multilevel algorithms considered as iterative methods on semidefinite systems for the representation of piecewise d linear functions instead of the usual finite element basis a generating system is introduced that contains the nodal basis functions of the finest level and of all coarser levels of discretization this approach enables the author to work directly with multilevel decompositions of a function for a partial differential equation the galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the one dimensional 1d case only about twice in the two dimensional 2d case about 4 3 times and in the three dimensional 3d case about 8 7 times as many unknowns as the usual system furthermore the semidefinite system possesses not just one but many solutions however the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem the author shows that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite sy

multilevel first order system least squares for nonlinear elliptic partial differential equations a fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods the equations are converted to a first order system that is then linearized via newton s method first order system least squares fosls is used to formulate and discretize the newton step and the resulting matrix equation is solved using algebraic multigrid amg the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation a general theory is developed that confirms the usual full multigrid efficiency accuracy comparable to the finest level discretization is achieved at a cost proportional to the number of finest level degrees of freedom in a companion paper the theory is applied to elliptic grid generation egg and supported by numerical results

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

a multilevel nonlinear method a multilevel nonlinear method mnm for the numerical solution of nonlinear partial differential equations is developed the mnm algorithm is motivated and analyzed using a simplified model which retains the essential features of the new approach it is thereby shown to combine the advantages of the two classical multigrid approaches to nonlinear problems the analysis is supported by numerical tests for nonlinear differential equations in one and two dimensions

spectral symmetry analysis we present a spectral approach for detecting and analyzing rotational and reflectional symmetries in n dimensions our main contribution is the derivation of a symmetry detection and analysis scheme for sets of points irn and its extension to image analysis by way of local features each object is represented by a set of points s irn where the symmetry is manifested by the multiple self alignments of s the alignment problem is formulated as a quadratic binary optimization problem with an efficient solution via spectral relaxation for symmetric objects this results in a multiplicity of eigenvalues whose corresponding eigenvectors allow the detection and analysis of both types of symmetry we improve the scheme s robustness by incorporating geometrical constraints into the spectral analysis our approach is experimentally verified by applying it to 2d and 3d synthetic objects as well as real images

skewed rotation symmetry group detection we present a novel and effective algorithm for affinely skewed rotation symmetry group detection from real world images we define a complete skewed rotation symmetry detection problem as discovering five independent properties of a skewed rotation symmetry group 1 the center of rotation 2 the affine deformation 3 the type of the symmetry group 4 the cardinality of the symmetry group and 5 the supporting region of the symmetry group in the image we propose a frieze expansion fe method that transforms rotation symmetry group detection into a simple 1d translation symmetry detection problem we define and construct a pair of rotational symmetry saliency maps complemented by a local feature method frequency analysis using discrete fourier transform dft is applied to the frieze expansion patterns feps to uncover the types cyclic dihedral and o 2 the cardinalities and the corresponding supporting regions concentric or otherwise of multiple rotation symmetry groups in an image the phase information of the fep is used to rectify affinely skewed rotation symmetry groups our result advances the state of the art in symmetry detection by offering a unique combination of region based feature based and frequency based approaches experimental results on 170 synthetic and natural images demonstrate superior performance of our rotation symmetry detection algorithm over existing methods

analysis of velocity flux least squares principles for the navier stokes equations part ii this paper continues the development of the least squares methodology for the solution of the incompressible navier stokes equations started in part i here we again use a velocity flux first order navier stokes system but our focus now is on a practical algorithm based on a discrete negative norm

preferred point geometry and the local differential geometry of the kullback leibler divergence a new preferred point geometric structure for statistical analysis closely related to amari s geometries is introduced the added preferred point structure is seen to resolve the problem that divergence measures do not obey the intuitively natural axioms for a distance function as commonly used in geometry using this tool two key results of amari which connect geodesics and divergence functions are developed the embedding properties of the kullback leibler divergence are considered and a strong curvature condition is produced under which it agrees with a statistically natural squared preferred point geodesic distance when this condition fails the choice of divergence may be crucial further amari s pythagorean result is shown to generalise in the preferred point context

local divergence and association summary in applications of differential geometry to problems of parametric statistical inference the notion of divergence is often used to measure the separation between two parametric densities in this paper we move away from the parametric framework of much of statistical differential geometry and instead employ divergence in a covariate setting many data analyses involve investigation of how the conditional distribution f y i x x of a random variable y changes with covariate values x we propose using local divergence between conditional distributions as a measure of this change and thus as a general measure of association between y and the covariates x under certain regularity conditions we define a class of divergences which are locally the rao distance lr the limiting lr divergence is bounded below by the signal to noise ratio with equality holding if and only if the conditional density comes from the natural exponential family the correlation curve and in particular the correlation coefficient are simply transformations of the signal to noise ratio and thus are transformations of local divergence we therefore obtain a differential geometric interpretation of standard ideas of association the class of lr divergences is broad and includes the kullback leibler divergence and renyi oc information measures we can therefore interpret local association as local utility or information gain of order cx we obtain comparable results without regularity conditions

unigrid for multigrid simulation this paper develops unigrid which is an algorithm that results from interpreting multigrid as it directly affects the fine grid approximation unigrid is theoretically equivalent to multigrid under certain assumptions yet has different computational characteristics though it is generally less efficient unigrid is very useful as a software tool for testing the feasibility of applying multigrid to a given problem this is illustrated with several numerical examples 1 introduction the subject of this paper is unigrid a method designed primarily for solving the equations arising from discretization of partial differential boundary value problems the method is closely related and under certain conditions theoretically equivalent to multigrid although all of the unigrid computations are performed on a single grid as the name implies unigrid is generally more computa tionally costly than multigrid yet its simplicity is reflected in its relative ease of code implementation and use with existing applications software this enables unigrid to act effectively as a software tool to simulate the multigrid approach that requires a

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

a technique for finding the symmetry axes of implicit polynomial curves under perspective projection this paper presents an algebraic technique for detecting the symmetry axes of a perspectively projected plane curve the procedure applies to implicit polynomials which has been fitted to 2d image data acquired by means of a photo camera or a tv set the effectiveness of the proposed method has been demonstrated experimentally by using both synthetic and real images

on the local geometry of mixture models summary despite the well known difficulties of undertaking inference with mixture models they are frequently used for modelling these inferential problems arise because the underlying geometry of a mixture family is very complicated this paper shows that by adding a simplifying assumption which frequently is natural statistically the geometric structure is reduced to a much more tractable form this enables standard inferential techniques to be applied successfully one result of studying the local geometry is that it unifies the convex and differential geometric theories of mixture models the techniques proposed are applied to prediction random effects and measurement error models

detection of rotational symmetry in curves represented by the slope chain code abstract we present a new approach based on the slope chain code to determine whether a curve is rotational symmetrical and its order of symmetry the proposed approach works for open and closed perfectly symmetrical or quasi symmetrical 2d curves simple operations on the scc and its invariant properties are central to our methodology to evaluate the proposed methodology we use 1400 curves from a public database for the symmetrical asymmetrical classification task a recall r of 0 86 a balanced accuracy ba of 0 92 and a precision p of 0 87 were obtained for the quasi symmetrical quasi asymmetrical classification task r 0 77 ba 0 83 and p 0 70 were obtained for the order of rotational symmetry detection task the following performance was achieved r 0 97 ba 0 98 and p 0 95 for a symmetrical set of curves and r 0 98 ba 0 98 and p 0 90 for a quasi symmetrical set of curves we conclude our presentation demonstrating the usefulness of our methodology with three practical applications

unified detection of skewed rotation reflection and translation symmetries from affine invariant contour features symmetry detection is significant for object detection and recognition since symmetries are salient cues for distinguishing geometrical structures from cluttered backgrounds this paper proposes a unified framework to detect rotation reflection and translation symmetries simultaneously on unsegmented real world images the detection is performed based on affine invariant contour features and is applicable under skewed imaging with distortions contours on a natural image are first matched to each other to find affine invariant matching pairs which are then classified into three groups using a sign change criterion corresponding to the three types of symmetries the three groups are used to vote for the corresponding symmetries the voting for rotation utilizes a scheme of short line segments the voting for reflection is based on a parameterization of axis equation and the voting for translation uses a cascade like approach experimental results of real world images are provided with quantitative evaluations validating that the proposed framework achieves desired performance

on multilevel iterative methods for optimization problems this paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces the construction of auxiliary problems as well as applications to elasto plastic model and linear programming are described the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates equivalents of this condition are presented in special cases

detection and localisation of reflectional and rotational symmetry under weak perspective projection in this paper a novel method is proposed for detecting and recovering the pose of a reflectional and rotational symmetry from a single weak perspective image in our paradigm an effective and novel hough based approach is proposed for extracting skew symmetry axes from edge contours a refinement algorithm is developed for verifying and pruning away the implausible and spurious hypothesised symmetry axes for each of the detected skew symmetry axes and its corresponding skew angle a mathematical formulation is devised for evaluating the geometric relationships between an orthogonal junction formed by a symmetry axis and its perpendicular axis in an original plane and its corresponding skew angle in an image plane the derived geometric formulation is integrated into an effective voting scheme for estimating the projective parameters of a reflectionally and rotationally symmetric planar shape using the information of the plausible skew symmetry axes experimental results are presented to verify the computational performance of the proposed paradigm the practical significance and generality of the proposed method are manifold

curved glide reflection symmetry detection we generalize the concept of bilateral reflection symmetry to curved glide reflection symmetry in 2d euclidean space such that classic reflection symmetry becomes one of its six special cases we propose a local feature based approach for curved glide reflection symmetry detection from real unsegmented 2d images furthermore we apply curved glide reflection axis detection for curved reflection surface detection in 3d images our method discovers groups and connects statistically dominant local glide reflection axes in an axis parameter space aps without preassumptions on the types of reflection symmetries quantitative evaluations and comparisons against state of the art algorithms on a diverse 64 test image set and 1 125 swedish leaf data images show a promising average detection rate of the proposed algorithm at 80 and 40 percent respectively and superior performance over existing reflection symmetry detection algorithms potential applications in computer vision particularly biomedical imaging include saliency detection from unsegmented images and quantification of deviations from normality we make our 64 test image set publicly available

projection multilevel methods for quasilinear elliptic partial differential equations theoretical results the goal of this paper is to introduce a new multilevel solver for two dimensional elliptic systems of nonlinear partial differential equations pdes where the nonlinearity is of the type u partial v the incompressible navier stokes equations are an important representative of this class and are the target of this study using a first order system least squares fosls approach and introducing a new variable for partial v for this class of pdes we obtain a formulation in which the nonlinearity appears as a product of two different dependent variables the result is a system that is linear within each variable but nonlinear in the cross terms in this paper we introduce a new multilevel method that treats the nonlinearities directly this approach is based on a projection multilevel pml method s f mccormick multilevel projection methods for partial differential equations siam philadelphia 1992 applied to the fosls functional the implementation of the discretization process relaxation coarse grid correction and cycling strategies is discussed and optimal performance is established numerically a companion paper t a manteuffel s f mccormick and o rohrle siam j numer anal 44 2006 pp 139 152 establishes a two level convergence proof for this new multilevel method

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach