isospectral conformal metrics on 3 manifolds theorem let g uj go be a sequence of conformal metrics satisfying the following conditions i vol m gj ao for some positive constant ao ii f r2 gj lp gj 12 dvj a 0 i e for each q defined on m we have

a local algorithm for the computation of image velocity via constructive interference of global fourier components a novel fourier based technique for local motion detection from image sequences is proposed in this method the instantaneous velocities of local image points are inferred directly from the global 3d fourier components of the image sequence this is done by selecting those velocities for which the superposition of the corresponding fourier gratings leads to constructive interference at the image point hence image velocities can be assigned locally even though position is computed from the phases and amplitudes of global fourier components spanning the whole image sequence that have been filtered based on the motion constraint equation reducing certain aperture effects typically arising from windowing in other methods regularization is introduced for sequences having smooth flow fields aperture effects and their effect on optic flow regularization are investigated in this context the algorithm is tested on both synthetic and real image sequences and the results are compared to those of other local methods finally we show that other motion features i e motion direction can be computed using the same algorithmic framework without requiring an intermediate representation of local velocity which is an important characteristic of the proposed method

numerical investigation on the stability of singular driven cavity flow by applying the singularity subtraction technique to the unsteady driven cavity problem the stability of the impulsively started flow is investigated without smoothing the corner singularity a second order spectral projection method allows localization of the critical reynolds number for the first hopf bifurcation in the interval 8017 6 8018 8

compactness of conformal metrics with integral bounds on curvature in this thesis we show that a sequence of conformal metrics on a compact n dimensional riemannian manifold n 4 which has an upper bound on volume and an upper bound on the lp norm of the curvature tensor for fixed p n 2 has a subsequence which converges in c if n 3 we have the same result if we assume in addition that the scalar curvature has an l2 bound as corollaries we have the compactness of a sequence of conformal metrics on a compact three manifold which are isospectral with respect to either the standard or conformal laplacian and the result of lelong ferrand that any compact manifold with non compact conformal group is conformally equivalent to the standard sphere

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

homotopy classes of harmonic maps of the stratified 2 spheres and applications to geometric flows abstract we show that the set of harmonic maps from the 2 dimensional stratified spheres with uniformly bounded energies contains only finitely many homotopy classes we apply this result to construct infinitely many harmonic map flows and mean curvature flows of 2 sphere in the connected sum of two closed 3 dimensional manifolds m 1 s 3 and m 2 s 3 r p 3 which must develop finite time singularity

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

a theoretical framework for convex regularizers in pde based computation of image motion many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy typically these energy functionals consist of two terms a data term which requires e g that a brightness constancy assumption holds and a regularizer that encourages global or piecewise smoothness of the flow field in this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images this taxonomy provides a unifying framework for data driven and flow driven isotropic and anisotropic as well as spatial and spatio temporal regularizers while some of these techniques are classic methods from the literature others are derived here for the first time we prove that all these methods are well posed they posses a unique solution that depends in a continuous way on the initial data an interesting structural relation between isotropic and anisotropic flow driven regularizers is identified and a design criterion is proposed for constructing anisotropic flow driven regularizers in a simple and direct way from isotropic ones its use is illustrated by several examples

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

analysis of superimposed oriented patterns estimation of local orientation in images may be posed as the problem of finding the minimum gray level variance axis in a local neighborhood in bivariate images the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2times2 tensor for an ideal single orientation the tensor is rank deficient i e the smaller eigenvalue vanishes a large minimal eigenvalue signals the presence of more than one local orientation what may be caused by non opaque additive or opaque occluding objects crossings bifurcations or corners we describe a framework for estimating such superimposed orientations our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions unlike in the single orientation case the eigensystem analysis does not directly yield the orientations rather it provides so called mixed orientation parameters mops we therefore show how to decompose the mops into the individual orientations we also show how to use tensor invariants to increase efficiency and derive a new feature for describing local neighborhoods which is invariant to rigid transformations applications are e g in texture analysis directional filtering and interpolation feature extraction for corners and crossings tracking and signal separation

a mixed basis spectral projection method a new grid less no collocation spectral projection method is presented the unsteady navier stokes equations are approximated according to the variational framework of guermond and quartapelle which accommodates two vector spaces for the velocity fields obtained in the two half steps of the fractional step method but retains only one in the final solution algorithm two different bases built on legendre polynomials are used for the velocity and pressure to solve the corresponding helmholtz and poisson equations by direct spectral elliptic solvers interpolations pn and pn 2 are employed for velocity and pressure to satisfy the lbb stability requirement and a gauss legendre quadrature formula with 3 2n integration points is used to prevent aliasing error in the pseudospectral evaluation of the nonlinear terms a bdf second order time stepping is implemented to provide accurate numerical results about the stability of the singular driven cavity problem

cavity flow dynamics at higher reynolds number and higher aspect ratio abstract a primitive variable unsteady viscous incompressible navier stokes flow simulation in a higher aspect ratio a depth width 2 driven cavity at a higher reynold s number re 10 000 exhibits dynamical features not apparent in previous studies of stationary solutions or at lower reynold s number or in the more usual unit cavity investigations the initial dynamics e g between t 0 sec and t 70 sec reveals transient bifurcations between states including two three and four interacting vortices in an anterior separation region following this there is an intermediate interval e g between t 70 sec and t 140 sec characterized by a vertical oscillation of the primary vortex related to the general activities of the secondary features of the flow the long time behavior e g between t 140 seconds and t 360 seconds is one of qualitative smoothing of all secondary features except a persistent oscillation indicating a hopf bifurcation

a weak energy identity and the length of necks for a sequence of sacks uhlenbeck α harmonic maps abstract in this paper we discuss the convergence behavior of a sequence of harmonic maps u with e u c from a compact surface m g into a compact riemannian manifold n h without boundary generally such a sequence converges weakly to a harmonic map in w 1 2 m n and strongly in c away from a finite set of points in m if energy concentration phenomena appears we show a generalized energy identity and discover a direct convergence relation between the blow up radius and the parameter which ensures the energy identity and no neck property we show that the necks converge to some geodesics moreover in the case there is only one bubble a length formula for the neck is given in addition we also give an example which shows that the necks contain at least a geodesic of infinite length

highly accurate optic flow computation with theoretically justified warping in this paper we suggest a variational model for optic flow computation based on non linearised and higher order constancy assumptions besides the common grey value constancy assumption also gradient constancy as well as the constancy of the hessian and the laplacian are proposed since the model strictly refrains from a linearisation of these assumptions it is also capable to deal with large displacements for the minimisation of the rather complex energy functional we present an efficient numerical scheme employing two nested fixed point iterations following a coarse to fine strategy it turns out that there is a theoretical foundation of so called warping techniques hitherto justified only on an experimental basis since our algorithm consists of the integration of various concepts ranging from different constancy assumptions to numerical implementation issues a detailed account of the effect of each of these concepts is included in the experimental section the superior performance of the proposed method shows up by significantly smaller estimation errors when compared to previous techniques further experiments also confirm excellent robustness under noise and insensitivity to parameter variations

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

accurate ω ψ spectral solution of the singular driven cavity problem this article provides accurate spectral solutions of the driven cavity problem calculated in the vorticity stream function representation without smoothing the corner singularities a prima facie impossible task as in a recent benchmark spectral calculation by primitive variables of botella and peyret closed form contributions of the singular solution for both zero and finite reynolds numbers are subtracted from the unknown of the problem tackled here numerically in biharmonic form the method employed is based on a split approach to the vorticity and stream function equations a galerkin legendre approximation of the problem for the perturbation and an evaluation of the nonlinear terms by gauss legendre numerical integration results computed for re 0 100 and 1000 compare well with the benchmark steady solutions provided by the aforementioned collocation chebyshev projection method the validity of the proposed singularity subtraction scheme for computing time dependent solutions is also established

on the optimality of solutions of the max product belief propagation algorithm in arbitrary graphs graphical models such as bayesian networks and markov random fields mrfs represent statistical dependencies of variables by a graph the max product belief propagation algorithm is a local message passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree furthermore when the graph is a tree the assignment based on the fixed point yields the most probable values of the unobserved variables given the observed ones good empirical performance has been obtained by running the max product algorithm or the equivalent min sum algorithm on graphs with loops for applications including the decoding of turbo codes except for two simple graphs cycle codes and single loop graphs there has been little theoretical understanding of the max product algorithm on graphs with loops here we prove a result on the fixed points of max product on a graph with arbitrary topology and with arbitrary probability distributions discrete or continuous valued nodes we show that the assignment based on a fixed point is a neighborhood maximum of the posterior probability the posterior probability of the max product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment the region includes all assignments that differ from the max product assignment in any subset of nodes that form no more than a single loop in the graph in some graphs this neighborhood is exponentially large we illustrate the analysis with examples

a phase based approach to the estimation of the optical flow field using spatial filtering we introduce a new technique for estimating the optical flow field starting from image sequences as suggested by fleet and jepson 1990 we track contours of constant phase over time since these are more robust to variations in lighting conditions and deviations from pure translation than contours of constant amplitude our phase based approach proceeds in three stages first the image sequence is spatially filtered using a bank of quadrature pairs of gabor filters and the temporal phase gradient is computed yielding estimates of the velocity component in directions orthogonal to the filter pairs orientations second a component velocity is rejected if the corresponding filter pair s phase information is not linear over a given time span third the remaining component velocities at a single spatial location are combined and a recurrent neural network is used to derive the full velocity we test our approach on several image sequences both synthetic and realistic

a computational framework and an algorithm for the measurement of visual the robust measurement of visual motion from digitized image sequences has been an important but difficult problem in computer vision this paper describes a hierarchical computational framework for the determination of dense displacement fields from a pair of images and an algorithm consist ent with that framework our framework is based on the separation of the image intensity information as well as the process of measuring motion according to scale the large scale intensity information is first used to obtain rough estimates of image motion which are then refined by using intensity information at smaller scales the estimates are in the form of displacement or velocity vectors for pixels and are accompanied by a direction dependent confidence measure a smoothness constraint is employed to propagate the measurements with high confidence to their neigboring areas where the confidences are low at all levels the computations are pixel parallel uniform across the image and based on information from a small neighborhood of a pixel for our algorithm the local displacement vectors are determind by mini mizing the sum of squared differences ssd of intensities the confidence measures are derived from the shape of the ssd surface and the smoothness constraint is cast in the form of energy minimization results of applying our algorithm to pairs of real images are included in addition to our own

lucas kanade meets horn schunck combining local and global optic flow methods differential methods belong to the most widely used techniques for optic flow computation in image sequences they can be classified into local methods such as the lucas kanade technique or bigun s structure tensor method and into global methods such as the horn schunck approach and its extensions often local methods are more robust under noise while global techniques yield dense flow fields the goal of this paper is to contribute to a better understanding and the design of novel differential methods in four ways i we juxtapose the role of smoothing regularisation processes that are required in local and global differential methods for optic flow computation ii this discussion motivates us to describe and evaluate a novel method that combines important advantages of local and global approaches it yields dense flow fields that are robust against noise iii spatiotemporal and nonlinear extensions as well as multiresolution frameworks are presented for this hybrid method iv we propose a simple confidence measure for optic flow methods that minimise energy functionals it allows to sparsify a dense flow field gradually depending on the reliability required for the resulting flow comparisons with experiments from the literature demonstrate the favourable performance of the proposed methods and the confidence measure

determining optical flow optical flow cannot be computed locally since only one independent measurement is available from the image sequence at a point while the flow velocity has two components a second constraint is needed a method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image an iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences the algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time it is also insensitive to quantization of brightness levels and additive noise examples are included where the assumption of smoothness is violated at singular points or along lines in the image

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

role of the lbb condition in weak spectral projection methods 405 this paper investigates the relevance of the ladyshenskaya babuska brezzi con dition in spectral projection methods we consider the stability and convergence properties for a first order nonincremental projection method and a second order incremental projection method both based on a spectral galerkin legendre spatial discretization we show that the convergence of both projection methods is controlled by the ability of the spectral framework to approximate correctly the steady stokes problem c 2001 elsevier science

the uncertainty principle in image processing the uncertainty principle is recognized as one of the fundamental results in signal processing its role in inference is however less well known outside of quantum mechanics it is the aim of this paper to provide a unified approach to the problem of uncertainty in image processing it is shown that uncertainty can be derived from the fundamental constraints on the process of vision the requirements for class defining operations which are both shift invariant and insensitive to changes in illumination it is thus shown that uncertainty plays a key role in the language of vision since it affects the choice of both the alphabet the elementary signals and the syntax the inferential structure of vision the report is concluded with a number of practical illustrations of these ideas taken from such image processing tasks as enhancement data compression and segmentation

bubble tree convergence for conformal metrics with m r n2dvg bounds abstract given a sequence of conformal metrics g k u k 4 n 2 g 0 on a smooth compact boundaryless riemannian manifold m n g 0 assume the volume of g k and l n 2 norm of scalar curvatures both are bounded we prove that after passing to a subsequence u k weakly converges to the bubble tree limit u u 1 1 u i u l l 1 i l 1 i in w 2 p for some p n 2 where u w 2 p m g 0 and u i w 2 p r n g r n moreover after passing to a subsequence the sequence of metric spaces m d k defined by g k converges to a connected metric space m d in the gromov hausdorff topology sense and lim k vol m g k h n m d where h n is the n dimensional hausdorff measure defined by d

hopf bifurcation in the driven cavity abstract incompressible two dimensional calculations are reported for the impulsively started lid driven cavity with aspect ratio two the algorithm is based on the time dependent stream function equation with a crank nicolson differencing scheme for the diffusion terms and with an adams bashforth scheme for the convection terms a multigrid method is used to solve the linear implicit equations at each time step periodic asymptotic solutions have been found for re 10000 and for re 5000 the re 5000 results are validated by grid refinement calculations the solutions are shown to be precisely periodic and care is taken to demonstrate that asymptotic states have been reached a discussion is included about the indicators that are used to show that an asymptotic state has been reached and to show that the asymptotic state is indeed periodic

bubble tree convergence for harmonic maps let be a compact riemann surface any sequence fn m of harmonic maps with bounded energy has a bubble tree limit consisting of a harmonic map o m and a tree of bubbles fk s2 m we give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class and that the images of the fn converge pointwise we then give explicit counterexamples showing that bubble tree convergence fails i for harmonic maps fn when the conformal structure of varies with n and ii when the conformal structure is fixed and n is a palais smale sequence for the harmonic map energy