hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

mean curvature flow orbits moment maps given a compact riemannian manifold together with a group of isometries we discuss mcf of the orbits and some applications e g finding minimal orbits we then specialize to lagrangian orbits in kaehler manifolds in particular in the kaehler einstein case we find a relation between mcf and moment maps which for example proves that the minimal lagrangian orbits are isolated

elastically adaptive deformable models we present a technique for the automatic adaptation of a deformable model s elastic parameters within a kalman filter framework for shape estimation applications the novelty of the technique is that the model s elastic parameters are not constant but spatio temporally varying the variation of the elastic parameters depends on the distance of the model from the data and the rate of change of this distance each pass of the algorithm uses physics based modeling techniques to iteratively adjust both the geometric and the elastic degrees of freedom of the model in response to forces that are computed from the discrepancy between the model and the data by augmenting the state equations of an extended kalman filter to incorporate these additional variables we are able to significantly improve the quality of the shape estimation therefore the model s elastic parameters are always initialized to the same value and they are subsequently modified depending on the data and the noise distribution we present results demonstrating the effectiveness of our method for both two dimensional and three dimensional data

reductions of minimal lagrangian submanifolds with symmetries let m be a fano manifold equipped with a kahler form omega in 2 pi c 1 m and k a connected compact lie group acting on m as holomorphic isometries in this paper we show the minimality of a k invariant lagrangian submanifold l in m with respect to a globally conformal kahler metric is equivalent to the minimality of the reduced lagrangian submanifold l 0 l k in a kahler quotient m 0 with respect to the hsiang lawson metric furthermore we give some examples of kahler reductions by using a circle action obtained from a cohomogenenity one action on a kahler einstein manifold of positive ricci curvature applying these results we obtain several examples of minimal lagrangian submanifolds via reductions

reduced weyl asymptotics for pseudodifferential operators on bounded domains ii the compact group case abstract let g o n be a compact group of isometries acting on n dimensional euclidean space r n and x a bounded domain in r n which is transformed into itself under the action of g consider a symmetric classical pseudodifferential operator a 0 in l 2 r n that commutes with the regular representation of g and assume that it is elliptic on x we show that the spectrum of the friedrichs extension a of the operator res a 0 ext c c x l 2 x is discrete and using the method of the stationary phase we derive asymptotics for the number n of eigenvalues of a equal or less than and with eigenfunctions in the isotypic component of l 2 x as giving also an estimate for the remainder term for singular group actions since the considered critical set is a singular variety we recur to partial desingularization in order to apply the stationary phase theorem

hamiltonian minimal lagrangian submanifolds in kaehler manifolds with symmetries abstract by making use of the symplectic reduction and the cohomogeneity method we give a general method for constructing hamiltonian minimal lagrangian submanifolds in kaehler manifolds with symmetries as applications we construct infinitely many nontrivial complete hamiltonian minimal lagrangian submanifolds in c p n and c n

on lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres the n dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the n 1 dimensional complex projective space which is isometric to the real grassmann manifold of oriented 2 planes and is a compact hermitian symmetric space of rank 2 in this paper we study geometry of compact lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres from this viewpoint we provide a classification theorem of compact homogeneous lagrangian submanifolds in complex hyperquadrics by using the moment map technique moreover we determine the hamiltonian stability of compact minimal lagrangian submanifolds embedded in complex hyperquadrics which are obtained as gauss images of isoparametric hypersurfaces in spheres with g 1 2 3 distinct principal curvatures

viewpoint invariant recovery of visual surfaces from sparse data an algorithm for the reconstruction of visual surfaces from sparse data is proposed an important aspect of this algorithm is that the surface estimated from the sparse data is approximately invariant with respect to rigid transformation of the surface in 3d space the algorithm is based on casting the problem as an ill posed inverse problem that must be stabilized using a priori information related to the image and constraint formation to form a surface estimate that is approximately invariant with respect to viewpoint the stabilizing information is based on invariant surface characteristics with appropriate approximations this results in a convex functional to minimize which is then solved using finite element analysis the relationship of this algorithm to several previously proposed reconstruction algorithms is discussed and several examples that demonstrate its effectiveness in reconstructing viewpoint invariant surface estimates are given

semiclassical asymptotics gauge fields and quantum chaos a gauge field is given by a connection on a principal bundle p m we consider the semiclassical behavior of a family of schrodinger operators associated with a gauge field in the limit as h 0 we relate the spectral theory of such operators to behavior of the hamiltonian flow on the natural phase space associated to a gauge field examining in particular situations where this flow exhibits chaotic behavior

quantum ergodicity and symmetry reduction abstract we study the spectral and ergodic properties of schrodinger operators on a compact connected riemannian manifold m without boundary in case that the examined system possesses certain symmetries more precisely if m carries an isometric and effective action of a compact connected lie group g we prove a generalized equivariant version of the semiclassical weyl law with an estimate for the remainder using a semiclassical functional calculus for h dependent functions and relying on recent results on singular equivariant asymptotics we then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry reduced hamiltonian flow on the principal stratum of the singular symplectic reduction of m is ergodic in particular we obtain an equivariant version of the shnirelman zelditch colin de verdiere theorem as well as a representation theoretic equidistribution theorem if m g is an orbifold similar results were recently obtained by kordyukov when g is trivial one recovers the classical results

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension abstract carroll j a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension journal of computational and applied mathematics 46 1993 327 343 we apply the numerical method of lines to approximate the solution of a system of parabolic partial differential equations in one space dimension by approximating the spatial derivatives by discrete values of the solution at a set of mesh points in space we use these approximations to represent the partial differential equations at each mesh point giving a semidiscrete system of ordinary differential equations in the time direction the resulting ode system is integrated numerically using a second order l stable composite integration scheme the author this journal 1989 using variable stepsize integration an implementation of the composite scheme is applied to approximate the solution of some test problems selected from the literature using a uniform spatial mesh we present the results of a number of numerical experiments which allow an appraisal of the proposed approach with published works

residual analysis for feature detection it is shown that in a very simple form residual analysis achieves results that are at least as good as if not better than those obtained by other techniques there are many ways for extensions of the method for example moving average filters of regularization can be used to obtain the residual images also the strength of the correlation measured by d sub rr o can be used to eliminate noise weak edges etc a more ambitious extension is by considering smoothing filters that leave invariant the function representing the reflectance from smooth surfaces

a composite integration scheme for the numerical solution of systems of ordinary differential equations abstract a generalization of a composite linear multistep method 2 is developed and applied to the approximate integration of systems of ordinary differential equations the proposed scheme is second order accurate and l stable an algorithm based on the integration formula derived in this paper is applied to approximate the solutions of a number of standard test problems the numerical results indicate that the method is competitive with other fixed order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

ergodic billiards that are not quantum unique ergodic partially rectangular domains are compact two dimensional riemannian manifolds x either closed or with boundary that contain a flat rectangle or cylinder in this paper we are interested in partially rectangular domains with ergodic billiard flow examples are the bunimovich stadium the sinai billiard or donnelly surfaces we consider a one parameter family x t of such domains parametrized by the aspect ratio t of their rectangular part there is convincing theoretical and numerical evidence that the laplacian on x t with dirichlet or neumann boundary conditions is not quantum unique ergodic que we prove that this is true for all t in 1 2 excluding possibly a set of lebesgue measure zero this yields the first examples of ergodic billiard systems proven to be non que

efficient variable stiffness methods for cooling of hot rolled steel sections abstract an industrial finite element package for the simulation of alternative cooling strategies for hot rolled steel sections has been enhanced by the incorporation of a variable stiffness second order time integration scheme based on a specially developed family of extended stability explicit runge kutta methods and an l stable semi implicit formula the integration scheme uses local error estimation to vary step size and runge kutta method selection is achieved by monitoring the numerical stiffness through the solution period using a computationally inexpensive estimate for the spectral radius of the system jacobian numerical tests on a range of section cooling problems indicate that the variable stiffness rk code developed is considerably more efficient than standard first order integration methods for the levels of accuracy required for industrial simulations the efficiency compares favorably with other high efficiency codes such as the variable order backward differentiation formulae vode and the nearly stiff rkc explicit solver

coping with discontinuities in computer vision their detection classification and measurement the general principles of detection classification and measurement of discontinuities are studied the following issues are discussed detecting the location of discontinuities classifying discontinuities by their degrees measuring the size of discontinuities and coping with the random noise and designing optimal discontinuity detectors an algorithm is proposed for discontinuity detection from an input signal s for degree k discontinuity detection and measurement a detector p phi is used where p is the pattern and phi is the corresponding filter if there is a degree k discontinuity at location t sub 0 then in the filter response there is a scaled pattern alpha p at t sub 0 where alpha is the size of the discontinuity this reduces the problem to searching for the scaled pattern in the filter response a statistical method is proposed for the approximate pattern matching to cope with the random noise a study is made of optimal detectors which minimize the effects of noise

nonparametric fixed interval smoothing with vector splines a computationally efficient algorithm for nonparametric smoothing of vector signals with general measurement covariances is presented this algorithm provides an alternative to the optimal smoothing algorithms that hinge on possibly inaccurate parametric state space models automatic procedures that use the measurements to determine how much to smooth are developed and compared this adaptation allows the data to speak for itself without imposing a gauss markov model structure a nonparametric approach to covariance estimation for the case of independently identically distributed i i d measurement errors is presented monte carlo simulations demonstrate the performance of the algorithm

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

regularization of noisy isar images containing extended features we present a regularization method that employs a cross entropy functional and which deals with the three issues of complex valued data prior information and noise mitigation the basic model of this approach is similar to that used in usual maximum a posteriori analysis and allows for a more general relationship between the image and its configuration entropy than that usually employed

on discontinuity adaptive smoothness priors in computer vision a variety of analytic and probabilistic models in connection with markov random fields mrfs have been proposed in the last decade for solving low level vision problems involving discontinuities this paper presents a systematic study of these models and defines a general discontinuity adaptive da mrf model by analyzing the euler equation associated with the energy minimization it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points which is determined by the a priori smoothness constraint encoded into the energy function an important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing this forms the basis on which a class of adaptive interaction functions aifs is defined the da model is defined in terms of the euler equation constrained by this class of aifs its solution is c sup 1 continuous and allows arbitrarily large but bounded slopes in dealing with discontinuities because of the continuous nature it is stable to changes in parameters and data a good property for regularizing ill posed problems experimental results are shown

second derivative extended backward differentiation formulas for the numerical integration of stiff systems a class of second derivative extended backward differentiation formulas suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is examine

invariant surface segmentation through energy minimization with discontinuities the computational problems in segmenting range data into surface patches based on the invariant surface properties i e mean curvature h and gaussian curvature k are investigated the goal is to obtain reliable hk surface maps two commonly encountered problems are firstly the noise effect in computing derivative estimates and secondly the smoothing across discontinuities here the segmentation is formulated as finding minimization solutions of energy functionals involving discontinuities a two stage approach to the goal is presented stage 1 from a range image to curvature images and stage 2 from the curvature images to the hk maps in both stages solutions are found through minimizing energy functionals that measure the degree of bias of a solution from two constraints the closeness of the solution to the data and the smoothness of the solution controlled by predetermined discontinuities propagation across discontinuities is prevented during minimization which preserves the original surface shapes experimental results are given for a variety of test images

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

edge detection and surface reconstruction using refined regularization an edge detection and surface reconstruction algorithm in which the smoothness is controlled spatially over the image space is presented the values of parameters in the model are adaptively determined by an iterative refinement process hence the image dependent parameters such as the optimum value of the regularization parameter or the filter size are eliminated the algorithm starts with an oversmoothed regularized solution and iteratively refines the surface around discontinuities by using the structure exhibited in the error signal the spatial control of smoothness is shown to resolve the conflict between detection and localization criteria of edge detection by smoothing the noise in continuous regions while preserving discontinuities the performance of the algorithm is quantitatively and qualitatively evaluated on real and synthetic images and it is compared with those of marr hildreth and canny edge detectors

constructing intrinsic parameters with active models for invariant surface reconstruction a technique for constructing a canonical surface parameterization in terms of lines of curvature is presented two methods of computing the canonical invariant representation are also presented in the first method a static instance of the controlled continuity spline is used for the stabilizer ways to modify it to reflect a change of parameters to the lines of curvature are described in the second method the dynamic instance of the controlled continuity spline called the deformable model is used a force field defined in terms of the principal vectors is synthesized and applied to the parameter curves of the deformable model to coerce them along the lines of curvature in essence any transformation of parameters requires a modification of the stabilizer in the first method whereas in the second method it is tantamount to synthesizing a new force field experimental results with real and synthetic range data are included

equivariant spectral asymptotics for h pseudodifferential operators we prove equivariant spectral asymptotics for h pseudodifferential operators for compact orthogonal group actions generalizing results of el houakmi and helffer comportement semi classique en presence de symetries action d un groupe de lie compact asymp anal 5 2 91 113 1991 and cassanas reduced gutzwiller formula with symmetry case of a lie group j math pures appl 85 6 719 742 2006 using recent results for certain oscillatory integrals with singular critical sets p ramacher singular equivariant asymptotics and weyl s law on the distribution of eigenvalues of an invariant elliptic operator j reine angew math crelles j to be published we can deduce a weak equivariant weyl law furthermore we can prove a complete asymptotic expansion for the gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the hamilton flow

constructing simple stable descriptions for image partitioning a new formulation of the image partitioning problem is presented construct a complete and stable description of an image in terms of a specified descriptive language that is simplest in the sense of being shortest we show that a descriptive language limited to a low order polynomial description of the intensity variation within each region and a chain code like description of the region boundaries yields intuitively satisfying partitions for a wide class of images the advantage of this formulation is that it can be extended to deal with subsequent steps of the image understanding problem or to deal with other attributes such as texture in a natural way by augmenting the descriptive language experiments performed on a variety of both real and synthetic images demonstrate the superior performance of this approach over partitioning techniques based on clustering vectors of local image attributes and standard edge detection techniques

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

partial smoothing splines for noisy boundaries with corners investigates the estimation of 2 d boundary functions from sampled data sets where both noise and corners are present the approach is based on the partial smoothing spline in which the estimated boundary function consists of an ordinary smoothing spline and a parametric function that describes the discontinuities i e corners of the boundary prior knowledge about the boundary such as the number of corners their locations noise levels and the amount of smoothness is not required for the boundary estimate the smoothing parameter and the corner locations of the spline which are parts of the estimate are determined by the generalized cross validation method whereby statistical properties are gathered from the input sampled data rather than specified a priori this approach enables the smoothing of a noisy boundary while retaining an accurate description of the boundary corners extensive experiments were conducted to verify its ability to smooth noise while retaining a good representation of boundary corners and do not rely on any prior information

the viterbi optimal runlength constrained approximation nonlinear filter simple nonlinear filters are often used to enforce hard syntactic constraints while remaining close to the observation data e g in the binary case it is common practice to employ iterations of a suitable median or a one pass recursive median openclose or closeopen filter to impose a minimum symbol runlength constraint while remaining faithful to the observation unfortunately these filters are in general suboptimal motivated by this observation we pose the following optimization given a finite alphabet sequence of finite extent y y n sub n 0 sup n 1 find a sequence x x n sub n 0 sup n 1 that minimizes d x y spl sigma sub n 0 sup n 1 d sub n y n x n subject to the following x is piecewise constant of plateau run length spl ges m we show how a suitable reformulation of the problem naturally leads to a simple and efficient viterbi type optimal algorithmic solution we call the resulting nonlinear input output operator the viterbi optimal runlength constrained approximation vorca filter the method can be easily generalized to handle a variety of local syntactic constraints the vorca is optimal computationally efficient and possesses several desirable properties e g idempotence we therefore propose it as an attractive alternative to standard median stack and morphological filtering we also discuss some applications

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

surfaces with occlusions from layered stereo we propose a new binocular stereo algorithm that estimates scene structure as a collection of smooth surface patches the disparities within each patch are modeled by a continuous valued spline while the extent of each patch is represented via a pixelwise partitioning of the images disparities and extents are alternately estimated in an iterative energy minimization framework experimental results demonstrate that for scenes consisting of smooth surfaces the proposed algorithm significantly improves upon the state of the art

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected

optimal control of flow with discontinuities optimal control of the 1 d riemann problem of euler equations is studied with the initial values for pressure and density as control parameters the least squares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window existence of solutions for the optimal control problem is proven smooth and nonsmooth optimization methods employ the numerical gradient respectively a subgradient of the cost functional obtained from the adjoint of the discrete forward model the numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations the algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon except when the observations corresponding to shocks are detected and removed

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

optimal estimation of contour properties by cross validated regularization the problem of estimating the properties of smooth continuous contours from discrete noisy samples is used as vehicle to demonstrate the robustness of cross validated regularization applied to a vision problem a method for estimation of contour properties based on smoothing spline approximations is presented generalized cross validation is to devise an automatic algorithm for finding the optimal value of the smoothing regularization parameter from the data the cross validated smoothing splines are then used to obtain optimal estimates of the derivatives of quantized contours experimental results are presented which demonstrate the robustness of the method applied to the estimation of curvature of quantized contours under variable scale rotation and partial occlusion these results suggest the application of generalized cross validation to other computer vision algorithms involving regularization

one dimensional regularization with discontinuities regularization is equivalent to fitting smoothing splines to the data so that efficient and reliable numerical algorithms exist for finding solutions however the results exhibit poor performance along edges and boundaries to cope with such anomalies a more general class of smoothing splines that preserve corners and discontinuities is studied cubic splines are investigated in detail since they are easy to implement and produce smooth curves near all data points except those marked as discontinuities or creases a discrete regularization method is introduced to locate corners and discontinuities in the data points before the continuous regularization is applied