efficient block preconditioning for a c1 finite element discretisation of the dirichlet biharmonic problem we present an efficient block preconditioner for the two dimensional biharmonic dirichlet problem discretised by c1 bicubic hermite finite elements in this formulation each node in the mesh has four different degrees of freedom dofs grouping dofs of the same type together leads to a natural blocking of the galerkin coefficient matrix based on this block structure we develop two preconditioners a 2x2 block diagonal preconditioner bd and a block bordered diagonal bbd preconditioner we prove mesh independent bounds for the spectra of the bd preconditioned galerkin matrix under certain conditions the eigenvalue analysis is based on the fact that the proposed preconditioner like the coefficient matrix itself is symmetric positive definite and is assembled from element matrices we demonstrate the effectiveness of an inexact version of the bbd preconditioner which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size finally we study robustness of this preconditioner with respect to element stretching domain distortion and non convex domains

symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

multigrid preconditioning for the biharmonic dirichlet problem a multigrid preconditioning scheme for solving the ciarlet raviart mixed method equations for the biharmonic dirichlet problem is presented in particular a schur complement formulation for these equations which yields noninherited quadratic forms is considered the preconditioning scheme is compared with a standard w cycle multigrid iteration it is proved that a variable v cycle preconditioner leads to problems with uniformly bounded condition numbers however w cycle convergence is proved only if the number of smoothings m is sufficiently large an example is given in which the w cycle diverges unless m geq 8 divergent w cycles are also encountered when solving the morley equations for the biharmonic dirichlet problem although brenner has proved w cycle convergence for sufficiently large m 9 this is illustrated with additional computations while variable v cycles continue to produce excellent preconditioners in this setting certain approximate l 2 inner products are described and a modif

learning nonlinear image manifolds by global alignment of local linear models appearance based methods based on statistical models of the pixel values in an image region rather than geometrical object models are increasingly popular in computer vision in many applications the number of degrees of freedom dof in the image generating process is much lower than the number of pixels in the image if there is a smooth function that maps the dof to the pixel values then the images are confined to a low dimensional manifold embedded in the image space we propose a method based on probabilistic mixtures of factor analyzers to 1 model the density of images sampled from such manifolds and 2 recover global parameterizations of the manifold a globally nonlinear probabilistic two way mapping between coordinates on the manifold and images is obtained by combining several locally valid linear mappings we propose a parameter estimation scheme that improves upon an existing scheme and experimentally compare the presented approach to self organizing maps generative topographic mapping and mixtures of factor analyzers in addition we show that the approach also applies to finding mappings between different embeddings of the same manifold

manifold learning for object tracking with multiple nonlinear models this paper presents a novel manifold learning algorithm for high dimensional data sets the scope of the application focuses on the problem of motion tracking in video sequences the framework presented is twofold first it is assumed that the samples are time ordered providing valuable information that is not presented in the current methodologies second the manifold topology comprises multiple charts which contrasts to the most current methods that assume one single chart being overly restrictive the proposed algorithm gaussian process multiple local models gp mlm can deal with arbitrary manifold topology by decomposing the manifold into multiple local models that are probabilistic combined using gaussian process regression in addition the paper presents a multiple filter architecture where standard filtering techniques are integrated within the gp mlm the proposed approach exhibits comparable performance of state of the art trackers namely multiple model data association and deep belief networks and compares favorably with gaussian process latent variable models extensive experiments are presented using real video data including a publicly available database of lip sequences and left ventricle ultrasound images in which the gp mlm achieves state of the art results

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

dealing with label switching in mixture models summary in a bayesian analysis of finite mixture models parameter estimation and clustering are sometimes less straightforward than might be expected in particular the common practice of estimating parameters by their posterior mean and summarizing joint posterior distributions by marginal distributions often leads to nonsensical answers this is due to the so called label switching problem which is caused by symmetry in the likelihood of the model parameters a frequent response to this problem is to remove the symmetry by using artificial identifiability constraints we demonstrate that this fails in general to solve the problem and we describe an alternative class of approaches relabelling algorithms which arise from attempting to minimize the posterior expected loss under a class of loss functions we describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples

on the dirichlet problem for the harmonic vector fields equation abstract we show that any harmonic with respect to the bergman metric vector field tangent to the levi distribution of the foliation by level sets of the defining function z k z z 1 n 1 of a strictly pseudoconvex bounded domain c n which is smooth up to the boundary must vanish on if n 5 and u t is a harmonic vector field with u c 2 then u 0

a study of a numerical solution of the steady two dimensions navier stokes equations in a constricted channel problem by a compact fourth order method we present a numerical solution for the steady 2d navier stokes equations using a fourth order compact type method the geometry of the problem is a constricted symmetric channel where the boundary can be varied via a parameter from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular the set of non linear equations is solved by the newton method results have been obtained for reynolds number up to 500 estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method

compressive sensing on manifolds using a nonparametric mixture of factor analyzers algorithm and performance bounds nonparametric bayesian methods are employed to constitute a mixture of low rank gaussians for data x rn that are of high dimension n but are constrained to reside in a low dimensional subregion of rn the number of mixture components and their rank are inferred automatically from the data the resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds based on compressive sensing cs projection measurements the statistical cs inversion is performed analytically we derive the required number of cs random measurements needed for successful reconstruction based on easily computed quantities drawing on block sparsity properties the proposed methodology is validated on several synthetic and real datasets

convergence of a compact scheme for the pure streamfunction formulation of the unsteady navier stokes system this paper is devoted to the analysis of a new compact scheme for the navier stokes equations in pure streamfunction formulation numerical results using that scheme have been reported in m ben artzi e j comput phys 205 2005 pp 640 664 the scheme discussed here combines the stephenson scheme for the biharmonic operator and ideas from box scheme methodology consistency and convergence are proved for the full nonlinear system instead of customary periodic conditions the case of boundary conditions is addressed it is shown that in one dimension the truncation error for the biharmonic operator is o h 4 at interior points and o h at near boundary points in two dimensions the truncation error is o h 2 at interior points due to the cross terms and o h at near boundary points hence the scheme is globally of order four in the one dimensional periodic case and of order two in the two dimensional periodic case but of order 3 2 for one and two dimensional nonperiodic boundary conditions we emphasize in particular that there is no special treatment of the boundary thus allowing robust use of the scheme the finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications

higher order numerical discretizations for exterior and biharmonic type pdes a higher order numerical discretization technique based on minimum sobolev norm msn interpolation was introduced in our previous work in this article the discretization technique is presented as a tool to solve two hard classes of pdes namely the exterior laplace problem and the biharmonic problem the exterior laplace problem is compactified and the resultant near singular pde is solved using this technique this finite difference type method is then used to discretize and solve biharmonic type pdes a simple book keeping trick of using ghost points is used to obtain a perfectly constrained discrete system numerical results such as discretization error condition number estimate and solution error are presented for both classes of pdes variable coefficient examples on complicated geometries and irregular grids are considered the method is seen to have high order of convergence in all these cases through numerical evidence perhaps for the first time such a systematic higher order procedure for irregular grids and variable coefficient cases is now available though not discussed in the paper the idea seems to be easily generalizable to finite element type techniques as well

g2 structures and octonion bundles abstract we use a g 2 structure on a 7 dimensional riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non associative product we then define a metric compatible octonionic covariant derivative on this bundle that is compatible with the octonion product the torsion of the g 2 structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the riemann curvature that lies in the 7 dimensional representation of g 2 we also interpret the choice of a particular g 2 structure within the same metric class as a choice of gauge and show that under a change of this gauge the torsion does transform as an octonion valued connection 1 form finally we also show an explicit relationship between the octonion bundle and the spinor bundle define an octonionic dirac operator and explore an energy functional for octonion sections we then prove that critical points correspond to divergence free torsion which is shown to be an octonionic analog of the coulomb gauge

a new paradigm for solving navier stokes equations streamfunction velocity formulation in this paper we propose a new paradigm for solving navier stokes equations the proposed methodology is based on a streamfunction velocity formulation of the two dimensional steady state navier stokes equations representing incompressible fluid flows in two dimensional domains similar formulations are also possible for three dimensional fluid flows the main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values especially on solid boundaries encountered when solving the streamfunction vorticity formulations our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity pressure formulations of the navier stokes equations we describe the new formulation of the navier stokes equations and use this formulation to solve a couple of fluid flow problems we use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem it is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem in addition we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature in all cases we obtain high accuracy solutions with little additional cost

a kernel based low rank klr model for low dimensional manifold recovery in highly accelerated dynamic mri while many low rank and sparsity based approaches have been developed for accelerated dynamic magnetic resonance imaging dmri they all use low rankness or sparsity in input space overlooking the intrinsic nonlinear correlation in most dmri data in this paper we propose a kernel based framework to allow nonlinear manifold models in reconstruction from sub nyquist data within this framework many existing algorithms can be extended to kernel framework with nonlinear models in particular we have developed a novel algorithm with a kernel based low rank model generalizing the conventional low rank formulation the algorithm consists of manifold learning using kernel low rank enforcement in feature space and preimaging with data consistency extensive simulation and experiment results show that the proposed method surpasses the conventional low rank modeled approaches for dmri

bayesian gaussian copula factor models for mixed data gaussian factor models have proven widely useful for parsimoniously characterizing dependence in multivariate data there is rich literature on their extension to mixed categorical and continuous variables using latent gaussian variables or through generalized latent trait models accommodating measurements in the exponential family however when generalizing to non gaussian measured variables the latent variables typically influence both the dependence structure and the form of the marginal distributions complicating interpretation and introducing artifacts to address this problem we propose a novel class of bayesian gaussian copula factor models that decouple the latent factors from the marginal distributions a semiparametric specification for the marginals based on the extended rank likelihood yields straightforward implementation and substantial computational gains we provide new theoretical and empirical justifications for using this likelihood in bayesian inference we propose new default pri

single cell discretizations of order two and four for biharmonic problems abstract new difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain these methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions derivatives of the solution are obtained as a by product of the methods second order formulas are derived for both the first and second biharmonic problems in numerical experiments the new second order formulas compare favourably with the standard second order methods extensions to fourth order formulas are given the method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

total variation projection with first order schemes this article proposes a new algorithm to compute the projection on the set of images whose total variation is bounded by a constant the projection is computed through a dual formulation that is solved by first order non smooth optimization methods this yields an iterative algorithm that applies iterative soft thresholding to the dual vector field and for which we establish convergence rate on the primal iterates this projection algorithm can then be used as a building block in a variety of applications such as solving inverse problems under a total variation constraint or for texture synthesis numerical results are reported to illustrate the usefulness and potential applicability of our tv projection algorithm on various examples including denoising texture synthesis inpainting deconvolution and tomography problems we also show that our projection algorithm competes favorably with state of the art tv projection methods in terms of convergence speed

iterative multigrid regularization techniques for image matching in this paper we consider the problem of matching images i e to find a deformation u which transforms a digital image into another such that the images have nearly equal gray values in every image element the difference of the two images is measured by their l2 difference which should be minimized this yields a nonlinear ill conditioned inverse problem for u so the numerical solution is quite difficult a tikhonov regularization method is considered to rule out discontinuous and irregular solutions to the minimization problem an important problem is a proper choice of the regularization parameter alpha for the practical choice of alpha we use iterative regularization methods based on multigrid techniques to obtain a suitable initial guess we use an approach similar to the full multigrid fmg developed by brandt math comp 31 1977 pp 333 390 the algorithms have optimal complexity the amount of work is proportional to the number of picture elements finally we present some experimental results for synthetic and real images

classification via bayesian nonparametric learning of affine subspaces it has become common for datasets to contain large numbers of variables in studies conducted in areas such as genetics machine vision image analysis and many others when analyzing such data parametric models are often too inflexible while nonparametric procedures tend to be nonrobust because of insufficient data on these high dimensional spaces this is particularly true when interest lies in building efficient classifiers in the presence of many predictor variables when dealing with these types of data it is often the case that most of the variability tends to lie along a few directions or more generally along a much smaller dimensional submanifold of the data space in this article we propose a class of models that flexibly learn about this submanifold while simultaneously performing dimension reduction in classification this methodology allows the cell probabilities to vary nonparametrically based on a few coordinates expressed as linear combinations of the predictors also as opposed to man

a fast direct solver for the biharmonic problem in a rectangular grid we present a fast direct solver methodology for the dirichlet biharmonic problem in a rectangle the solver is applicable in the case of the second order stephenson scheme j w stephenson j comput phys 55 1984 pp 65 80 as well as in the case of a new fourth order scheme which is discussed in this paper and is based on the capacitance matrix method b l buzbee and f w dorr siam j numer anal 11 1974 pp 1136 1150 p bjorstad siam j numer anal 20 1983 pp 59 71 the discrete biharmonic operator is decomposed into two components the first is a diagonal operator in the eigenfunction basis of the laplacian to which the fft algorithm is applied the second is a low rank perturbation operator given by the capacitance matrix which is due to the deviation of the discrete operators from diagonal form the sherman morrison formula g h golub and c f van loan matrix computations 3rd ed john hopkins university press baltimore md 1996 is applied to obtain a fast solution of the resulting linear system of equations

a fast solver for the orthogonal spline collocation solution of the biharmonic dirichlet problem on rectangles a fast schur complement algorithm is presented for computing the piecewise hermite bicubic orthogonal spline collocation solution of the biharmonic dirichlet problem on a rectangular region on an n n uniform partition the algorithm which involves the preconditioned conjugate gradient method and fast fourier transforms requires o n2 log2 n arithmetic operations

computational acceleration for mr image reconstruction in partially parallel imaging in this paper we present a fast numerical algorithm for solving total variation and l1 tvl1 based image reconstruction with application in partially parallel magnetic resonance imaging our algorithm uses variable splitting method to reduce computational cost moreover the barzilai borwein step size selection method is adopted in our algorithm for much faster convergence experimental results on clinical partially parallel imaging data demonstrate that the proposed algorithm requires much fewer iterations and or less computational cost than recently developed operator splitting and bregman operator splitting methods which can deal with a general sensing matrix in reconstruction framework to get similar or even better quality of reconstructed images

mismatch in the classification of linear subspaces sufficient conditions for reliable classification this paper considers the classification of linear subspaces with mismatched classifiers in particular we assume a model where one observes signals in the presence of isotropic gaussian noise and the distribution of the signals conditioned on a given class is gaussian with a zero mean and a low rank covariance matrix we also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters by constructing an asymptotic low noise expansion of an upper bound to the error probability of such a mismatched classifier we provide sufficient conditions for reliable classification in the low noise regime that are able to sharply predict the absence of a classification error floor such conditions are a function of the geometry of the true signal distribution the geometry of the mismatched signal distributions as well as the interplay between such geometries namely the principal angles and the overlap between the true and the mismatched signal subspaces numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a motion segmentation and a hand written digit classification applications

a pure compact scheme for the streamfunction formulation of navier stokes equations a pure streamfunction formulation is introduced for the numerical simulation of the two dimensional incompressible navier stokes equations the idea is to replace the vorticity in the vorticity streamfunction evolution equation by the laplacian of the streamfunction the resulting formulation includes the streamfunction only thus no inter function relations need to be invoked a compact numerical scheme which interpolates streamfunction values as well as its first order derivatives is presented and analyzed a number of numerical experiments are presented including driven and double driven cavities where the reynolds numbers are sufficiently large leading to symmetry breaking of asymptotic solutions

volumes of vector fields on spheres in this paper we study the problem what is the unit vector field of smallest volume on an odd dimensional sphere we exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four we conjecture that this volume is the infimum for volumes of bona fide unit vector fields and is only achieved by the singular vector field we generalize the construction of the singular vector field to give a family of cycles in stiefel manifolds each of which is a smooth manifold except for one singular point except for some low dimensional cases the tangent cones at these singular points are volume minimizing and half of the cones are nonorientable thus we obtain a new family of nonorientable volume minimizing cones

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

a compact difference scheme for the biharmonic equation in planar irregular domains we present a finite difference scheme applicable to general irregular planar domains to approximate the biharmonic equation the irregular domain is embedded in a cartesian grid in order to approximate delta 2 phi at a grid point we interpolate the data on the irregular stencil by a polynomial of degree six the finite difference scheme is delta 2 q phi 0 0 where q phi is the interpolation polynomial the interpolation polynomial is not uniquely determined we present a method to construct such an interpolation polynomial and prove that our construction is second order accurate for a regular stencil m ben artzi j p croisille and d fishelov siam j sci comput 31 2008 pp 303 333 shows that the proposed interpolation polynomial is fourth order accurate we present some suitable numerical examples

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

an optimal order nonconforming multigrid method for the biharmonic equation an optimal order multigrid method for solving the biharmonic equation using morley nonconforming finite elements is developed

multigrid solution of high order discretisation for three dimensional biharmonic equation with dirichlet boundary conditions of second kind abstract in this paper we propose two compact finite difference approximations for three dimensional biharmonic equation with dirichlet boundary conditions of second kind in these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques the unknown solution and its second derivatives are carried as unknowns at grid points we derive second order and fourth order approximations on a 27 point compact stencil classical iteration methods such as gauss seidel and sor for solving the linear system arising from the second order and fourth order discretisation suffer from slow convergence in order to overcome this problem we use multigrid method which exhibit grid independent convergence and solve the linear system of equations in small amount of computer time the fourth order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions

multigrid solution of automatically generated high order discretizations for the biharmonic equation in this work we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil the solution and its first derivatives are carried as unknowns at the grid points dirichlet boundary conditions are thus incorporated naturally since the approximations use the 9 point compact stencil no special formulas are needed near the boundaries both second order and fourth order discretizations are derived the fourth order approximation produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled with the boundary condition the method suffers from slow convergence when classical iteration methods such as gauss seidel or sor are employed in order to alleviate this problem we propose several multigrid techniques that exhibit grid independent convergence and solve the biharmonic equation in a small amount of computer time test results from three different problems including stokes flow in a driven cavity are reported

an efficient primal dual interior point method for minimizing a sum of euclidean norms the problem of minimizing a sum of euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature this nonsmooth optimization problem arises in many different kinds of modern scientific applications we derive a primal dual interior point algorithm for the problem by applying newton s method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition the main work at each step consists of solving a system of linear equations the schur complement equations this schur complement matrix is not symmetric unlike in linear programming we incorporate a mehrotra type predictor corrector scheme and present some experimental results comparing several variations of the algorithm including as one option explicit symmetrization of the schur complement with a skew corrector term we also present results obtained from a code implemented to solve large sparse problems using a symmetrized schur complement this has been applied to problems arising in plastic collapse analysis with hundreds of thousands of variables and millions of nonzeros in the constraint matrix the algorithm typically finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable

bayesian analysis of mixtures of factor analyzers for bayesian inference on the mixture of factor analyzers natural conjugate priors on the parameters are introduced and then a gibbs sampler that generates parameter samples following the posterior is constructed in addition a deterministic estimation algorithm is derived by taking modes instead of samples from the conditional posteriors used in the gibbs sampler this is regarded as a maximum a posteriori estimation algorithm with hyperparameter search the behaviors of the gibbs sampler and the deterministic algorithm are compared on a simulation experiment

fast transformation invariant component analysis dimensionality reduction techniques such as principal component analysis and factor analysis are used to discover a linear mapping between high dimensional data samples and points in a lower dimensional subspace previously frey and jojic introduced transformation invariant component analysis tca to learn a linear mapping invariant to a set of known form of global transformations however parameter estimation in that model using the previously proposed expectation maximization em algorithm required scalar operations in the order of n2 where n is the dimensionality of each training example this is prohibitive for many applications of interest such as modeling mid to large size images where for instance i n may be as high as 786432 512 512 rgb image in this paper we present an efficient algorithm that reduces the computational requirements to order of nlogn with this speedup we show the effectiveness of transformation invariant component analysis in various applications including tracking learning video textures clustering object recognition and object detection in images software for tca can be downloaded from http www psi toronto edu fasttca htm

an algorithm for image removals and decompositions without inverse matrices partial differential equation pde based methods in image processing have been actively studied in the past few years one of the effective methods is the method based on a total variation introduced by rudin oshera and fatemi rof l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms physica d 60 1992 259 268 this method is a well known edge preserving model and an useful tool for image removals and decompositions unfortunately this method has a nonlinear term in the equation which may yield an inaccurate numerical solution to overcome the nonlinearity a fixed point iteration method has been widely used the nonlinear system based on the total variation is induced from the rof model and the fixed point iteration method to solve the rof model is introduced by dobson and vogel d c dobson c r vogel convergence of an iterative method for total variation denoising siam j numer anal 34 5 1997 1779 1791 however some methods had to compute inverse matrices which led to roundoff error to address this problem we developed an efficient method for solving the rof model we make a sequence like richardson s method by using a fixed point iteration to evade the nonlinear equation this approach does not require the computation of inverse matrices the main idea is to make a direction vector for reducing the error at each iteration step in other words we make the next iteration to reduce the error from the computed error and the direction vector we describe that our method works well in theory in numerical experiments we show the results of the proposed method and compare them with the results by d dobson and c vogel and then we confirm the superiority of our method

projection design for statistical compressive sensing a tight frame based approach in this paper we develop a framework to design sensing matrices for compressive sensing applications that lead to good mean squared error mse performance subject to sensing cost constraints by capitalizing on the mse of the oracle estimator whose performance has been shown to act as a benchmark to the performance of standard sparse recovery algorithms we use the fact that a parseval tight frame is the closest design in the frobenius norm sense to the solution of a convex relaxation of the optimization problem that relates to the minimization of the mse of the oracleestimator with respect to the equivalent sensing matrix subject to sensing energy constraints based on this result we then propose two sensing matrix designs that exhibit two key properties the designs are closed form rather than iterative the designs exhibit superior performance in relation to other designs in the literature which is revealed by our numerical investigation in various scenarios with different sparse recovery algorithms including basis pursuit de noise bpdn the dantzig selector and orthogonal matching pursuit omp

a more robust multigrid algorithm for diffusion type registration models abstract registration refers to the useful process of aligning two similar but different intensity image functions in order to either track changes or combine information variational models are capable of finding transform maps containing large and non uniform deformations between such a pair of images since finding a transform map is an inverse problem as with all models suitable regularisation is necessary to overcome the non uniqueness of the problem in the case of diffusion type models regularisation terms impose smoothness on the transformation by minimising the gradient of the flow field the diffusion model also coincides with the basic model for optical flow frameworks of horn and schunck 1981 the biggest drawback with variational models is the large computational cost required to solve the highly non linear system of pdes chumchob and chen 2011 developed a non linear multigrid nmg method to address this cost problem however a closer look at the analysis of the nmg scheme highlighted omissions which affected the convergence of the nmg scheme moreover the nmg method proposed by chumchob and chen did not impose any control of non physical folding which invalidates a map this paper has proposed several key ideas first we re evaluate the analysis of the nmg method to show how the omissions in chumchob and chen 2011 have a noticeable impact on the convergence of the nmg method in addition we also provide a way of estimating the convergence rate of a solver on the coarsest grid in order to estimate the number of iterations that will be required to obtain a solution with appropriate accuracy second we propose an extension to the chumchob chen nmg method which controls any folding within the deformation experimental results on the proposed multigrid framework demonstrate improvements in convergence and the accuracy of registrations compared with previous methods

enhancing information discriminant analysis in this paper we develop a novel feature transformation method for supervised linear dimensionality reduction existing methods e g information discriminant analysis ida estimate the first and second order statistics of the data in the original high dimensional space and then design the transformation matrix based on the information theoretic criteria unfortunately such transformation methods are sensitive to the accuracy of the statistics estimation to overcome this disadvantage our method describes the statistical structure of the transformed low dimensional subspace via a linear statistical model which can reduce the number of unknown parameters while simultaneously maximizes the mutual information mi between the transformed data and their class labels which can ensure the between class separability according to the information theory the key idea is that we seek the optimal model parameters including the transformation matrix via the joint optimization of mi function and log likelihood function therefore this method can not only reduce the estimation errors but also maximize the between class separability experimental results based on synthetic dataset and benchmark datasets demonstrate the better performance of our method over other related methods we develop a novel feature transformation method for linear dimensionality reduction the statistics in the transformed subspace are learned to reduce unknown parameters the transformation matrix is obtained via joint optimization of mi and likelihood our method can maximize between class separability as well as reduce estimation errors experimental results show our method performs better than other related methods

the role of principal angles in subspace classification subspace models play an important role in a wide range of signal processing tasks and this paper explores how the pairwise geometry of subspaces influences the probability of misclassification when the mismatch between the signal and the model is vanishingly small the probability of misclassification is determined by the product of the sines of the principal angles between subspaces when the mismatch is more significant the probability of misclassification is determined by the sum of the squares of the sines of the principal angles reliability of classification is derived in terms of the distribution of signal energy across principal vectors larger principal angles lead to smaller classification error motivating a linear transform that optimizes principal angles the transform presented here trait preserves some specific characteristic of each individual class and this approach is shown to be complementary to a previously developed transform lrt that enlarges inter class distance while suppressing intraclass dispersion theoretical results are supported by demonstration of superior classification accuracy on synthetic and measured data even in the presence of significant model mismatch

a fourth order spatial accurate and practically stable compact scheme for the cahn hilliard equation abstract we present a fourth order spatial accurate and practically stable compact difference scheme for the cahn hilliard equation the compact scheme is derived by combining a compact nine point formula and linearly stabilized splitting scheme the resulting system of discrete equations is solved by a multigrid method numerical experiments are conducted to verify the practical stability and fourth order accuracy of the proposed scheme we also demonstrate that the compact scheme is more robust and efficient than the non compact fourth order scheme by applying to parallel computing and adaptive mesh refinement

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

infinite max margin factor analysis via data augmentation this paper addresses the bayesian estimation of the discriminative probabilistic latent models especially the mixture models we develop the max margin factor analysis mmfa model which utilizes the latent variable support vector machine lvsvm as the classification criterion in the latent space to learn a discriminative subspace with max margin constraint furthermore to deal with multimodally distributed data we further extend mmfa to infinite gaussian mixture model and develop the infinite max margin factor analysis immfa model via the consideration of dirichlet process mixtures dpm it jointly learns clustering max margin classifiers and the discriminative latent space in a united framework to improve the prediction performance moreover both of mmfa and immfa are natural to handle outlier rejection problem since the observations are described by a single or a mixture of gaussian distributions additionally thanks to the conjugate property the parameters in the two models can be inferred efficiently via the simple gibbs sampler finally we implement our models on synthesized and real world data including multimodally distributed datasets and measured radar echo data to validate the classification and rejection performance of the proposed models jointly learning fa and svm mmfa is proposed to get a discriminative subspace clustering the dataset in the subspace by dpm mmfa is extended to immfa thanks to the jointly learning framework they gain good prediction performance having the data description ability the proposed models can reject outlier samples in bayesian framework parameters can be inferred efficiently by the gibbs sampler

variational problems on contact riemannian manifolds we define the generalized tanaka connection for contact riemannian manifolds generalizing one for nondegenerate integrable cr manifolds then the torsion and the generalized tanaka webster scalar curvature are defined properly furthermore we define gauge transformations of contact riemannian structure and obtain an invariant under such transformations concerning the integral related to the invariant we define a functional and study its first and second variational formulas as an example we study this functional on the unit sphere as a standard contact manifold

a hyperelastic regularization energy for image registration image registration is one of the most challenging problems in image processing where ill posedness arises due to noisy data as well as nonuniqueness and hence the choice of regularization is crucial this paper presents hyperelasticity as a regularizer and introduces a new and stable numerical implementation on one hand hyperelastic registration is an appropriate model for large and highly nonlinear deformations for which a linear elastic model needs to fail on the other hand the hyperelastic regularizer yields very regular and diffeomorphic transformations while hyperelasticity might be considered as just an additional outstanding regularization option for some applications it becomes inevitable for applications involving higher order distance measures like mass preserving registration the paper gives a short introduction to image registration and hyperelasticity the hyperelastic image registration problem is phrased in a variational setting and an existence proof is provided the focus of th

anisotropic second and fourth order diffusion models based on convolutional virtual electric field for image denoising this paper proposes novel second order and fourth order anisotropic diffusion models for image denoising these two models are based on convolutional virtual electric field convef the proposed second order anisotropic diffusion model was introduced by introducing the convef into the perona malik p m equation and the fourth order one is coined by combining convef and the you kaveh y k equation the employment of the convef model allows to improve the estimation of high order derivatives and makes the proposed methods more robust to noise in addition the proposed convef based y k model is an anisotropic diffusion equation which can avoid the over smoothness of edges and provide better edge protection capability except for better denoising some experiments are conducted to demonstrate the efficiency of the proposed methods

subelliptic harmonic maps we study a nonlinear harmonic map type system of subelliptic pde in particular we solve the dirichlet problem with image contained in a convex ball

stability of the reeb vector field of h contact manifolds it is well known that a hopf vector field on the unit sphere s 2n 1 is the reeb vector field of a natural sasakian structure on s 2n 1 a contact metric manifold whose reeb vector field is a harmonic vector field is called an h contact manifold sasakian and k contact manifolds generalized k spaces and contact metric three manifolds with strongly normal are h contact manifolds in this paper we study in dimension three the stability with respect to the energy of the reeb vector field for such special classes of h contact manifolds and with respect to the volume when is also minimal in terms of webster scalar curvature finally we extend for the reeb vector field of a compact k contact 2n 1 manifold the obtained results for the hopf vector fields to minimize the energy functional with mean curvature correction

planar crack identification in 3d linear elasticity by the reciprocity gap method abstract we study the reciprocity gap method andrieux et al 1999 for the identification of planar cracks in the framework of three dimensional linear elasticity in order to achieve better accuracy on thick domains we propose to use polynomial test functions instead of fourier series we also propose to improve the solution by various regularization techniques that are tested and compared numerical assessments are provided in 2d and 3d

bayesian manifold regression there is increasing interest in the problem of nonparametric regression with high dimensional predictors when the number of predictors d is large one encounters a daunting problem in attempting to estimate a d dimensional surface based on limited data fortunately in many applications the support of the data is concentrated on a d dimensional subspace with d ll d manifold learning attempts to estimate this subspace our focus is on developing computationally tractable and theoretically supported bayesian nonparametric regression methods in this context when the subspace corresponds to a locally euclidean compact riemannian manifold we show that a gaussian process regression approach can be applied that leads to the minimax optimal adaptive rate in estimating the regression function under some conditions the proposed model bypasses the need to estimate the manifold and can be implemented using standard algorithms for posterior computation in gaussian processes finite sample performance is illustrated in a data analysis example

area minimizing vector fields on round 2 spheres a vector field v on a n dimensional round sphere sn r defines a submanifold v sn of the tangent bundle tsn the gluck and ziller question is to find the infimum of the n dimensional volume of v sn among unit vector fields this volume is computed with respect to the natural metric on the tangent bundle as defined by sasaki surprisingly the problem is only solved for dimension three 10 in this article we tackle the question for the 2 sphere since there is no globally defined vector field on s the infimum is taken on singular unit vector fields without boundary these are vector fields defined on a dense open set and such that the closure of their image is a surface without boundary in particular if the vector field is area minimizing it defines a minimal surface of t s r we prove that if this minimal surface is homeomorphic to rp 2 then it must be the pontryagin cycle it is the closure of unit vector fields with one singularity obtained by parallel translating a given vector along any great circle passing through a given point we show that pontryagin fields of the unit 2 sphere are area minimizing 2000 mathematics subject classification 53c20

a robust multigrid approach for variational image registration models variational registration models are non rigid and deformable imaging techniques for accurate registration of two images as with other models for inverse problems using the tikhonov regularization they must have a suitably chosen regularization term as well as a data fitting term one distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable this paper first reviews the current state of the art numerical methods for such models and observes that the nonlinear fitting term is mostly avoided in developing fast multigrid methods it then proposes a unified approach for designing fixed point type smoothers for multigrid methods the diffusion registration model second order equations and a curvature model fourth order equations are used to illustrate our robust methodology analysis of the proposed smoothers and comparisons to other methods are given as expected of a multigrid method being many orders of magnitude faster than the unilevel gradient descent approach the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images

conformal energy in four dimension the conformal energy for 4 manifolds using the paneitz operator is introduced in this article the conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume we also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator

a central difference scheme for a pure stream function formulation of incompressible viscous flow we present a numerical scheme for incompressible viscous flow formulated as an equation for the stream function the pure stream function formulation obviates the difficulty associated with vorticity boundary conditions the resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver the advection of vorticity is implemented with a high resolution central scheme that remains stable and accurate in the presence of large gradients the accuracy and robustness of the method are demonstrated for high reynolds number flows in a lid driven cavity

vectorial total variation based regularization for variational image registration to use interdependence between the primary components of the deformation field for smooth and non smooth registration problems the channel by channel total variation or standard vectorial total variation svtv based regularization has been extended to a more flexible and efficient technique allowing high quality regularization procedures based on this method this paper proposes a fast nonlinear multigrid nmg method for solving the underlying euler lagrange system of two coupled second order nonlinear partial differential equations numerical experiments using both synthetic and realistic images not only confirm that the recommended vtv based regularization yields better registration qualities for a wide range of applications than those of the svtv based regularization but also that the proposed nmg method is fast accurate and reliable in delivering visually pleasing registration results

a critical radius for unit hopf vector fields on spheres the volume of a unit vector field v of the sphere n odd is the volume of its image v in the unit tangent bundle unit hopf vector fields that is unit vector fields that are tangent to the fibre of a hopf fibration are well known to be critical for the volume functional moreover gluck and ziller proved that these fields achieve the minimum of the volume if n 3 and they opened the question of whether this result would be true for all odd dimensional spheres it was shown to be inaccurate on spheres of radius one indeed pedersen exhibited smooth vector fields on the unit sphere with less volume than hopf vector fields for a dimension greater than five in this article we consider the situation for any odd dimensional spheres but not necessarily of radius one we show that the stability of the hopf field actually depends on radius instability occurs precisely if and only if in particular the hopf field cannot be minimum in this range on the contrary for r small a computation shows that the volume of vector fields built by pedersen is greater than the volume of the hopf one thus in this case the hopf vector field remains a candidate to be a minimizer we then study the asymptotic behaviour of the volume for small r it is ruled by the first term of the taylor expansion of the volume we call this term the twisting of the vector field the lower this term is the lower the volume of the vector field is for small r it turns out that unit hopf vector fields are absolute minima of the twisting this fact together with the stability result gives two positive arguments in favour of the gluck and ziller conjecture for small r

multichannel blind deconvolution of spatially misaligned images existing multichannel blind restoration techniques assume perfect spatial alignment of channels correct estimation of blur size and are prone to noise we developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral this stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise we observe that the exact knowledge of the blur size is not necessary and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process

a cubic hermite finite element continuation method for numerical solutions of the von karman equations abstract we study a cubic hermite finite element method for numerical solutions of the von karman equations defined in a rectangular domain with totally clamped boundary conditions a novel iterative method combined with a predictor corrector continuation algorithm is exploited to trace solution curves of the von karman equations the fourth order finite element approximations compute accurate numerical solutions for the deformation and the airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives in this regard the classical predictor corrector continuation method is interpreted in a different way our numerical results show that the bifurcation scenario of the von karman equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions

an effective method for solving nonlinear equations and its application the linearized partial differential equation from the nonlinear partial differential equation which was proposed by rudin osher and fatemi l i rudin s osher and e fatemi nonlinear total variation based noise removal algorithms for solving image decomposition was introduced by chambolle a chambolle an algorithm for total variation minimization and applications and r acar and c r vogel r acar and c r vogel analysis of bounded variation penalty methods for ill posed problems in this paper we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works

harmonicity of sections of sphere bundles we consider the energy functional on the space of sections of a sphere bundle over a riemannian manifold m langle cdot cdot rangle equipped with the sasaki metric and discuss the characterising condition for critical points furthermore we provide a useful method for computing the tension field in some particular situations such a method is shown to be adequate for many tensor fields defined on manifolds m equipped with a g structure compatible with langle cdot cdot rangle this leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from m langle cdot cdot rangle into its sphere bundle

second variation of volume and energy of vector fields stability of hopf vector fields in this paper we compute the hessian of the volume of unit vector fields at a minimal one we also find the hessians of a family of functionals thus generalizing the known results concerning second variation of the energy or total bending we use them to study the stability of hopf vector fields on s 2m 1 r and to show that they are stable for m 1 but that for m 1 there is r 0 m le 1 such that for r r 0 the index is at least 2m 2

image restoration based on the minimized surface regularization abstract we propose an image restoration model based on the minimized surface regularization the proposed model closely relates to the classical smoothing rof model rudin et al 1992 we deduce two different conjugation forms via coupling the gradient operator with the smoothing parameter or not and then provide the existence of the minimizer in the continuous setting in order to efficiently solving the proposed model we employ the primal dual method by reformulating the proposed model as a min max problem relying on the convex conjugation the convergence of the algorithm is provided as well numerical implementations mainly emphasize the effectiveness of the proposed method by comparing it to other two well known methods in terms of the cpu time and restoration quality

on the numerical solution of high order multi dimensional elliptic pdes abstract using the idea of differential quadrature two new methods are constructed to approximate the solution of elliptic partial differential equations in higher dimensions to obtain the weighting coefficients of the first method a mixture of grid points and mid points of the uniform partition is used the order of convergence obtained by the first algorithm is non optimal so we mixed the idea with spline collocation to obtain higher order approximations using the weighting coefficients of the non optimal algorithm some new weighting coefficients are obtained which are used to obtain higher accuracy for the first time a sixth order approximation is obtained to the solution of well known high order multi dimensional elliptic pdes such as biharmonic and triharmonic problems we found a way to increase the order of convergence and the accuracy without increasing the cpu runtime the block form of the coefficients matrix for the linear case is presented to have a better vision on the system arised from the method some examples of biharmonic and triharmonic problems are solved to show the good performance and applicability of the proposed algorithms

a multigrid method for a fourth order diffusion equation with application to image processing in this paper we are concerned with the efficient solution of the elliptic system alpha l nobreak i u nobreak f where l is the biharmonic operator supplemented by higher order boundary conditions this is a core task in image processing arising from the minimization of rotation invariant energy functionals for the purpose of numerical solution we approximate the partial differential equation by finite differences in order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods finally experimental results are reported that demonstrate the effectiveness of the multigrid approach

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

retrospective markov chain monte carlo methods for dirichlet process hierarchical models inference for dirichlet process hierarchical models is typically performed using markov chain monte carlo methods which can be roughly categorized into marginal and conditional methods the former integrate out analytically the infinite dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the gibbs sampler conditional methods impute the dirichlet process and update it as a component of the gibbs sampler since this requires imputation of an infinite dimensional process implementation of the conditional method has relied on finite approximations in this paper we show how to avoid such approximations by designing two novel markov chain monte carlo algorithms which sample from the exact posterior distribution of quantities of interest the approximations are avoided by the new technique of retrospective sampling we also show how the algorithms can obtain samples from functionals of the dirichlet process the marginal and the conditional methods are compared and a careful simulation study is included which involves a non conjugate model different datasets and prior specifications

compressed gaussian process for manifold regression nonparametric regression for large numbers of features p is an increasingly important problem if the sample size n is massive a common strategy is to partition the feature space and then separately apply simple models to each partition set this is not ideal when n is modest relative to p and we propose an alternative approach relying on random compression of the feature vector combined with gaussian process regression the proposed approach is particularly motivated by the setting in which the response is conditionally independent of the features given the projection to a low dimensional manifold conditionally on the random compression matrix and a smoothness parameter the posterior distribution for the regression surface and posterior predictive distributions are available analytically running the analysis in parallel for many random compression matrices and smoothness parameters model averaging is used to combine the results the algorithm can be implemented rapidly even in very large p and moderately large n nonparametric regression has strong theoretical justification and is found to yield state of the art predictive performance

new implementation of elliptic systems method for time dependent diffusion tomography with back reflected and transmitted boundary data abstract a problem of coefficient recovery from incomplete boundary data in inverse problems is considered it uses a new implementation of the elliptic systems method esm in time dependent diffusion tomography the basic formulation of the esm involves solving a system of coupled fourth order partial differential equations with the time variable integrated out using legendre polynomials here we use c 1 bogner fox schmit bi cubic elements over rectangles with a new treatment of boundary conditions in the common case of incomplete boundary data this new method is fourth order accurate for sufficiently smooth functions the new boundary condition approach allows the use of homogeneous natural boundary conditions on parts of the boundary where no measured data is available we will focus on a comparison with three previously published examples using back reflected or transmitted data with one or two inclusions the new implementation gives markedly improved results for inclusion recovery all of which are achieved without use of additional aids such as weight functions which previously have been thought to be essential and is shown to be surprisingly robust with respect to noise we conclude with two examples illustrating the effect of increasing levels of noise

modeling the manifolds of images of handwritten digits this paper describes two new methods for modeling the manifolds of digitized images of handwritten digits the models allow a priori information about the structure of the manifolds to be combined with empirical data accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models one of the methods is grounded in principal components analysis the other in factor analysis both methods are based on locally linear low dimensional approximations to the underlying data manifold links with other methods that model the manifold are discussed