a multi level method for transmission eigenvalues of anisotropic media in this paper we propose a multi level finite element method for the transmission eigenvalue problem of anisotropic media the problem is non standard and non self adjoint with important applications in inverse scattering theory we employ a suitable finite element method to discretize the problem the resulting generalized matrix eigenvalue problem is large sparse and non hermitian to compute the smallest real transmission eigenvalue which is usually an interior eigenvalue we devise a multi level method using arnoldi iteration at the coarsest mesh the eigenvalue is obtained using arnoldi iteration with an adaptive searching technique this value is used as the initial guess for arnoldi iteration at the next mesh level this procedure is then repeated until the finest mesh level numerical examples are presented to show the viability of the method

flexible skew symmetric shape model for shape representation classification and sampling skewness of shape data often arises in applications e g medical image analysis and is usually overlooked in statistical shape models in such cases a gaussian assumption is unrealistic and a formulation of a general shape model which accounts for skewness is in order in this paper we present a novel statistical method for shape modeling which we refer to as the flexible skew symmetric shape model fssm the model is sufficiently flexible to accommodate a departure from gaussianity of the data and is fairly general to learn a mean shape template with a potential for classification and random generation of new realizations of a given shape robustness to skewness results from deriving the fssm from an extended class of flexible skew symmetric distributions in addition we demonstrate that the model allows us to extract principal curves in a point cloud the idea is to view a shape as a realization of a spatial random process and to subsequently learn a shape distribution which captures the inherent variability of realizations provided they remain with high probability within a certain neighborhood range around a mean specifically given shape realizations fssm is formulated as a joint bimodal distribution of angle and distance from the centroid of an aggregate of random points mean shape is recovered from the modes of the distribution while the maximum likelihood criterion is employed for classification

a projection method for generalized eigenvalue problems using numerical integration in this paper we propose a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane the proposed method projects the matrix pencil onto a subspace associated with the eigenvalues that are located in the domain via numerical integration

a robust comparison of biological shapes summary localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than least squares the repeated median resistant fitting algorithm is introduced the methods are tested by comparing hominid skulls with those of other primates and the male ostracode with the female in each case the resistant fit is superior to the least squares fit in that the difierences are more readily identified and agl ee more closely with the structural differences perceived on biological grounds le llltroductioll studies of allometric growth and of morphological change often run into difficulties in identifying and measuring the location and extent of relative shape deformations a reference system is needed for measuring these deformations and in the absence of an inherent fixed reference system one must be estimated the problem of estimating shape change in animal morphology provides an elegant setting for the comparison of least squares with recently developed robust and resistant statistical techniques being robust these new methods are protected against departures from the assumptions of analysis such as independent identically and normally distributed errors being resistant they are also protected against the potentially strong influences of atypical or incorrect data values two shapes can rarely be superimposed perfectly different fitting criteria will generally yield different results by allowing regions with large deformations to have a large impact on the fit least squares methods can minimize true shape differences and thereby obscure them a resistant technique however limits the influence of large deformations and the resulting fit is close in similar regions and not close in relatively deformed regions in this way resistant techniques can help to identify similarities and diflerences in form more effectively than least squares methods when sneath 1967 used least squares as a basis for some of the first quantitative estimates of relative deformation and so extended the fundamental descriptive work of thompson 1917 useful robust methods were not generally available sneath did in fact consider using the available robust methods but concluded that alternative methods that are less affected by wild values are not attractive in practice since then however the development of new theory and methods as reported by huber 1972 1981 hampel

finite element methods for maxwell s transmission eigenvalues the transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory and more importantly are parts of inverse algorithms for estimating material properties in this paper we propose two finite element methods to compute a few lowest maxwell s transmission eigenvalues which are of interest in applications since the discrete matrix eigenvalue problem is large sparse and in particular non hermitian due to the fact that the problem is neither elliptic nor self adjoint we devise an adaptive method which combines the arnoldi iteration and estimation of transmission eigenvalues exact transmission eigenvalues for balls are derived and used as a benchmark numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues

asymptotic global confidence regions for 3 d parametric shape estimation in inverse problems this paper derives fundamental performance bounds for statistical estimation of parametric surfaces embedded in ropf3 unlike conventional pixel based image reconstruction approaches our problem is reconstruction of the shape of binary or homogeneous objects the fundamental uncertainty of such estimation problems can be represented by global confidence regions which facilitate geometric inference and optimization of the imaging system compared to our previous work on global confidence region analysis for curves two dimensional 2 d shapes computation of the probability that the entire surface estimate lies within the confidence region is more challenging because a surface estimate is an inhomogeneous random field continuously indexed by a 2 d variable we derive an asymptotic lower bound to this probability by relating it to the exceedence probability of a higher dimensional gaussian random field which can in turn be evaluated using the tube formula due to sun simulation results demonstrate the tightness of the resulting bound and the usefulness of the three dimensional global confidence region approach

second chern class of fano manifolds and anti canonical geometry let x be a fano manifold of picard number one we establish a lower bound for the second chern class of x in terms of its index and degree as an application if y is a n dimensional fano manifold with k y n 3 h for some ample divisor h we prove that h 0 y h ge n 2 moreover we show that the rational map defined by vert mh vert is birational for m ge 5 and the linear system vert mh vert is basepoint free for m ge 7 as a by product the pluri anti canonical systems of singular weak fano varieties of dimension at most 4 are also investigated

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

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

mixed methods for the helmholtz transmission eigenvalues the ciarlet raviart mixed finite element method is popular for the biharmonic equations in this paper we apply this method to the helmholtz transmission eigenvalue problem which is quadratic and

an integral method for solving nonlinear eigenvalue problems we propose a numerical method for computing all eigenvalues and the corresponding eigenvectors of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane the method uses complex integrals of the resolvent operator applied to at least k column vectors where k is the number of eigenvalues inside the contour the theorem of keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k no initial approximations of eigenvalues and eigenvectors are needed the method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension we also give an extension of the method to the case where k is larger than the matrix dimension the quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

on birational geometry of minimal threefolds with numerically trivial canonical divisors for a minimal threefold x with k x equiv 0 and a nef and big weil divisor l on x we investigate the birational geometry inspired by l we prove that ml and k x ml give birational maps for all m ge 17 the result remains true under weaker assumption that l is big and has no stable base components

embedding gestalt laws in markov random fields the goal of this paper is to study a mathematical framework of 2d object shape modeling and learning for middle level vision problems such as image segmentation and perceptual organization for this purpose we pursue generic shape models which characterize the most common features of 2d object shapes in this paper shape models are learned from observed natural shapes based on a minimax entropy learning theory the learned shape models are gibbs distributions defined on markov random fields mrfs the neighborhood structures of these mrfs correspond to gestalt laws colinearity cocircularity proximity parallelism and symmetry thus both contour based and region based features are accounted for stochastic markov chain monte carlo mcmc algorithms are proposed for learning and model verification furthermore this paper provides a quantitative measure for the so called nonaccidental statistics and thus justifies some empirical observations of gestalt psychology by information theory our experiments also demonstrate that global shape properties can arise from interactions of local features

regularization of inverse visual problems involving discontinuities inverse problems such as the reconstruction problems that arise in early vision tend to be mathematically ill posed through regularization they may be reformulated as well posed variational principles whose solutions are computable standard regularization theory employs quadratic stabilizing functionals that impose global smoothness constraints on possible solutions discontinuities present serious difficulties to standard regularization however since their reconstruction requires a precise spatial control over the smoothing properties of stabilizers this paper proposes a general class of controlled continuity stabilizers which provide the necessary control over smoothness these nonquadratic stabilizing functionals comprise multiple generalized spline kernels combined with noncontinuous continuity control functions in the context of computational vision they may be thought of as controlled continuity constraints these generic constraints are applicable to visual reconstruction problems that involve both continuous regions and discontinuities for which global smoothness constraints fail

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

a coupled bem and fem for the interior transmission problem in acoustics abstract the interior transmission problem itp is a boundary value problem arising in inverse scattering theory and it has important applications in qualitative methods in this paper we propose a coupled boundary element method bem and a finite element method fem for the itp in two dimensions the coupling procedure is realized by applying the direct boundary integral equation method to define the so called dirichlet to neumann dtn mappings we show the existence of the solution to the itp for the anisotropic medium numerical results are provided to illustrate the accuracy of the coupling method

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

active appearance models we describe a new method of matching statistical models of appearance to images a set of model parameters control modes of shape and gray level variation learned from a training set we construct an efficient iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

shape matching and object recognition using shape contexts we present a novel approach to measuring similarity between shapes and exploit it for object recognition in our framework the measurement of similarity is preceded by 1 solving for correspondences between points on the two shapes 2 using the correspondences to estimate an aligning transform in order to solve the correspondence problem we attach a descriptor the shape context to each point the shape context at a reference point captures the distribution of the remaining points relative to it thus offering a globally discriminative characterization corresponding points on two similar shapes will have similar shape contexts enabling us to solve for correspondences as an optimal assignment problem given the point correspondences we estimate the transformation that best aligns the two shapes regularized thin plate splines provide a flexible class of transformation maps for this purpose the dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points together with a term measuring the magnitude of the aligning transform we treat recognition in a nearest neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image results are presented for silhouettes trademarks handwritten digits and the coil data set

shape statistics in kernel space for variational image segmentation we present a variational integration of nonlinear shape statistics into a mumford shah based segmentation process the nonlinear statistics are derived from a set of training silhouettes by a novel method of density estimation which can be considered as an extension of kernel pca to a probabilistic framework we assume that the training data forms a gaussian distribution after a nonlinear mapping to a higher dimensional feature space due to the strong nonlinearity the corresponding density estimate in the original space is highly non gaussian applications of the nonlinear shape statistics in segmentation and tracking of 2d and 3d objects demonstrate that the segmentation process can incorporate knowledge on a large variety of complex real world shapes it makes the segmentation process robust against misleading information due to noise clutter and occlusion 2003 pattern recognition society published by elsevier science ltd all rights reserved

recursive integral method for transmission eigenvalues the transmission eigenvalue problem arose in the inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing the problem is numerically challenging because of the nonself adjointness and complicated spectrum in this talk we introduce a novel recursive contour integral method using the spectrum projection the method can compute both real and complex eigenvalues and does not require any a priori information numerical examples show that the method is effective and robust

iterative methods for transmission eigenvalues transmission eigenvalues have important applications in inverse scattering theory they can be used to obtain useful information of the physical properties such as the index of refraction of the scattering target despite considerable effort devoted to the existence and estimation for the transmission eigenvalues the numerical treatment is limited since the problem is nonstandard classical finite element methods result in non hermitian matrix eigenvalue problems in this paper we focus on the computation of a few lowest transmission eigenvalues which are of practical importance instead of a non hermitian problem we work on a series of generalized hermitian problems we first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem the roots of these functions are the transmission eigenvalues then we apply iterative methods to compute the transmission eigenvalues we show the convergence of the numerical schemes the effectiveness of the methods is demonstrated using various numerical examples

asymptotic global confidence regions in parametric shape estimation problems we introduce confidence region techniques for analyzing and visualizing the performance of two dimensional parametric shape estimators assuming an asymptotically normal and efficient estimator for a finite parameterization of the object boundary cramer rao bounds are used to define an asymptotic confidence region centered around the true boundary computation of the probability that an entire boundary estimate lies within the confidence region is a challenging problem because the estimate is a two dimensional nonstationary random process we derive lower bounds on this probability using level crossing statistics the same bounds also apply to asymptotic confidence regions formed around the estimated boundaries lower bounding the probability that the entire true boundary lies within the confidence region the results make it possible to generate asymptotic confidence regions for arbitrary prescribed probabilities these asymptotic global confidence regions conveniently display the uncertainty in various geometric parameters such as shape size orientation and position of the estimated object and facilitate geometric inferences numerical simulations suggest that the new bounds are quite tight

a c0ip method of transmission eigenvalues for elastic waves abstract this paper develops a discontinuous galerkin method to compute a few smallest elasticity transmission eigenvalues which are of practical importance in inverse elastic scattering theory for high order problems comparing with classical conforming finite element methods discontinuous galerkin methods use simpler basis functions which make the numerical implementation much easier in this paper we propose an interior penalty discontinuous galerkin method using c 0 lagrange elements c 0 ip for the elastic transmission eigenvalue problem and prove the optimal convergence numerical examples are presented to validate its effectiveness both real and complex eigenvalues can be obtained

cramer rao bounds for parametric shape estimation in inverse problems we address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems when the boundaries are parameterized by a finite number of unknown variables our model applies to multiple unknown objects each with its own unknown gray level or color and boundary parameterization on an arbitrary known background while such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the cramer rao lower bounds very few results have been reported due to the difficulty of computing the derivatives of a functional with respect to shape deformation we provide a general formula for computing cramer rao lower bounds in inverse problems where the observations are related to the object by a general linear transform followed by a possibly nonlinear and noisy measurement system as an illustration we derive explicit formulas for computed tomography fourier imaging and deconvolution problems the bounds reveal that highly accurate parametric reconstructions are possible in these examples using severely limited and noisy data