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

on critical phenomena in interacting growth systems part ii bounded growth this paper completes the classification of some infinite and finite growth systems which was started in part i components whose states are integer numbers interact in a local deterministic way in addition to which every component s state grows by a positive integerk with a probability ek 1 e at every moment of the discrete time proposition 1 says that in the infinite system which starts from the state all zeros percentages of elements whose states exceed a given valuek 0 never exceed ce k wherec const proposition 2 refers to finite systems it states that the same inequalities hold during a time which depends exponentially on the system size

moments of the position of the maximum for gue characteristic polynomials and for log correlated gaussian processes we study three instances of log correlated processes on the interval the logarithm of the gaussian unitary ensemble gue characteristic polynomial the gaussian log correlated potential in presence of edge charges and the fractional brownian motion with hurst index h rightarrow 0 fbm0 in previous collaborations we obtained the probability distribution function pdf of the value of the global minimum equivalently maximum for the first two processes using the freezing duality conjecture fdc here we study the pdf of the position of the maximum x m through its moments using replica this requires calculating moments of the density of eigenvalues in the beta jacobi ensemble using jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary beta 0 and positive integer n in terms of sums over partitions for positive moments this expression agrees with a very recent independent derivation by mezzadri and reynolds we check our results against a contour integral formula derived recently by borodin and gorin presented in the appendix 1 from these authors the duality necessary for the fdc to work is proved and on our expressions found to correspond to exchange of partitions with their dual performing the limit n rightarrow 0 and to negative dyson index beta rightarrow 2 we obtain the moments of x m and give explicit expressions for the lowest ones numerical checks for the gue polynomials performed independently by n simm indicate encouraging agreement some results are also obtained for moments in laguerre hermite gaussian as well as circular and related ensembles the correlations of the position and the value of the field at the minimum are also analyzed

diagonal approximation of the form factor of the unitary group the form factor of the unitary group u n endowed with the haar measure characterizes the correlations within the spectrum of a typical unitary matrix it can be decomposed into a sum over pairs of periodic orbits where by periodic orbit we understand any sequence of matrix indices from here the diagonal approximation can be defined in the usual fashion as a sum only over pairs of identical orbits we prove that as we take the dimension n to infinity the diagonal approximation becomes exact that is converges to the full form factor in the interval 0 1

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

full counting statistics of chaotic cavities from classical action correlations we present a trajectory based semiclassical calculation of the full counting statistics of quantum transport through chaotic cavities in the regime of many open channels our method to obtain the mth moment of the density of transmission eigenvalues requires two correlated sets of m classical trajectories therefore generalizing previous works on conductance and shot noise the semiclassical results agree for all values of m with the corresponding predictions from random matrix theory

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

lyndon word decompositions and pseudo orbits on q nary graphs abstract a foundational result in the theory of lyndon words words that are strictly earlier in lexicographic order than their cyclic permutations is the chen fox lyndon theorem which states that every word has a unique non increasing decomposition into lyndon words this article extends this factorization theorem obtaining the proportion of these decompositions that are strictly decreasing this result is then used to count primitive pseudo orbits sets of primitive periodic orbits on q nary graphs as an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients of q nary quantum graphs

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

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

a hypergeometric generating function approach to charge counting statistics in ballistic chaotic cavities we apply the method of multivariate hypergeometric generating function to study cumulants of the charge counting statistics ccs of a ballistic chaotic cavity coupled ideally to two electron reservoirs via perfect conducting leads with an arbitrary number of scattering channels the underlying chaotic dynamics causes each cumulant of ccs denoted charge transfer cumulant ctc to behave like a random variable which we describe via exact calculations of its statistical moments and cumulants besides reproducing several known results of the literature for the first few statistical cumulants of conductance and shot noise power which are the first and second ctc respectively we obtain new exact results for the first four statistical cumulants of the third and fourth ctc all analytical results are supported by numerical simulations of the circular ensembles

semiclassics for chaotic systems with tunnel barriers the addition of tunnel barriers to open chaotic systems as well as representing more general physical systems leads to a much richer semiclassical dynamics in particular we present here a complete semiclassical treatment for these systems in the regime where ehrenfest time effects are negligible and for times shorter than the heisenberg time to start we explore the trajectory structures which contribute to the survival probability and find results that are also in agreement with random matrix theory then we progress to the treatment of the probability current density and are able to show using recursion relation arguments that the continuity equation connecting the current density to the survival probability is satisfied to all orders in the semiclassical approximation following on we also consider a correlation function of the scattering matrix for which we have to treat a new set of possible trajectory diagrams by simplifying the contributions of these diagrams we show that the results obtained here are consistent with the known properties of the scattering matrix the correlation function can be trivially connected to the ac and dc conductances quantities of particular interest for which finally we present a semiclassical expansion

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

optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension in binary images the distance transformation dt and the geometrical skeleton extraction are classic tools for shape analysis in this paper we present time optimal algorithms to solve the reverse euclidean distance transformation and the reversible medial axis extraction problems for d dimensional images we also present a d dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

transmission eigenvalue densities and moments in chaotic cavities from random matrix theory we point out that the transmission eigenvalue density and higher order correlation functions in chaotic cavities for an arbitrary number of incoming and outgoing leads n1 n 2 are analytically known from the jacobi ensemble of random matrix theory using this result and a simple linear statistic we give an exact and non perturbative expression for moments of the form m for m n1 n2 1 and 2 thus improving the existing results in the literature secondly we offer an independent derivation of the average density and higher order correlation function for 2 4 which does not make use of the orthogonal polynomials technique this result may be relevant for an efficient numerical implementation avoiding determinants pacs numbers 73 23 b 73 50 td 05 45 mt 73 63 kv

combinatorial theory of the semiclassical evaluation of transport moments i equivalence with the random matrix approach to study electronic transport through chaotic quantum dots there are two main theoretical approachs one involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging the other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories there are established evaluation procedures within the semiclassical evaluation that for several linear and non linear transport moments to which they were applied have always resulted in the agreement with random matrix predictions we prove that this agreement is universal any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory the equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs maps with certain properties and exhibiting systematic cancellations among their contributions remaining trajectory sets can be identified with primitive palindromic factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices the equivalence is proved for systems with and without time reversal symmetry

on the intrinsic reconstruction of shape from its symmetries the main question we address is what is the minimal information required to generate closed nonintersecting planar boundaries for this paper we restrict shape to this meaning more precisely we examine whether the medial axis together with dynamics can serve as a language to design shapes and to effect shape changes we represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points a sub 1 sup 2 a sub 1 sup 3 a sub 3 and the associated six types of shock points first we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary this implies that a rather coarse sampling of the symmetry axis its tangent curvature speed and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points a sub 1 sup 2 second we examine the reconstruction of shape at branch points a sub 1 sup 3 where three regular branches are joined we show that the three pairs of geometry that is curvature and dynamics that is acceleration must satisfy certain constraints finally we derive similar results for the end points of shock branches a sub 3 points these formulas completely specify the local reconstruction of a shape from its shock graph or medial axis and the conditions required to form a coherent shape from the medial axis

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

computing skeletons in three dimensions skeletonization will probably become as valuable a tool for shape analysis in 3d as it is in 2d we present a topology preserving 3d skeletonization method which computes both surface and curve sk

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

simple one dimensional interaction systems with superexponential relaxation times finite one dimensional random processes with local interaction are presented which keep some information of a topological nature about their initial conditions during time the logarithm of whose expectation grows asymptotically at least asm3 wherem is the size of the setr m of states of one component actuallyr m is a circle of lengthm at every moment of the discrete time every component turns into some kind of average of its neighbors after which it makes a random step along this circle all these steps are mutually independent and identically distributed in the present version the absolute values of the steps never exceed a constant the processes are uniform in space time and the set of states this estimation contributes to our awareness of what kind of stable behavior one can expect from one dimensional random processes with local interaction

modeling multi object configurations via medial skeletal linking structures we introduce a method for modeling a configuration of objects in 2d or 3d images using a mathematical skeletal linking structure which will simultaneously capture the individual shape features of the objects and their positional information relative to one another the objects may either have smooth boundaries and be disjoint from the others or share common portions of their boundaries with other objects in a piecewise smooth manner these structures include a special class of blum medial linking structures which are intrinsically associated to the configuration and build upon the blum medial axes of the individual objects we give a classification of the properties of blum linking structures for generic configurations the skeletal linking structures add increased flexibility for modeling configurations of objects by relaxing the blum conditions and they extend in a minimal way the individual skeletal structures which have been previously used for modeling individual objects and capturing their geometric properties this allows for the mathematical methods introduced for single objects to be significantly extended to the entire configuration of objects these methods not only capture the internal shape structures of the individual objects but also the external structure of the neighboring regions of the objects in the subsequent second paper damon and gasparovic in shape and positional geometry of multi object configurations we use these structures to identify specific external regions which capture positional information about neighboring objects and we develop numerical measures for closeness of portions of objects and their significance for the configuration this allows us to use the same mathematical structures to simultaneously analyze both the shape properties of the individual objects and positional properties of the configuration this provides a framework for analyzing the statistical properties of collections of similar configurations such as for applications to medical imaging

leading off diagonal approximation for the spectral form factor for uniformly hyperbolic systems we consider the semiclassical approximation to the spectral form factor k for two dimensional uniformly hyperbolic systems with time reversal symmetry and derive the first off diagonal correction for small the result agrees with the 2 term of the form factor for the goe random matrix ensemble

spectral statistics in chaotic systems with a point interaction we consider quantum systems with a chaotic classical limit that are perturbed by a point like scatterer the spectral form factor k for these systems is evaluated semiclassically in terms of periodic and diffractive orbits it is shown for order 2 and 3 that off diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits implying that the perturbation by the scatterer does not change the spectral statistic we further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

nonlinear random matrix statistics symmetric functions and hyperdeterminants nonlinear statistics i e statistics of permanents on the eigenvalues of invariant random matrix models are considered for the three dyson s symmetry classes 1 2 4 general formulas in terms of hyperdeterminants are found for 2 for specific cases and all s more computationally efficient results are obtained based on symmetric functions expansions as an application we consider the case of quantum transport in chaotic cavities extending results from savin et al 2008 phys rev b 77 125332

three dimensional descriptions based on the analysis of the invariant and quasi invariant properties of some curved axis generalized cylinders we address the recovery of object level 3d descriptions of some classes of curved axis generalized cylinders for this the first part of the paper analyzes the projective properties of two common generic shapes planar right constant generalized cylinders prcgcs and circular planar right generalized cylinders circular prgcs the properties we analyze include new geometric invariant and quasi invariant properties of the orthographic projection of the above shapes and a useful classification of their structural properties as functions of their pose the second part of the paper describes an implemented system which detects and recovers prcgcs and circular prgcs from an intensity image in the presence of noise surface markings shadows and partial occlusion the methods exploit the projective properties to hypothesize and verify relevant curved axis objects thus explicitly using the three dimensionality of the objects and of the desired descriptions this work extends past work on the recovery of volumetric shapes from an intensity image by addressing new primitives deriving new properties and by developing a system that recovers them from an intensity image we demonstrate our method on several real intensity images

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

statistics of quantum transport in weakly nonideal chaotic cavities we consider statistics of electronic transport in chaotic cavities where time reversal symmetry is broken and one of the leads is weakly nonideal that is it contains tunnel barriers characterized by tunneling probabilities i using symmetric function expansions and a generalized selberg integral we develop a systematic perturbation theory in 1 i valid for an arbitrary number of channels and obtain explicit formulas up to second order for the average and variance of the conductance and for the average shot noise higher moments of the conductance are considered to leading order

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

on critical phenomena in interacting growth systems part i general components which are placed in a finite or infinite space have integer numbers as possible states they interact in a discrete time in a local deterministic way in addition to which all the components states are incremented at every time step by independent identically distributed random variables we assume that the deterministic interaction function is translation invariant and monotonic and that its values are between the minimum and the maximum of its arguments theorems 1 and 2 based on propositions which we give in a separate part ii give sufficient conditions for a system to have an invariant distribution or a bounded mean other statements proved herein provide background for them by giving conditions when a system has no invariant distribution or the mean of its components states tends to infinity all our main results use one and the same geometrical criterion

moments of the wigner delay times the wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system for chaotic systems the statistics of the individual delay times whose average is the wigner time delay are thought to be well described by random matrix theory here we present a semiclassical derivation showing the validity of random matrix results in order to simplify the semiclassical treatment we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies in the semiclassical approximation the elements of the scattering matrix are given in terms of the classical scattering trajectories requiring one to study correlations between sets of such trajectories we describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number this allows us to derive a polynomial equation satisfied by the generating function of the moments along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory we infer that the scattering matrix is unitary to all orders in the semiclassical approximation

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

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos

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

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

large n expansions for the laguerre and jacobi β ensembles from the loop equations the ensembles of random matrix theory with classical weights have many special properties one is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at the general order of the correlator via aomoto s method from the theory of the selberg integral we use aomoto s method to derive the full hierarchy of loop equations for laguerre and jacobi ensembles and use these to systematically construct the explicit form of the 1 n expansion at low orders this allows us to give the explicit form of corrections to the global density and allows various moments to be computed complementing results available in the literature motivated by problems in quantum transport

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

quantum conductance problems and the jacobi ensemble in one dimensional transport problems the scattering matrix s is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends for s a random unitary matrix the singular value probability distribution function of these blocks is calculated the same is done when s is constrained to be symmetric or to be self dual quaternion real as is relevant in the presence of particular time reversal symmetries the latter results are shown to be very similar to those obtained under the unphysical assumption that s has real elements or has real quaternion elements respectively three methods are used metric forms a variant of the ingham seigel matrix integral and a theorem specifying the jacobi random matrix ensemble in terms of wishart distributed matrices

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

construction of iso contours bisectors and voronoi diagrams on triangulated surfaces in the research of computer vision and machine perception 3d objects are usually represented by 2 manifold triangular meshes m in this paper we present practical and efficient algorithms to construct iso contours bisectors and voronoi diagrams of point sites on m based on an exact geodesic metric compared to euclidean metric spaces the voronoi diagrams on m exhibit many special properties that fail all of the existing euclidean voronoi algorithms to provide practical algorithms for constructing geodesic metric based voronoi diagrams on m this paper studies the analytic structure of iso contours bisectors and voronoi diagrams on m after a necessary preprocessing of model m practical algorithms are proposed for quickly obtaining full information about iso contours bisectors and voronoi diagrams on m the complexity of the construction algorithms is also analyzed finally three interesting applications surface sampling and reconstruction 3d skeleton extraction and point pattern analysis are presented that show the potential power of the proposed algorithms in pattern analysis

irregular shape symmetry analysis theory and application to quantitative galaxy classification this paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the amount of symmetry of arbitrary shapes the definitions of both bilateral symmetricity and rotational symmetricity give new insight into analyzing the geometrical property of a shape and enable characterizing arbitrary shapes in a new way we developed a set of criteria for quantitative galaxy classification using our proposed irregular shape symmetry measures our study has demonstrated the effectiveness of the proposed method for the characterization of the shape of the celestial bodies the concepts described in the paper are applicable to many fields such as mathematics artificial intelligence digital image processing robotics biomedicine etc

semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems we present a semiclassical calculation of the generalized form factor kab tau which characterizes the fluctuations of matrix elements of the operators a and b in the eigenbasis of the hamiltonian of a chaotic system our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f 2 degrees of freedom that allow us to go beyond the diagonal approximation first we extend these techniques to systems with f 2 then we use these results to calculate kab tau we show that the dependence on the rescaled time tau time in units of the heisenberg time is universal for both the spectral and the generalized form factor furthermore we derive a relation between kab tau and the classical time correlation function of the weyl symbols of a and b