quadratic optimization with orthogonality constraint explicit łojasiewicz exponent and linear convergence of retraction based line search and stochastic variance reduced gradient methods the problem of optimizing a quadratic form over an orthogonality constraint qp oc for short is one of the most fundamental matrix optimization problems and arises in many applications in this paper we characterize the growth behavior of the objective function around the critical points of the qp oc problem and demonstrate how such characterization can be used to obtain strong convergence rate results for iterative methods that exploit the manifold structure of the orthogonality constraint i e the stiefel manifold to find a critical point of the problem specifically our primary contribution is to show that the ojasiewicz exponent at any critical point of the qp oc problem is 1 2 such a result is significant as it expands the currently very limited repertoire of optimization problems for which the ojasiewicz exponent is explicitly known moreover it allows us to show in a unified manner and for the first time that a large family of retraction based line search methods will converge linearly to a critical point of the qp oc problem then as our secondary contribution we propose a stochastic variance reduced gradient svrg method called stiefel svrg for solving the qp oc problem and present a novel ojasiewicz inequality based linear convergence analysis of the method an important feature of stiefel svrg is that it allows for general retractions and does not require the computation of any vector transport on the stiefel manifold as such it is computationally more advantageous than other recently proposed svrg type algorithms for manifold optimization

gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

least squares estimation of transformation parameters between two point patterns in many applications of computer vision the following problem is encountered two point patterns sets of points x sub i and x sub i i 1 2 n are given in m dimensional space and the similarity transformation parameters rotation translation and scaling that give the least mean squared error between these point patterns are needed recently k s arun et al 1987 and b k p horn et al 1987 presented a solution of this problem their solution however sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted the proposed theorem is a strict solution of the problem and it always gives the correct transformation parameters even when the data is corrupted

an exact forward backward maximum likelihood autoregressive parameter estimation method a method for obtaining an exact maximum likelihood estimate mle of the autoregressive ar parameters is proposed the method is called the forward backward maximum likelihood algorithm based on a new form of the log likelihood function for a gaussian ar process an iterative maximization is used to obtain an mle of the inverse covariance matrix the ar parameters are then determined via the normal equations experimental results comparing the new method with other popular ar spectrum estimation methods indicate the new method achieves low bias and low variance ar parameter estimates comparable with the existing methods

extrema of sums of heterogeneous quadratic forms abstract we analyze the following problem arising in various situations in multivariate statistical analysis we are given k symmetric positive definite n n matrices a1 a2 ak k n and we would like to maximize the function ki 1 xtiaixi under the constraint that x1 x2 xk rn form an orthonormal system some theoretical results as well as an algorithm are presented

a framework of constraint preserving update schemes for optimization on stiefel manifold this paper considers optimization problems on the stiefel manifold x mathsf t x i p xtx ip where x in mathbb r n times p x rn p is the variable and i p ip is the p p by p p identity matrix a framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of x x and the null space of x mathsf t xt while this general framework can unify many existing schemes a new update scheme with low complexity cost is also discovered then we study a feasible barzilai borwein like method under the new update scheme the global convergence of the method is established with an adaptive nonmonotone line search the numerical tests on the nearest low rank correlation matrix problem the kohn sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method in particular the new method performs remarkably well for the nearest low rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used scf iteration for the kohn sham total energy minimization

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

dynamical systems that perform the singular value decomposition abstract a dynamical system is presented whose solutions are shown to converge exponentially to the singular value decomposition of a given matrix provided that its singular values are distinct this system has potential application to certain problems in adaptive control and adaptive filtering it is one of several techniques that employ dynamical systems to perform tasks done conventionally with the techniques of numerical linear algebra

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

qr type factorization the yang baxter equation and an eigenvalue problem of control theory abstract it is shown that every solution to the yang baxter equation corresponds with the o r type algorithm the structure of solutions of the yang baxter equation is described and examples relevant to the algebraic riccati equation are considered

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

differential equations and the qr algorithm in this paper we consider a variety of isospectral flows on the set of n times n matrices these flows arise from lax pairs and can all be interpreted in terms of the qr decomposition for nonsingular matrices the asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem

unsupervised bayesian convex deconvolution based on a field with an explicit partition function this paper proposes a non gaussian markov field with a special feature an explicit partition function to the best of our knowledge this is an original contribution moreover the explicit expression of the partition function enables the development of an unsupervised edge preserving convex deconvolution method the method is fully bayesian and produces an estimate in the sense of the posterior mean numerically calculated by means of a monte carlo markov chain technique the approach is particularly effective and the computational practicability of the method is shown on a simple simulated example

augmented lagrangian applied to convex quadratic problems an algorithm based on the augmented lagrangian method is proposed to solve convex quadratic programming problem the quadratic penalty is considered here hence the augmented lagrangian function is quadratic when applied to quadratic programming problem for this penalty we show that if the lagrangian function associated with the original problem is strict convex or convex then the hessian matrix of augmented lagrangian function is positive definite or positive semi definite numerical experiments are presented illustrating the performance of the algorithm for the cute test set

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

a gradient algorithm for stereo matching without correspondence in this paper we study the problem of feature point matching via a technique well known in geometry leading to a new gradient algorithm in machine vision the procedure does not require one to solve explicit correspondences between feature points but relies on a specific form of spatial averaging in this sense the estimation procedure is robust in this paper we describe the algorithm and report simulation results

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

scaled toda like flows abstract this paper discusses the class of isospectral flows x x a x where denotes the hadamard product and is the lie bracket the presence of a allows arbitrary and independent scaling for each element in the matrix x the time 1 mapping of the scaled toda like flow still enjoys a qr like iteration the scaled structure includes the classical toda flow brockett s double bracket flow and other interesting flows as special cases convergence proof is thus unified and simplified the effect of scaling on a variety of applications is demonstrated by examples

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

maximal matching of 3 d points for multiple object motion estimation abstract determining the relative motion position between an observer and its environment is an important problem in computer vision a major task is to find corresponding object features in this paper an efficient algorithm for matching 3 d points of multiple rigid objects is presented the point matching algorithm determines the correspondence by initiating a pairing of a triplet of noncollinear sensed points with a triplet of reference points and searching for new pairs of corresponding points one at a time using local distance and angular constraints the pairing of each subsequent sensed point with a reference point is determined if the tetrahedron formed by the sensed point and the initial triplet is congruent to that formed by the corresponding reference points only simple computations are required in the algorithm global consistency of the pairings found by the algorithm is ensured without using model tests the algorithm can be easily extended to incorporate other geometrical or non geometrical object attributes to further prune the matching results of running the algorithm on synthetic and real data are given

efficient point pattern matching algorithm for planar point sets under transform of translation rotation and scale abstract point pattern matching is an important topic in computer vision and pattern recognition and finds many applications such as image registration motion detection object tracking and pose estimation in this paper we propose an efficient algorithm for determining correspondence between two planar point sets under transform of translation rotation and scale this algorithm randomly selects some points of a set and extracts their neighbor points it views the selected points and their neighbor points as local point patterns and finds the local matched patterns in the other set point pattern matching is finally achieved by counting the unique point number of those local matched point patterns with the same transform parameters many experiments are conducted to validate efficiency of the proposed algorithm running time comparisons with a well known point pattern matching algorithm are also done and the results show that the proposed algorithm is faster than the compared algorithm

algebraic aspects of brockett type equations abstract it is well known that soliton equations can be written in lax form where the bracket of two linear differential n kdv or pseudo differential kp operators appears in this work we introduce a new hierarchy each equation of which is defined by a double bracket of two pseudo differential operators these double bracket equations arose originally in the study by brockett of the steepest descent equations corresponding to certain least squares matching and sorting problems we deal with some algebraic properties of these equations in particular we show that as in the classical case they are related to the presence of an infinite sequence of first integrals

classification of rotated and scaled textured images using gaussian markov random field models consideration is given to the problem of classifying a test textured image that is obtained from one of c possible parent texture classes after possibly applying unknown rotation and scale changes to the parent texture the training texture images parent classes are modeled by gaussian markov random fields gmrfs to classify a rotated and scaled test texture the rotation and scale changes are incorporated in the texture model through an appropriate transformation of the power spectral density of the gmrf for the rotated and scaled image a bona fide likelihood function that shows the explicit dependence of the likelihood function on the gmrf parameters as well as on the rotation and scale parameters is derived although in general the scaled and or rotated texture does not correspond to a finite order gmrf it is possible nonetheless to write down a likelihood function for the image data the likelihood function of the discrete fourier transform of the image data corresponds to that of a white nonstationary gaussian random field with the variance at each pixel i j being a known function of the rotation the scale the gmrf model parameters and i j the variance is an explicit function of the appropriately sampled power spectral density of the gmrf the estimation of the rotation and scale parameters is performed in the frequency domain by maximizing the likelihood function associated with the discrete fourier transform of the image data cramer rao error bounds on the scale and rotation estimates are easily computed a modified bayes decision rule is used to classify a given test image into one of c possible texture classes the classification power of the method is demonstrated through experimental results on natural textures from the brodatz album

super brockett equations a graded gradient integrable system rather recently equations of lax type defined by a double commutator the so called brockett equations have received considerable attention in this paper we prove that a supersymmetric version of a brockett hierarchy is an infinite dimensional integrable gradient system as far as we know this is the only graded system of this type existing in the literature

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

model indexing and object recognition using 3d viewpoint invariance object recognition and location of 3d objects from range data are described based on invariant representations of characteristic views of single objects depth data is processed to remove outliers then smoothed and segmented into surface patches fitted by polynomial equations mathematical invariants are derived from groups of these surface patches and used for indexing a model database stored in the form of a kd tree the pose of the known object may then be determined from the euclidean centre and orientation of the higher order surfaces formed from the feature groups the approach is complete scalable and extendable the robustness of the method is evaluated in terms of the stability and discriminatory power of the representation and the accuracy of the determined pose

convergence rate analysis of nonquadratic proximal methods for convex and linear programming the divergence proximal method is an extension of the proximal minimization algorithm where the usual quadratic proximal term is substituted by a class of convex statistical distances called divergences in this paper we study the convergence rate of this nonquadratic proximal method for convex and particularly linear programming we identify a class of divergences for which superlinear convergence is attained both for optimization problems with strongly convex objectives at the optimum and linear programming problems when the regularization parameters tend to zero we prove also that with regularization parameters bounded away from zero convergence is at least linear for a wider class of divergences when the method is applied to the same kinds of problems we further analyze the associated class of augmented lagrangian methods for convex programming with nonquadratic penalty terms and prove convergence of the dual generated by these methods for linear programming problems under a weak nondegeneracy assumption

a grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

model based deformable surface finding for medical images describes a new global shape parameterization for smoothly deformable three dimensional 3 d objects such as those found in biomedical images whose diversity and irregularity make them difficult to represent in terms of fixed features or parts this representation is used for geometric surface matching to 3 d medical image data such as from magnetic resonance imaging mri the parameterization decomposes the surface into sinusoidal basis functions four types of surfaces are modeled tori open surfaces closed surfaces and tubes this parameterization allows a wide variety of smooth surfaces to be described with a small number of parameters extrinsic model based information is incorporated by introducing prior probabilities on the parameters surface finding is formulated as an optimization problem results of the method applied to synthetic images and 3 d medical images of the heart and brain are presented

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

the dynamics of rayleigh quotient iteration this paper analyzes the rayleigh quotient iteration algorithm for symmetric matrices dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector it is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues

simple parallel hierarchical and relaxation algorithms for segmenting noncausal markovian random fields the modeling and segmentation of images by mrf s markov random fields is treated these are two dimensional noncausal markovian stochastic processes two conceptually new algorithms are presented for segmenting textured images into regions in each of which the data are modeled as one of c mrf s the algorithms are designed to operate in real time when implemented on new parallel computer architectures that can be built with present technology a doubly stochastic representation is used in image modeling here a gaussian mrf is used to model textures in visible light and infrared images and an autobinary or autoternary etc mrf to model a priori information about the local geometry of textured image regions for image segmentation the true texture class regions are treated either as a priori completely unknown or as a realization of a binary or ternary etc mrf in the former case image segmentation is realized as true maximum likelihood estimation in the latter case it is realized as true maximum a posteriori likelihood segmentation in addition to providing a mathematically correct means for introducing geometric structure the autobinary or ternary etc mrf can be used in a generative mode to generate image geometries and artificial images and such simulations constitute a very powerful tool for studying the effects of these models and the appropriate choice of model parameters the first segmentation algorithm is hierarchical and uses a pyramid like structure in new ways that exploit the mutual dependencies among disjoint pieces of a textured region

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

total linear least squares and the algebraic riccati equation abstract it is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation which turns into an algebraic riccati equation when the matrix of unknowns is square if there is an additional symmetry constraint on the solution the optimal solution is given by the anti stabilizing solution of this riccati equation

robust algorithms for object localization model based localization the task of estimating an object s pose from sensed and corresponding model features is a fundamental task in machine vision exact constant time localization algorithms have been developed for the case where the sensed features and the model features are the same type still it is not uncommon for the sensed features and the model features to be of different types i e sensed data points may correspond to model faces or edges previous localization approaches have handled different model and sensed features of different types via sampling and synthesizing virtual features to reduce the problem of matching features of dissimilar types to the problem of matching features of similar types unfortunately these approaches may be suboptimal because they introduce artificial errors other localization approaches have reformulated object localization as a nonlinear least squares problem where the error is between the sensed data and model features in image coordinates the euclidean image error metric unfortunately all of the previous approaches which minimized the euclidean image error metric relied on gradient descent methods to find the global minima and gradient descent methods may suffer from problems of local minima in this paper we describe an exact efficient solution to the nonlinear least squares minimization problem based upon resultants linear algebra and numerical techniques on a sparc 20 our localization algorithm runs in a few microseconds for rectilinear polygonal models a few milliseconds for generic polygonal models and one second for generalized polygonal models models composed of linear edges and circular arcs

recognition of occluded polyhedra from range images abstract occlusion remains a major hindrance for automatic recognition of 3 d objects in this paper we address the occlusion problem in the context of polyhedral object recognition from range data a novel approach is presented for object recognition based on sound occlusion guided reasoning for feature distortion analysis and perceptual organization this type of reasoning enables us to maximize the amount of information extracted from the scene data thus leading to robust and efficient recognition the proposed approach is based on a multi stage matching process which attempts to recognize scene objects according to their order in the occlusion hierarchy i e an object is recognized before those that are occluded by it such a strategy helps in resolving some occlusion induced ambiguities in feature distortion analysis furthermore it leads to verification of object pose hypotheses with greater confidence matching is based on a hypothesize cluster and verify approach hypotheses are generated using an occlusion tolerant composite feature a fork which is a pair of non parallel edges that belong to the same surface generated hypotheses are then clustered and verified using a robust pixel based technique indexing is performed using distortion adaptive bounds on a rich set of viewpoint invariant fork attributes for high selectivity even in the presence of heavy occlusion performance of the system is demonstrated using complex multi object scenes

on minimization on stiefel manifolds abstract the minimization of a smooth function f r kn r under the constraint that vectors x 1 x 2 x k r n k n form an orthonormal system seems to be a new and interesting global optimization problem with important theoretical and practical applications the set of feasible points determines a differentiable manifold introduced by stiefel in 1935 based on the nice geometric structure the optimality conditions are obtained by the global lagrange multiplier rule and global optimality conditions based on local information which make the advantages of using the riemannian geometry in difficult smooth optimization problems clear

bayesian segmentation via asymptotic partition functions asymptotic approximations to the partition function of gaussian random fields are derived textures are characterized via gaussian random fields induced by stochastic difference equations determined by finitely supported stationary linear difference operators adjusted to be nonstationary at the boundaries it is shown that as the scale of the underlying shape increases the log normalizer converges to the integral of the log spectrum of the operator inducing the random field fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator induced random field model matrix analysis techniques are proposed for handling textures with variable orientation examples of texture parameters estimated from training data via asymptotic maximum likelihood are shown isotropic models involving powers of the laplacian and directional models involving partial derivative mixtures are explored parameters are estimated for mitochondria and actin myocin complexes in electron micrographs and clutter in forward looking infrared images deformable template models are used to infer the shape of mitochondria in electron micrographs with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds

texture synthesis using 2 d noncausal autoregressive models the purpose of this paper is to illustrate the usefulness of two dimensional noncausal autoregressive ncar models for the synthesis of textures these models characterize the gray level at a pixel as a linear combination of gray levels at nearby locations in all directions and an additive white noise variable we first show that the class of ncar models is capable of generating a wide variety of image patterns posessing the local replication attribute an essential ingredient of many natural textures it is also shown that the theoretical variograms of many ncar models possess an oscillatory behavior a characteristic of the variograms of many natural textures next we give experimental results of synthesis of 64 64 textures resembling several real textures in the brodatz album the synthetic textures generated by 16 parameter ncar models retain most the visual characteristics of the original textures the variograms of the original and synthetic textures are also similar

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

codimension and connectedness of degeneracy loci over local rings we deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules f m rightarrow n over a local noetherian ring a mathfrak m we show for instance that the expected determinantal bounds on the dimension of the t th degeneracy locus of f hold if f in mathfrak m mathrm hom m n and that this degeneracy locus is connected in the expected dimension provided hat a is a domain

estimation principal components and hamiltonian systems in this paper we show how symplectic geometry can be used to analyse a fundamental problem in estimation theory the fitting of lines and planes to a data set in n dimensions symplectic methods yield information not only on the critical point structure of the maximum likelihood estimation mle function see byrnes and willems 7 but also on the asymmetries in the distribution of the data points in n space

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

a dynamical systems approach to weighted graph matching graph matching is a fundamental problem that arises frequently in the areas of distributed control computer vision and facility allocation in this paper we consider the optimal graph matching problem for weighted graphs which is computationally challenging due the combinatorial nature of the set of permutations contrary to optimization based relaxations to this problem in this paper we develop a novel relaxation by constructing dynamical systems on the manifold of orthogonal matrices in particular since permutation matrices are orthogonal matrices with nonnegative elements we define two gradient flows in the space of orthogonal matrices the first minimizes the cost of weighted graph matching over orthogonal matrices whereas the second minimizes the distance of an orthogonal matrix from the finite set of all permutations the combination of the two dynamical systems converges to a permutation matrix which provides a suboptimal solution to the weighted graph matching problem finally our approach is shown to be promising by illustrating it on nontrivial problems

a comparison of approaches to high level image interpretation abstract a critical comparison of methods of high level interpretation of segmented images is presented and illustrated by examples drawn from two dimensional image analysis the methods discussed include boundary correlation generalized hough transformation relational distance measures graph matching heuristic search and relaxation labelling each technique is considered in terms of its reliability in interpreting noisy and cluttered scenes and in terms of the dynamic complexity of the algorithm as a function of the input data although 2 d analysis of linearly segmented scenes is used as a mechanism for comparison many of the arguments are applicable to three dimensional image interpretation based on surface modelling for example

least squares fitting of two 3 d point sets two point sets pi and p i i 1 2 n are related by p i rpi t ni where r is a rotation matrix t a translation vector and ni a noise vector given pi and p i we present an algorithm for finding the least squares solution of r and t which is based on the singular value decomposition svd of a 3 3 matrix this new algorithm is compared to two earlier algorithms with respect to computer time requirements

a control theoretic approach to the design of zero finding numerical methods in this paper it is shown how standard iterative methods for solving linear and nonlinear equations can be designed from the point of view of control appropriate choices of control lyapunov functions clfs lead to both continuous and discrete time versions of the newton raphson and conjugate gradient algorithms as well as new variants

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

nonlinear rescaling and proximal like methods in convex optimization the nonlinear rescaling principle nrp consists of transforming the objective function and or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides a nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints the methods based on nrp consist of sequential unconstrained minimization of the classical lagrangian for the equivalent problem followed by an explicit formula updating the lagrange multipliers we first show that the nrp leads naturally to proximal methods with an entropy like kernel which is defined by the conjugate of the scaling function and establish that the two methods are dually equivalent for convex constrained minimization problems we then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy like proximal methods for convex constrained optimization problems special cases of the nonlinear rescaling algorithm are presented in particular a new class of exponential penalty modified barrier functions methods is introduced

orthogonalization of vectors with minimal adjustment two transformations are proposed that give orthogonal components with a one to one correspondence between the original vectors and the components the aim is that each component should be close to the vector with which it is paired orthogonality imposing a constraint the transformations lead to a variety of new statistical methods including a unified approach to the identification and diagnosis of collinearities a method of setting prior weights for bayesian model averaging and a means of calculating an upper bound for a multivariate chebychev inequality one transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors and is determined using a new algorithm that also provides the decomposition of a positive definite matrix in terms of a diagonal matrix and a correlation matrix the algorithm is shown to converge to a global optimum

an unbiased and computationally efficient ls estimation method for identifying parameters of 2d noncausal sar models an unbiased and computationally efficient modified least squares ls estimation method for identifying parameters of two dimensional noncausal simultaneous autoregressive models is presented some intuitive and mathematical proof of the unbiasedness of the method are given and a recursive in order fast algorithm to implement it is introduced computer simulation results are given to sustain the theoretical analysis both the theoretical analysis and the computer simulation show that the method possesses much higher estimation accuracy and lower computational complexity than the conventional ls estimation method compared to the approximate maximum likelihood method of kashyap and chellappa 1983 the scheme is much faster has the same estimation accuracy and is parallelizable

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

steepest descent algorithms for optimization under unitary matrix constraint in many engineering applications we deal with constrained optimization problems with respect to complex valued matrices this paper proposes a riemannian geometry approach for optimization of a real valued cost function t of complex valued matrix argument w under the constraint that w is an n times n unitary matrix we derive steepest descent sd algorithms on the lie group of unitary matrices u n the proposed algorithms move towards the optimum along the geodesics but other alternatives are also considered we also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic sd algorithms armijo step size 1 adaptation rule is used similarly to 2 but with reduced complexity the theoretical results are validated by computer simulations the proposed algorithms are applied to blind source separation in mimo systems by using the joint diagonalization approach 3 we show that the proposed algorithms outperform other widely used algorithms

filters random fields and maximum entropy frame towards a unified theory for texture modeling this article presents a statistical theory for texture modeling this theory combines filtering theory and markov random field modeling through the maximum entropy principle and interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view our theory characterizes the ensemble of images i with the same texture appearance by a probability distribution f i on a random field and the objective of texture modeling is to make inference about f i given a set of observed texture examples in our theory texture modeling consists of two steps 1 a set of filters is selected from a general filter bank to capture features of the texture these filters are applied to observed texture images and the histograms of the filtered images are extracted these histograms are estimates of the marginal distributions of f i this step is called feature extraction 2 the maximum entropy principle is employed to derive a distribution p i which is restricted to have the same marginal distributions as those in 1 this p i is considered as an estimate of f i this step is called feature fusion a stepwise algorithm is proposed to choose filters from a general filter bank the resulting model called frame filters random fields and maximum entropy is a markov random field mrf model but with a much enriched vocabulary and hence much stronger descriptive ability than the previous mrf models used for texture modeling gibbs sampler is adopted to synthesize texture images by drawing typical samples from p i thus the model is verified by seeing whether the synthesized texture images have similar visual appearances to the texture images being modeled experiments on a variety of 1d and 2d textures are described to illustrate our theory and to show the performance of our algorithms these experiments demonstrate that many textures which are previously considered as from different categories can be modeled and synthesized in a common framework

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

self equivalent flows associated with the generalized eigenvalue problem abstract we present a family of flows which includes continuous analogues of the unshifted and shifted lz and qz algorithms for the generalized eigenvalue problem in order to do this we use elementary lie theory to create a general family of algorithms of which the lz and qz algorithms are special cases for each such algorithm we construct a family of associated flows some of which are interpolants of the algorithm we do not restrict our attention to hessenberg triangular forms we consider arbitrary pairs of nonsingular matrices

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

the connectedness of symmetric and skew symmetric degeneracy loci even ranks a degeneracy locus is the set of points where a vector bundle map has rank at most a given integer such a set is symmetric or skew symmetric according as whether the vector bundle map is symmetric or skew symmetric we prove a connectedness result first conjectured by fulton and lazarsfeld for skew symmetric degeneracy loci and for symmetric degeneracy loci of even ranks

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

on the convergence of the exponential multiplier method for convex programming in this paper we analyze the exponential method of multipliers for convex constrained minimization problems which operates like the usual augmented lagrangian method except that it uses an exponential penalty function in place of the usual quadratic we also analyze a dual counterpart the entropy minimization algorithm which operates like the proximal minimization algorithm except that it uses a logarithmic entropy proximal term in place of a quadratic we strengthen substantially the available convergence results for these methods and we derive the convergence rate of these methods when applied to linear programs