finite element methods for active contour models and balloons for 2 d and 3 d images the use of energy minimizing curves known as snakes to extract features of interest in images has been introduced by kass witkin and terzopoulos 1987 a balloon model was introduced by cohen 1991 as a way to generalize and solve some of the problems encountered with the original method a 3 d generalization of the balloon model as a 3 d deformable surface which evolves in 3 d images is presented it is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential we also show properties of energy minimizing surfaces concerning their relationship with 3 d edge points to solve the minimization problem for a surface two simplified approaches are shown first defining a 3 d surface as a series of 2 d planar curves then after comparing finite element method and finite difference method in the 2 d problem we solve the 3 d model using the finite element method yielding greater stability and faster convergence this model is applied for segmenting magnetic resonance images

boundary reconstruction in binary images using splines in image analysis it is often required to reconstruct the boundary of an object in a noisy image this paper presents a new method which relies on flexibility and computational simplicity of b spline curves to reconstruct a smooth connected boundary in a noisy binary image boundary inference is based on oriented distance functions yielding the estimator which is interpreted as a posterior expected boundary of the underlying random set the performance of the method and its dependence on the image quality and model specification are studied on simulated data the method is applied to reconstruct the skin air boundary in digitised analogue mammogram images highlightswe present a new method for boundary reconstruction in noisy images using b spline curves the method is bayesian and uses oriented distance functions for inference the resulting estimator can be interpreted as a posterior expected boundary we study the performance of the method on simulated data we apply the method to reconstruct the skin air boundary in mammogram images

a multistage optimal active contour model energy minimizing active contour models or snakes can be used in many applications such as edge detection motion tracking image matching computer vision and three dimensional 3 d reconstruction we present a novel snake that is superior both in accuracy and convergence speed over previous snake algorithms high performance is achieved by using spline representation and dividing the energy minimization process into multiple stages the first stage is designed to optimize the convergence speed in order to allow the snake to quickly approach the minimum energy state the second stage is devoted to snake refinement and to local minimization of energy thereby driving the snake to a quasiminimum energy state the third stage uses the bellman 1957 optimality principle to fine tune the snake to the global minimum energy state this three stage scheme is optimized for both accuracy and speed

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

a limit theorem for shifted schur measures to each partition 1 2 with distinct parts we assign the probability q x p y z where q and p are the schur q functions and z is a normalization constant this measure which we call the shifted schur measure is analogous to the much studied schur measure for the specialization of the first m coordinates of x and the first n coordinates of y equal to 0 1 and the rest equal to zero we derive a limit law for 1 as m n with m n fixed for the schur measure the specialization limit law was derived by johansson our main result implies that the two limit laws are identical

using dynamic programming for solving variational problems in vision dynamic programming is discussed as an approach to solving variational problems in vision dynamic programming ensures global optimality of the solution is numerically stable and allows for hard constraints to be enforced on the behavior of the solution within a natural and straightforward structure as a specific example of the approach s efficacy applying dynamic programming to the energy minimizing active contours is described the optimization problem is set up as a discrete multistage decision process and is solved by a time delayed discrete dynamic programming algorithm a parallel procedure for decreasing computational costs is discussed

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

commutation relations and markov chains it is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain markov chains examples studied include random walk on irreducible representations a local random walk on partitions whose stationary distribution is the ewens distribution and some birth death chains

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

new analogues of clausen s identities arising from the theory of modular forms abstract around 1828 t clausen discovered that the square of certain hypergeometric f 1 2 function can be expressed as a hypergeometric f 2 3 function special cases of clausen s identities were later used by s ramanujan in his derivation of 17 series for 1 since then there were several attempts to find new analogues of clausen s identities with the hope to derive new classes of series for 1 unfortunately none were successful in this article we will present three new analogues of clausen s identities their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

a growth model in a random environment we consider a model of interface growth in two dimensions given by a height function on the sites of the one dimensional integer lattice according to the discrete time update rule the height above the site x increases to the height above x 1 if the latter height is larger otherwise the height above x increases by 1 with probability p x we assume that p x are chosen independently at random with a common distribution f and that the initial state is such that the origin is far above the other sites we explicitly identify the asymptotic shape and prove that in the pure regime the fluctuations about that shape normalized by the square root of time are asymptotically normal this contrasts with the quenched version conditioned on the environment and normalized by the cube root of time the fluctuations almost surely approach a distribution known from random matrix theory

domb s numbers and ramanujan sato type series for 1 π abstract in this article we construct a general series for 1 we indicate that ramanujan s 1 series are all special cases of this general series and we end the paper with a new class of 1 series our work is motivated by series recently discovered by takeshi sato

limit theorems for height fluctuations in a class of discrete space and time growth models we introduce a class of one dimensional discrete space discrete time stochastic growth models described by a height function ht x with corner initialization we prove with one exception that the limiting distribution function of ht x suitably centered and normalized equals a fredholm determinant previously encountered in random matrix theory in particular in the universal regime of large x and large t the limiting distribution is the fredholm determinant with airy kernel in the exceptional case called the critical regime the limiting distribution seems not to have previously occurred the proofs use the dual rsk algorithm gessel s theorem the borodin okounkov identity and a novel rigorous saddle point analysis in the fixed x large t regime we find a brownian motion representation this model is equilvalent to the seppalainen johansson model hence some of our results are not new but the proofs are

b spline contour representation and symmetry detection the detection of edges is only one of many steps in the understanding of images further processing necessarily involves grouping operations between contours we present a representation of edge contours by approximating b splines and show that such a representation facilitates the extraction of symmetries between contours our representation is rich compact stable and does not critically depend on feature extraction we turn our attention to the detection of three types of symmetries skew symmetries and parallel symmetries which have proven to be of great importance in inferring shape from contour and smooth local symmetries which have been used for planar shape description we show that our representation facilitates the computation of these symmetries

monodromy of picard fuchs differential equations for calabi yau threefolds abstract in this paper we are concerned with the monodromy of picardfuchs differential equations associated with one parameter families of calabiyau threefolds our results show that in the hypergeometric cases the matrix representations of monodromy relative to the frobenius bases can be expressed in terms of the geometric invariants of the underlying calabi yau threefolds this phenomenon is also verified numerically for other families of calabi yau threefolds in the paper furthermore we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of sp 4 z of finite index and whose levels are related to the geometric invariants of the calabi yau threefolds

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

stein s method and random character ratios stein s method is used to prove limit theorems for random character ratios tools are developed for four types of structures finite groups gelfand pairs twisted gelfand pairs and association schemes as one example an error term is obtained for a central limit theorem of kerov on the spectrum of the cayley graph of the symmetric group generated by i cycles other main examples include an error term for a central limit theorem of ivanov on character ratios of random projective representations of the symmetric group and a new central limit theorem for the spectrum of certain random walks on perfect matchings the results are obtained with very little information a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk the limit theorems stated in this paper are for normal approximation but many of the tools developed are applicable for arbitrary distributional approximation

using an mri distortion transfer function to characterize the ghosts in motion corrupted images motion artefact suppression remains an active topic in mri in this paper the authors suggest that certain nonrigid or spatially variant characteristics of motion of an object can be represented by extending the work of mitsa et al 1989 this empirical extension uses a ghost distortion transfer function gtdf applied to the k space frequency domain data the authors demonstrate the variety of ghost characteristics that can be generated from various two dimensional 2 d gtdf s the distortion transfer function for periodic motion along the z axis can be determined from the nonoverlapped portions of the ghost and central image it required a gdtf with the shape of a belt bandpass filter to produce an image corresponding to the ghosts of a volunteer s abdomen image corrupted by unknown respiratory motion artefacts the preliminary results of a composite method of motion artefact suppression are presented the artefact suppression was successful for ghost images described by a gdtf have a low pass nature but less successful with ghosts have a gdtf of a bandpass nature

on differential equations satisfied by modular forms we use the theory of modular functions to give a new proof of a result of p f stiller which asserts that if t is a non constant meromorphic modular function of weight 0 and f is a meromorphic modular form of weight k with respect to a discrete subgroup of sl 2 commensurable with sl 2 then f as a function of t satisfies a k 1 st order linear differential equation with algebraic functions of t as coefficients furthermore we show that the schwarzian differential equation for the modular function t can be extracted from any given k 1 st order linear differential equation of this type one advantage of our approach is that every coefficient in the differential equations can be relatively easily determined

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

on the distributions of the lengths of the longest monotone subsequences in random words we consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length n from an alphabet of k letters in the limit as k these become the corresponding distributions for permutations on n letters we find toeplitz determinant representations for the exponential generating functions on n of these distribution functions and show that they are expressible in terms of solutions of painleve v equations we show further that in the weakly increasing case the generating unction gives the distribution of the smallest eigenvalue in the k k laguerre random matrix ensemble and that the distribution itself has after centering and normalizing an n limit which is equal to the distribution function for the largest eigenvalue in the gaussian unitary ensemble of k k hermitian matrices of trace zero

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

α pfaffian pfaffian point process and shifted schur measure abstract for any complex number and any even size skew symmetric matrix b we define a generalization pf b of the pfaffian pf b which we call the pfaffian the pfaffian is a pfaffian analogue of the determinant studied in t shirai and y takahashi j funct anal 205 2003 414 463 and d vere jones linear algebra appl 111 1988 119 124 it gives the pfaffian at 1 we give some formulas for pfaffians and study the positivity further we define point processes determined by the pfaffian also we provide a linear algebraic proof of the explicit pfaffian expression obtained in s matsumoto correlation functions of the shifted schur measure math co 0312373 for the correlation function of the shifted schur measure

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

precise segmentation of multimodal images we propose new techniques for unsupervised segmentation of multimodal grayscale images such that each region of interest relates to a single dominant mode of the empirical marginal probability distribution of grey levels we follow the most conventional approaches in that initial images and desired maps of regions are described by a joint markov gibbs random field mgrf model of independent image signals and interdependent region labels however our focus is on more accurate model identification to better specify region borders each empirical distribution of image signals is precisely approximated by a linear combination of gaussians lcg with positive and negative components we modify an expectation maximization em algorithm to deal with the lcgs and also propose a novel em based sequential technique to get a close initial lcg approximation with which the modified em algorithm should start the proposed technique identifies individual lcg models in a mixed empirical distribution including the number of positive and negative gaussians initial segmentation based on the lcg models is then iteratively refined by using the mgrf with analytically estimated potentials the convergence of the overall segmentation algorithm at each stage is discussed experiments show that the developed techniques segment different types of complex multimodal medical images more accurately than other known algorithms

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

limiting distributions for a polynuclear growth model with external sources the purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by prahofer and spohn depending on the strength of the sources the limiting distribution functions are either the tracy widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero moreover we obtain transition functions between pairs of the above distribution functions in suitably scaled limits there are also similar results for a discrete totally asymmetric exclusion process