose a higher order potential is defined which decays faster with distance than the inverse of distance it is shown that as the potential order becomes arbitrarily large the axes approach those computed using the shortest distance to the border algorithms are given for the computation of axes which can run in linear parallel time for part of the axes having initial guesses experimental results are presented for a number of examples

skeletonisation of three dimensional object using generalized potential field in this paper the potential based skeletonization approach for 2d medial axis transform mat which identifies object skeleton as potential valleys using a newtonian potential model in place of the distance function is generalized to three dimensions the generalized potential functions given by chung 1998 which decay faster with distance than the newtonian potential is used for the 3d case the efficiency of the proposed approach results from the fact that these functions and their gradients can be obtained in closed forms for polyhedral surfaces according to the simulation results the skeletons obtained with the proposed approach are closely related to the corresponding mat skeletons while the medial axis surface is 2d in general for a 3d object the potential valleys being one dimensional form a more realistic skeleton other desirable attributes of the algorithm include stability against perturbations of the object boundary the flexibility to obtain partial skeleton directly and low time complexity

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

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

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

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

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

transformation of uniformly distributed particle ensembles abstract this paper introduces a new method to construct asymptotically f distributed sequences of points in the r d the algorithm is based on a transformation proposed by hlawka and muck by exploitation of the special structure the method proves to be very efficient

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

actions of taft s algebras on finite dimensional algebras abstract let f be a field containing a primitive m th root of the unit we characterize the actions of a taft s algebra h m of a certain order m on finite dimensional arbitrary algebras we describe the action in terms of gradings and actions by skew derivations moreover we prove the associative algebra u t 2 of 2 2 upper triangular matrices with entries from f does not generate a variety of h m module algebras of almost polynomial growth

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

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

non semisimple hopf algebras of dimension p2 abstract let h be a finite dimensional hopf algebra with antipode s of dimension pq over an algebraically closed field of characteristic 0 where p q are odd primes if h is not semisimple then the order of s 4 is p and tr s 2 p is an integer divisible by p 2 in particular if dim h p 2 we prove that h is isomorphic to a taft algebra this completes the classification for the hopf algebras of dimension p 2

gradings on the algebra of upper triangular matrices and their graded identities abstract let k be an infinite field and let ut n k denote the algebra of n n upper triangular matrices over k we describe all elementary gradings on this algebra further we describe the generators of the ideals of graded polynomial identities of ut n k and we produce linear bases of the corresponding relatively free graded algebras we prove that one can distinguish the elementary gradings by their graded identities we describe bases of the graded polynomial identities in several typical cases although in these cases we consider elementary gradings by cyclic groups the same methods serve for elementary gradings by any finite group

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

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

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

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