generalized overlap measures for evaluation and validation in medical image analysis measures of overlap of labelled regions of images such as the dice and tanimoto coefficients have been extensively used to evaluate image registration and segmentation algorithms modern studies can include multiple labels defined on multiple images yet most evaluation schemes report one overlap per labelled region simply averaged over multiple images in this paper common overlap measures are generalized to measure the total overlap of ensembles of labels defined on multiple test images and account for fractional labels using fuzzy set theory this framework allows a single figure of merit to be reported which summarises the results of a complex experiment by image pair by label or overall a complementary measure of error the overlap distance is defined which captures the spatial extent of the nonoverlapping part and is related to the hausdorff distance computed on grey level images the generalized overlap measures are validated on synthetic images for which the overlap can be computed analytically and used as similarity measures in nonrigid registration of three dimensional magnetic resonance imaging mri brain images finally a pragmatic segmentation ground truth is constructed by registering a magnetic resonance atlas brain to 20 individual scans and used with the overlap measures to evaluate publicly available brain segmentation algorithms

shape recognition using eigenvalues of the dirichlet laplacian the eigenvalues of the dirichlet laplacian are used to generate three different sets of features for shape recognition and classification in binary images the generated features are rotation translation and size invariant the features are also shown to be tolerant of noise and boundary deformation these features are used to classify hand drawn synthetic and natural shapes with correct classification rates ranging from 88 9 to 99 2 the classification was done using few features only two features in some cases and simple feedforward neural networks or minimum euclidian distance

k classes for matroids and equivariant localization to every matroid we associate a class in the k theory of the grassmannian we study this class using the method of equivariant localization in particular we provide a geometric interpretation of the tutte polynomial we also extend results of the second author concerning the behavior of such classes under direct sum series and parallel connection and two sum these results were previously only established for realizable matroids and their earlier proofs were more difficult r esum e a chaque matronous associons une classe dans la k thde la grassmannienne nous

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

a gromov hausdorff framework with diffusion geometry for topologically robust non rigid shape matching in this paper the problem of non rigid shape recognition is studied from the perspective of metric geometry in particular we explore the applicability of diffusion distances within the gromov hausdorff framework while the traditionally used geodesic distance exploits the shortest path between points on the surface the diffusion distance averages all paths connecting the points the diffusion distance constitutes an intrinsic metric which is robust in particular to topological changes such changes in the form of shortcuts holes and missing data may be a result of natural non rigid deformations as well as acquisition and representation noise due to inaccurate surface construction the presentation of the proposed framework is complemented with examples demonstrating that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non rigid shapes

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

orbits in pr n and equivariant quantum cohomology abstract we compute the g l r 1 equivariant chow class of the g l r 1 orbit closure of any point x 1 x n p r n in terms of the rank polytope of the matroid represented by x 1 x n p r using these classes and generalizations involving point configurations in higher dimensional projective spaces we define for each d n matrix m an n ary operation m on the small equivariant quantum cohomology ring of p r which is the n ary quantum product when m is an invertible matrix we prove that m m is a valuative matroid polytope association like the quantum product these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting as an application we compute the number of line sections with given moduli of a general degree 2 r 1 hypersurface in p r generalizing the known case of quintic plane curves

hadamard inverses square roots and products of almost semidefinite matrices abstract let a aij be an n n symmetric matrix with all positive entries and just one positive eigenvalue bapat proved then that the hadamard inverse of a given by a 1 1 a ij is positive semidefinite we show that if moreover a is invertible then a 1 is positive definite we use this result to obtain a simple proof that with the same hypotheses on a except that all the diagonal entries of a are zero the hadamard square root of a given by a 1 2 a ij 1 2 has just one positive eigenvalue and is invertible finally we show that if a is any positive semidefinite matrix and b is almost positive definite and invertible then a b 1 e t b 1 e a

generalized schur complements and p complementable operators abstract let a be a selfadjoint operator and p be an orthogonal projection both operating on a hilbert space h we say that a is p complementable if a p 0 holds for some r in this case we define i p a max r a p 0 as a tool for computing ip a we introduce a natural generalization of the schur complement or shorted operator of a to s r p denoted by a p we give expressions and a characterization for ip a that generalize some known results for particular choices of p we also study some aspects of the shorted operator a p for p complementable a under the hypothesis of compatibility of the pair a s we give some applications in the finite dimensional context

geometry in grassmannians and a generalization of the dilogarithm abstract in this paper we introduce some polyhedra in grassman manifolds which we call grassmannian simplices we study two aspects of these polyhedra their combinatorial structure section 2 and their relation to harmonic differential forms on the grassmannian section 3 using this we obtain results about some new differential forms one of which is the classical dilogarithm section 1 the results here unite two threads of mathematics that were much studied in the 19th century the analytic one concerning the dilogarithm goes back to leibnitz 1696 and euler 1779 and the geometric one concerning grassmannian simplices can be traced to binet 1811 in section 4 we give some of this history along with some recent related results and open problems in section 0 we give as an introduction an account in geometric terms of the simplest cases

equivariant k theory and equivariant cohomology for t an abelian compact lie group we give a description of t equivariant k theory with complex coefficients in terms of equivariant cohomology in the appendix we give applications of this by extending results of chang skjelbred and goresky kottwitz macpherson from equivariant cohomology to equivariant k theory

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

a diffusion mechanism for obstacle detection from size change information a mechanism fur the visual detection of obstacles is presented a new immediacy measure representing the imminence of collision between an object and a moving observer is defined a diffusion process on the image domain whose initial condition is determined by the motion field normal to the object s boundary is shown to converge asymptotically to the immediacy measure a network of locally connected cells derived from a finite difference approximation of the diffusion equation estimates the immediacy measure from normal velocity and boundary information provided by a motion measurement and segmentation stage the algorithm s performance on real image sequences is demonstrated

xi and eta function resummation of finite series general case given a spectrum of positive numbers lambda m from which a xi function z s sigma m lambda sm can be constructed the reorganization of series of the type f s t identical to m lambda smf lambda mt into power series in t is examined in detail using the method of xi function resummation for summand functions f lambda mt having power series expansions in lambda mt with infinite radius of convergence and which satisfy other conditions of a rather general nature the author finds that f s t can be reorganized to f s t n antbn k cktdk in t r s t where r s t vanishes exponentially as t to 0 the numbers an bn cn dn can all be computed in terms of the xi function z s r s t is difficult to evaluate but important general features of this function can be determined the power series expansion of f s t can be regarded as a generalization of the heat kernel expansion for which f lambda mt exp lambda mt and s 0 to nonzero complex variable s which is useful and to many other summand functions f lambda mt remarkably the xi function resummation method can be applied as easily to divergent series f s t as it can to convergent ones the method is therefore both a rearrangement procedure for convergent series and a summation prescription for divergent series

the number of critical points of a product of powers of linear functions be the complement of a given a complex n vector 2 21 2n e cn consider the multivalued holomorphic function defined on m by studying bethe vectors in statistical mechanics a varchenko 18 conjectured that for generic 2 all critical points of qs are nondegenerate and the number of critical points is equal to the absolute value of the euler characteristic of m m he proved the conjecture for complexified real arrangements in 18 a similar result was known to k aomoto 1 example 1 who stated without proof that for positive 2 the number of critical points of b equals the number of bounded components of the complement of the real arrangement see 2 theorem 4 4 1 1 for details in this paper we prove varchenko s conjecture for all arrangements more generally we give a formula for the number of critical points of the multivalued holomorphic function

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

torus orbits on homogeneous varieties and kac polynomials of quivers in this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the kac polynomials of supernova quivers which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line we also prove that these polynomials can be expressed as a specialization of tutte polynomials of certain graphs providing a new way to compute them we also discuss connections with the gel fand macpherson correspondence

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

wesd weighted spectral distance for measuring shape dissimilarity this paper presents a new distance for measuring shape dissimilarity between objects recent publications introduced the use of eigenvalues of the laplace operator as compact shape descriptors here we revisit the eigenvalues to define a proper distance called weighted spectral distance wesd for quantifying shape dissimilarity the definition of wesd is derived through analyzing the heat trace this analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects we analyze the resulting distance definition present and prove its important theoretical properties some of these properties include 1 wesd is defined over the entire sequence of eigenvalues yet it is guaranteed to converge 2 it is a pseudometric 3 it is accurately approximated with a finite number of eigenvalues and 4 it can be mapped to the 0 1 interval last experiments conducted on synthetic and real objects are presented these experiments highlight the practical benefits of wesd for applications in vision and medical image analysis

an inverse problem of the wave equation for a general doubly connected region in r2 with a finite number of piecewise smooth robin boundary conditions the spectral distribution m t w 1 exp ite w 1 2 where e w w 1 are the eigenvalues of the negative laplacian d k 1 2 x k 2in the x 1 x 2 plane is studied for a variety of domains where 0

optimal bounds for ratios of eigenvalues of one dimensional schrodinger operators with dirichlet boundary conditions and positive potentials consider the schrodinger equation u v x u u on the intervali wherev x 0 forx i and where dirichlet boundary conditions are imposed at the endpoints ofi we prove the optimal bound frac lambda n lambda 1 leqq n 2 for n 2 3 4 on the ratio of thenth eigenvalue to the first eigenvalue for this problem this leads to a complete treatment of bounds on ratios of eigenvalues for such problems extensions of these results to singular problems are also presented a modified prufer transformation and comparison techniques are the key elements of the proof

higher algebraic k theory for actions of diagonalizable groups we study the k theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces our main result shows how to reconstruct the k theory ring of such an action from the k theory rings of the loci where the stabilizers have constant dimension we apply this to the calculation of the equivariant k theory of toric varieties and give conditions under which the merkurjev spectral sequence degenerates so that the equivariant k theory ring determines the ordinary k theory ring we also prove a very refined localization theorem for actions of this type

analysis of two dimensional non rigid shapes analysis of deformable two dimensional shapes is an important problem encountered in numerous pattern recognition computer vision and computer graphics applications in this paper we address three major problems in the analysis of non rigid shapes similarity partial similarity and correspondence we present an axiomatic construction of similarity criteria for deformation invariant shape comparison based on intrinsic geometric properties of the shapes and show that such criteria are related to the gromov hausdorff distance next we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole and present a construction of set valued distances based on the notion of pareto optimality finally we show that the correspondence between non rigid shapes can be obtained as a byproduct of the non rigid similarity problem as a numerical framework we use the generalized multidimensional scaling gmds method which is the numerical core of the three problems addressed in this paper

discriminative segmentation based evaluation through shape dissimilarity segmentation based scores play an important role in the evaluation of computational tools in medical image analysis these scores evaluate the quality of various tasks such as image registration and segmentation by measuring the similarity between two binary label maps commonly these measurements blend two aspects of the similarity pose misalignments and shape discrepancies not being able to distinguish between these two aspects these scores often yield similar results to a widely varying range of different segmentation pairs consequently the comparisons and analysis achieved by interpreting these scores become questionable in this paper we address this problem by exploring a new segmentation based score called normalized weighted spectral distance nwsd that measures only shape discrepancies using the spectrum of the laplace operator through experiments on synthetic and real data we demonstrate that nwsd provides additional information for evaluating differences between segmentations which is not captured by other commonly used scores our results demonstrate that when jointly used with other scores such as dice s similarity coefficient the additional information provided by nwsd allows richer more discriminative evaluations we show for the task of registration that through this addition we can distinguish different types of registration errors this allows us to identify the source of errors and discriminate registration results which so far had to be treated as being of similar quality in previous evaluation studies

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

valuative invariants for polymatroids abstract many important invariants for matroids and polymatroids such as the tutte polynomial the billera jia reiner quasi symmetric function and the invariant g introduced by the first author are valuative in this paper we construct the z modules of all z valued valuative functions for labeled matroids and polymatroids on a fixed ground set and their unlabeled counterparts the z modules of valuative invariants we give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes and explicit formulas for their ranks our results confirm a conjecture of the first author that g is universal for valuative invariants

voxel based morphometry should not be used with imperfectly registered images abstract john ashburner and karl friston 2000 introduced a standardized method of voxel based morphometry vbm for comparisons of local concentrations of gray matter between two groups of subjects segmented images of gray matter from grossly normalized high resolution images are smoothed and their group differences analyzed by the now conventional voxelwise worsley approach to gaussian random fields of differences this comment concerns an unfortunate interaction between the algorithm s spatial normalization and voxelwise comparison steps whereby several obvious quantitative confounds are injected at the core of the inference engine the authors put forward specifically the statistics of the resulting voxelwise comparisons are uninformative about group differences wherever the spatial normalization algorithm has failed to register on any robustly appearing image gradient the method of ashburner and friston is defensible only far from all image gradients

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

an observation on the hadamard product of hermitian matrices abstract we give the best possible bound from below in the lowener ordering for the hadamard product a b in terms of a multiple of a when a is positive semidefinite nonzero and b is positive definite

best possible upper and lower bounds for the zeros of the bessel function _ let j k denote the k th positive zero of the bessel function j x in this paper we prove that for 0 and k 1 2 3 ak 21 3 j k ak 21 3 3 20 ak 21 3 1 3 these bounds coincide with the first few terms of the well known asymptotic expansion j k ak 21 3 3 20 ak 21 3 1 3 as k being fixed where ak is the k th negative zero of the airy function ai x and so are best possible

a matroid invariant via the k theory of the grassmannian abstract let g d n denote the grassmannian of d planes in c n and let t be the torus c n diag c which acts on g d n let x be a point of g d n and let t x be the closure of the t orbit through x then the class of the structure sheaf of t x in the k theory of g d n depends only on which plucker coordinates of x are nonzero combinatorial data known as the matroid of x in this paper we will define a certain map of additive groups from k g d n to z t letting g x t denote the image of 1 n dim t x o t x g x behaves nicely under the standard constructions of matroid theory such as direct sum two sum duality and series and parallel extensions we use this invariant to prove bounds on the complexity of kapranov s lie complexes m kapranov chow quotients of grassmannians i adv soviet math 16 2 1993 29 110 hacking keel and tevelev s very stable pairs p hacking s keel e tevelev compactification of the moduli space of hyperplane arrangements j algebraic geom 15 2006 657 680 and the author s tropical linear spaces when they are realizable in characteristic zero d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 namely in characteristic zero a lie complex or the underlying d 1 dimensional scheme of a very stable pair can have at most n i 1 d i n d i i 1 strata of dimensions n i and d i respectively this prove the author s f vector conjecture from d speyer tropical linear spaces siam j discrete math 22 4 2008 1527 1558 in the case of a tropical linear space realizable in characteristic 0

brownian cuboid and its geometric characteristics abstract a class of random cuboid model which is defined brownian cuboid is constructed in this paper based on the structure characteristics of brownian cuboid and the properties of brownian motion geometric characteristics of brownian cuboid are discussed by which the probability estimates of the length of the space diagonal surface area and volume are obtained and moreover the small deviation estimates and some relevant results for the geometric characteristics of brownian cuboid are given additionally we directly derive the upper and lower bounds for the smallest positive zero of the bessel functions of the first kind

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros