on the convergence of gradient descent for finding the riemannian center of mass we study the problem of finding the global riemannian center of mass of a set of data points on a riemannian manifold specifically we investigate the convergence of constant step size gradient de

pixelwise adaptive blind optical flow assuming nonstationary statistics we address some of the major issues in optical flow within a new framework assuming nonstationary statistics for the motion field and for the errors problems addressed include the preservation of discontinuities model data errors outliers confidence measures and performance evaluation in solving these problems we assume that the statistics of the motion field and the errors are not only spatially varying but also unknown we thus derive a blind adaptive technique based on generalized cross validation for estimating an independent regularization parameter for each pixel our formulation is pixelwise and combines existing first and second order constraints with a new second order temporal constraint we derive a new confidence measure for an adaptive rejection of erroneous and outlying motion vectors and compare our results to other techniques in the literature a new performance measure is also derived for estimating the signal to noise ratio for real sequences when the ground truth is unknown

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

on the unification of line processes outlier rejection and robust statistics with applications in early vision the modeling of spatial discontinuities for problems such as surface recovery segmentation image reconstruction and optical flow has been intensely studied in computer vision while line process models of discontinuities have received a great deal of attention there has been recent interest in the use of robust statistical techniques to account for discontinuities this paper unifies the two approaches to achieve this we generalize the notion of a line process to that of an analog outlier process and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics we also characterize a class of robust statistical problems for which an equivalent outlier process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier process formulation we show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier process formulation these results indicate that the outlier process approach provides a general framework which subsumes the traditional line process approaches as well as a wide class of robust estimation problems examples in surface reconstruction image segmentation and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited an appendix provides a catalog of common robust error norms and their equivalent outlier process formulations

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

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

adaptive grid refinement procedures for efficient optical flow computation two approaches are described that improve the efficiency of optical flow computation without incurring loss of accuracy the first approach segments images into regions of moving objects the method is based on a previously defined galerkin finite element method on a triangular mesh combined with a multiresolution segmentation approach for object flow computation images are automatically segmented into subdomains of moving objects by an algorithm that employs a hierarchy of mesh coarseness for the flow computation and these subdomains are reconstructed over a finer mesh on which to recompute flow more accurately the second approach uses an adaptive mesh in which the resolution increases where motion is found to occur optical flow is computed over a reasonably coarse mesh and this is used to construct an optimal adaptive mesh in a way that is different from the gradient methods reported in the literature the finite element mesh facilitates a reduction in computational effort by enabling processing to focus on particular objects of interest in a scene i e those areas where motion is detected the proposed methods were tested on real and synthetic image sequences and promising results are reported

a riemannian bfgs method without differentiated retraction for nonconvex optimization problems in this paper a riemannian bfgs method for minimizing a smooth function on a riemannian manifold is defined based on a riemannian generalization of a cautious update and a weak line search condition it is proven that the riemannian bfgs method converges i globally to stationary points without assuming the objective function to be convex and ii superlinearly to a nondegenerate minimizer using the weak line search condition removes the need for information from differentiated retraction the joint matrix diagonalization problem is chosen to demonstrate the performance of the algorithms with various parameters line search conditions and pairs of retraction and vector transport a preliminary version can be found in numerical mathematics and advanced applications enumath 2015 lect notes comput sci eng 112 springer new york 2016 pp 627 634

robust optical flow estimation based on a sparse motion trajectory set this paper presents an approach to the problem of estimating a dense optical flow field the approach is based on a multiframe irregularly spaced motion trajectory set where each trajectory describes the motion of a given point as a function of time from this motion trajectory set a dense flow field is estimated using a process of interpolation a set of localized motion models are estimated with each pixel labeled as belonging to one of the motion models a markov random field framework is adopted allowing the incorporation of contextual constraints to encourage region like structures the approach is compared with a number of conventional optical flow estimation algorithms taken over a number of real and synthetic sequences results indicate that the method produces more accurate results for sequences with known ground truth flow also applying the method to real sequences with unknown flow results in lower dfd for all of the sequences tested

a broyden class of quasi newton methods for riemannian optimization this paper develops and analyzes a generalization of the broyden class of quasi newton methods to the problem of minimizing a smooth objective function f on a riemannian manifold a condition on vector transport and retraction that guarantees convergence and facilitates efficient computation is derived experimental evidence is presented demonstrating the value of the extension to the riemannian broyden class through superior performance for some problems compared to existing riemannian bfgs methods in particular those that depend on differentiated retraction

un lemme matriciel effectif we show an almost optimal effective version of masser s matrix lemma lecture notes math 1290 109 148 1987 giving a lower bound of the height of an abelian variety in terms of its period lattices

cooperative robust estimation using layers of support we present an approach to the problem of representing images that contain multiple objects or surfaces rather than using an edge based approach to represent the segmentation of a scene we propose a multilayer estimation framework which uses support maps to represent the segmentation of the image into homogeneous chunks this support based approach can represent objects that are split into disjoint regions or have surfaces that are transparently interleaved our framework is based on an extension of robust estimation methods that provide a theoretical basis for support based estimation we use a selection criteria derived from the minimum description length principle to decide how many support maps to use in describing an image our method has been applied to a number of different domains including the decomposition of range images into constituent objects the segmentation of image sequences into homogeneous higher order motion fields and the separation of tracked motion features into distinct rigid body motions

scene segmentation from visual motion using global optimization this paper presents results from computer experiments with an algorithm to perform scene disposition and motion segmentation from visual motion or optic flow the maximum a posteriori map criterion is used to formulate what the best segmentation or interpretation of the scene should be where the scene is assumed to be made up of some fixed number of moving planar surface patches the bayesian approach requires first specification of prior expectations for the optic flow field which here is modeled as spatial and temporal markov random fields and secondly a way of measuring how well the segmentation predicts the measured flow field the markov random fields incorporate the physical constraints that objects and their images are probably spatially continuous and that their images are likely to move quite smoothly across the image plane to compute the flow predicted by the segmentation a recent method for reconstructing the motion and orientation of planar surface facets is used the search for the globally optimal segmentation is performed using simulated annealing

pade approximations and diophantine geometry abstract using methods of pade approximations we prove a converse to eisenstein s theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function for classes of functions parametrized by meromorphic functions this result is applied to the tate conjecture on the effective description of isogenies for elliptic curves

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

robust optic flow computation this paper formulates the optic flow problem as a set of over determined simultaneous linear equations it then introduces and studies two new robust optic flow methods the first technique is based on using the least median of squares lmeds to detect the outliers then the inlier group is solved using the least square technique the second method employs a new robust statistical method named the least median of squares orthogonal distances lmsod to identify the outliers and then uses total least squares to solve the optic flow problem the performance of both methods are studied by experiments on synthetic and real image sequences these methods outperform other published methods both in accuracy and robustness

motion estimation in the 3 d gabor domain motion estimation methods can be broadly classified as being spatiotemporal or frequency domain in nature the gabor representation is an analysis framework providing localized frequency information when applied to image sequences the 3 d gabor representation displays spatiotemporal spatiotemporal frequency st stf information enabling the application of robust frequency domain methods with adjustable spatiotemporal resolution in this work the 3 d gabor representation is applied to motion analysis we demonstrate that piecewise uniform translational motion can be estimated by using a uniform translation motion model in the st stf domain the resulting motion estimation method exhibits both good spatiotemporal resolution and substantial noise resistance compared to existing spatiotemporal methods to form the basis of this model we derive the signature of the translational motion in the 3 d gabor domain finally to obtain higher spatiotemporal resolution for more complex motions a dense motion field estimation method is developed to find a motion estimate for every pixel in the sequence

formal subgroups of abelian varieties in this paper we generalize the result of 12 in the following sense let a be an abelian variety over a number field k let be the neron model of a over the ring of integers o k of k completing along its zero section defines a formal group widehat mathcal a over o k we prove that any formal subgroup of the generic fiber of widehat mathcal a whose closure in widehat mathcal a is smooth over an open subset of spec o k arises in fact from an abelian subvariety of a the proof is of a transcendental nature and uses the arakelovian formalism introduced by bost 3

bayesian estimation of motion vector fields a stochastic approach to the estimation of 2d motion vector fields from time varying images is presented the formulation involves the specification of a deterministic structural model along with stochastic observation and motion field models two motion models are proposed a globally smooth model based on vector markov random fields and a piecewise smooth model derived from coupled vector binary markov random fields two estimation criteria are studied in the maximum a posteriori probability map estimation the a posteriori probability of motion given data is maximized whereas in the minimum expected cost mec estimation the expectation of a certain cost function is minimized both algorithms generate sample fields by means of stochastic relaxation implemented via the gibbs sampler two versions are developed one for a discrete state space and the other for a continuous state space the map estimation is incorporated into a hierarchical environment to deal efficiently with large displacements

a riemannian symmetric rank one trust region method the well known symmetric rank one trust region method where the hessian approximation is generated by the symmetric rank one update is generalized to the problem of minimizing a real valued function over a d d dimensional riemannian manifold the generalization relies on basic differential geometric concepts such as tangent spaces riemannian metrics and the riemannian gradient as well as on the more recent notions of first order retraction and vector transport the new method called rtr sr1 is shown to converge globally and d 1 d 1 step q superlinearly to stationary points of the objective function a limited memory version referred to as lrtr sr1 is also introduced in this context novel efficient strategies are presented to construct a vector transport on a submanifold of a euclidean space numerical experiments rayleigh quotient minimization on the sphere and a joint diagonalization problem on the stiefel manifold illustrate the value of the new methods

projection like retractions on matrix manifolds this paper deals with constructing retractions a key step when applying optimization algorithms on matrix manifolds for submanifolds of euclidean spaces we show that the operation consisting of taking a tangent step in the embedding euclidean space followed by a projection onto the submanifold is a retraction we also show that the operation remains a retraction if the projection is generalized to a projection like procedure that consists of coming back to the submanifold along admissible directions and we give a sufficient condition on the admissible directions for the generated retraction to be second order this theory offers a framework in which previously proposed retractions can be analyzed as well as a toolbox for constructing new ones illustrations are given for projection like procedures on some specific manifolds for which we have an explicit easy to compute expression

a theoretical and experimental study of the symmetric rank one update this paper first discusses computational experience using the sri update in conventional line search and trust region algorithms for unconstrained optimization the experiments show that the sri is very competitive with the widely used bfgs method they also indicate two interesting features the final hessian approximations produced by the sri method are not generally appreciably better than those produced by the bfgs and the sequences of steps produced by the sri do not usually seem to have the uniform linear independence property that is assumed in recent convergence analysis this paper presents a new analysis that shows that the sri method with a line search is n 1 step q superlinearly convergent without the assumption of linearly independent iterates this analysis assumes that the hessian approximations are positive definite and bounded asymptotically which from computational experience are reasonable assumptions

the application of mean field theory to image motion estimation previously markov random field mrf model based techniques have been proposed for image motion estimation since motion estimation is usually an ill posed problem various constraints are needed to obtain a unique and stable solution the main advantage of the mrf approach is its capacity to incorporate such constraints for instance motion continuity within an object and motion discontinuity at the boundaries between objects in the mrf approach motion estimation is often formulated as an optimization problem and two frequently used optimization methods are simulated annealing sa and iterative conditional mode icm although the sa is theoretically optimal in the sense of finding the global optimum it usually takes many iterations to converge the icm on the other hand converges quickly but its results are often unsatisfactory due to its hard decision nature previously the authors have applied the mean field theory to image segmentation and image restoration problems it provides results nearly as good as sa but with much faster convergence the present paper shows how the mean field theory can be applied to mrf model based motion estimation this approach is demonstrated on both synthetic and real world images where it produced good motion estimates

dense motion estimation using regularization constraints on local parametric models this paper presents a method for dense optical flow estimation in which the motion field within patches that result from an initial intensity segmentation is parametrized with models of different order we propose a novel formulation which introduces regularization constraints between the model parameters of neighboring patches in this way we provide the additional constraints for very small patches and for patches whose intensity variation cannot sufficiently constrain the estimation of their motion parameters in order to preserve motion discontinuities we use robust functions as a regularization mean we adopt a three frame approach and control the balance between the backward and forward constraints by a real valued direction field on which regularization constraints are applied an iterative deterministic relaxation method is employed in order to solve the corresponding optimization problem experimental results show that the proposed method deals successfully with motions large in magnitude motion discontinuities and produces accurate piecewise smooth motion fields

optimization methods on riemannian manifolds and their application to shape space we extend the scope of analysis for linesearch optimization algorithms on possibly infinite dimensional riemannian manifolds to the convergence analysis of the bfgs quasi newton scheme and the fletcher reeves conjugate gradient iteration numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods

analysis of a symmetric rank one trust region method several recent computational studies have shown that the symmetric rank one sr1 update is a very competitive quasi newton update in optimization algorithms this paper gives a new analysis of a trust region sr1 method for unconstrained optimization and shows that the method has an n 1 step q superlinear rate of convergence the analysis makes neither of the assumptions of uniform linear independence of the iterates nor positive definiteness of the hessian approximations that have been made in other recent analyses of sr1 methods the trust region method that is analyzed is fairly standard except that it includes the feature that the hessian approximation is updated after all steps including rejected steps we also present computational results that show that this feature safeguarded in a way that is consistent with the convergence analysis does not harm the efficiency of the sr1 trust region method

bias in robust estimation caused by discontinuities and multiple structures when fitting models to data containing multiple structures such as when fitting surface patches to data taken from a neighborhood that includes a range discontinuity robust estimators must tolerate both gross outliers and pseudo outliers pseudo outliers are outliers to the structure of interest but inliers to a different structure they differ from gross outliers because of their coherence such data occurs frequently in computer vision problems including motion estimation model fitting and range data analysis the focus in this paper is the problem of fitting surfaces near discontinuities in range data to characterize the performance of least median of the squares least trimmed squares m estimators hough transforms ransac and minpran on this type of data the pseudo outlier bias metric is developed using techniques from the robust statistics literature and it is used to study the error in robust fits caused by distributions modeling various types of discontinuities the results show each robust estimator to be biased at small but substantial discontinuities they also show the circumstances under which different estimators are most effective most importantly the results imply present estimators should be used with care and new estimators should be developed

regularization of optical flow with m band wavelet transform abstract the optical flow is an important tool for problems arising in the analysis of image sequences flow fields generated by various existing solving techniques are often noisy and partially incorrect especially near occlusions or motion boundaries therefore the additional information on the scene gained from a sequence of images is usually worse in this paper discrete wavelet transform has been adopted in order to enhance the reliability of optical flow estimation a generalization of the well known dyadic orthonormal wavelets to the case of the dilation scale factor m 2 with n vanishing moments has been used and it has proved to be a useful regularizing tool the advantages in the computations have been shown by experimental results performed on real image sequences

robust adaptive scale parametric model estimation for computer vision robust model fitting essentially requires the application of two estimators the first is an estimator for the values of the model parameters the second is an estimator for the scale of the noise in the inlier data indeed we propose two novel robust techniques the two step scale estimator tsse and the adaptive scale sample consensus assc estimator tsse applies nonparametric density estimation and density gradient estimation techniques to robustly estimate the scale of the inliers the assc estimator combines random sample consensus ransac and tsse using a modified objective function that depends upon both the number of inliers and the corresponding scale assc is very robust to discontinuous signals and data with multiple structures being able to tolerate more than 80 percent outliers the main advantage of assc over ransac is that prior knowledge about the scale of inliers is not needed assc can simultaneously estimate the parameters of a model and the scale of the inliers belonging to that model experiments on synthetic data show that assc has better robustness to heavily corrupted data than least median squares lmeds residual consensus resc and adaptive least kth order squares alks we also apply assc to two fundamental computer vision tasks range image segmentation and robust fundamental matrix estimation experiments show very promising results

hermitian vector bundles and extension groups on arithmetic schemes i geometry of numbers we define and investigate extension groups in the context of arakelov geometry the arithmetic extension groups ext xi f g we introduce are extensions by groups of analytic types of the usual extension groups extxi f g attached to ox modules f and g over an arithmetic scheme x in this paper we focus on the first arithmetic extension group ext x1 f g the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over x and we especially consider the case when x is an arithmetic curve namely the spectrum specok of the ring of integers in some number field k then the study of arithmetic extensions over x is related to old and new problems concerning lattices and the geometry of numbers namely for any two hermitian vector bundles f and g over x specok we attach a logarithmic sizesf g to any element of ext x1 f g and we give an upper bound on sf g in terms of slope invariants of f and g we further illustrate this notion by relating the sizes of restrictions to points in p1 z of the universal extension over pz1 to the geometry of psl2 z acting on poincare s upper half plane and by deducing some quantitative results in reduction theory from our previous upper bound on sizes finally we investigate the behaviour of size by base change i e under extension of the ground field k to a larger number field k when the base field k is q we establish that the size which cannot increase under base change is actually invariant when the field k is an abelian extension of k or when f g is a direct sum of root lattices and of lattices of voronoi s first kind the appendices contain results concerning extensions in categories of sheaves on ringed spaces and lattices of voronoi s first kind which might also be of independent interest

image flow segmentation and estimation by constraint line clustering image flow is the velocity field in the image plane caused by the motion of the observer objects in the scene or apparent motion and can contain discontinuities due to object occlusion in the scene an algorithm that can estimate the image flow velocity field when there are discontinuities due to occlusions is described the constraint line clustering algorithm uses a statistical test to estimate the image flow velocity field in the presence of step discontinuities in the image irradiance or velocity field particular emphasis is placed on motion estimation and segmentation in situations such as random dot patterns where motion is the only cue to segmentation experimental results on a demanding synthetic test case and a real image are presented a smoothing algorithm for improving the velocity field estimate is also described the smoothing algorithm constructs a smooth estimate of the velocity field by approximating a surface between step discontinuities it is noted that the velocity field estimate can be improved using surface reconstruction between velocity field boundaries

image registration using wavelet based motion model an image registration algorithm is developed to estimate dense motion vectors between two images using the coarse to fine wavelet based motion model this motion model is described by a linear combination of hierarchical basis functions proposed by cai and wang siam numer anal 33 3 937 970 1996 the coarser scale basis function has larger support while the finer scale basis function has smaller support with these variable supports in full resolution the basis functions serve as large to small windows so that the global and local information can be incorporated concurrently for image matching especially for recovering motion vectors containing large displacements to evaluate the accuracy of the wavelet based method two sets of test images were experimented using both the wavelet based method and a leading pyramid spline based method by szeliski et al international journal of computer vision 22 3 199 218 1996 one set of test images taken from barron et al international journal of computer vision 12 43 77 1994 contains small displacements the other set exhibits low texture or spatial aliasing after image blurring and contains large displacements the experimental results showed that our wavelet based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation

statistical computations on grassmann and stiefel manifolds for image and video based recognition in this paper we examine image and video based recognition applications where the underlying models have a special structure the linear subspace structure we discuss how commonly used parametric models for videos and image sets can be described using the unified framework of grassmann and stiefel manifolds we first show that the parameters of linear dynamic models are finite dimensional linear subspaces of appropriate dimensions unordered image sets as samples from a finite dimensional linear subspace naturally fall under this framework we show that an inference over subspaces can be naturally cast as an inference problem on the grassmann manifold to perform recognition using subspace based models we need tools from the riemannian geometry of the grassmann manifold this involves a study of the geometric properties of the space appropriate definitions of riemannian metrics and definition of geodesics further we derive statistical modeling of inter and intraclass variations that respect the geometry of the space we apply techniques such as intrinsic and extrinsic statistics to enable maximum likelihood classification we also provide algorithms for unsupervised clustering derived from the geometry of the manifold finally we demonstrate the improved performance of these methods in a wide variety of vision applications such as activity recognition video based face recognition object recognition from image sets and activity based video clustering

a feasible method for optimization with orthogonality constraints minimization with orthogonality constraints e g x top x i and or spherical constraints e g vert x vert 2 1 has wide applications in polynomial optimization combinatorial optimization eigenvalue problems sparse pca p harmonic flows 1 bit compressive sensing matrix rank minimization etc these problems are difficult because the constraints are not only non convex but numerically expensive to preserve during iterations to deal with these difficulties we apply the cayley transform a crank nicolson like update scheme to preserve the constraints and based on it develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics the efficiency of the proposed algorithms is demonstrated on a variety of test problems in particular for the maxcut problem it exactly solves a decomposition formulation for the sdp relaxation for polynomial optimization nearest correlation matrix estimation and extreme eigenvalue problems the proposed algorithms run very fast and return solutions no worse than those from their state of the art algorithms for the quadratic assignment problem a gap 0 842 to the best known solution on the largest problem tai256c in qaplib can be reached in 5 min on a typical laptop

consistency of robust estimators in multi structural visual data segmentation a theoretical framework is presented to study the consistency of robust estimators used in vision problems involving extraction of fine details a strong correlation between asymptotic performance of a robust estimator and the asymptotic bias of its scale estimate is mathematically demonstrated where the structures are assumed to be linear corrupted by gaussian noise a new measure for the inconsistency of scale estimators is defined and formulated by deriving the functional forms of four recent high breakdown robust estimators for each estimator the inconsistency measures are numerically evaluated for a range of mutual distances between structures and inlier ratios and the minimum mutual distance between the structures for which each estimator returns a non bridging fit is calculated

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

design and use of linear models for image motion analysis linear parameterized models of optical flow particularly affine models have become widespread in image motion analysis the linear model coefficients are straightforward to estimate and they provide reliable estimates of the optical flow of smooth surfaces here we explore the use of parameterized motion models that represent much more varied and complex motions our goals are threefold to construct linear bases for complex motion phenomenas to estimate the coefficients of these linear modelss and to recognize or classify image motions from the estimated coefficients we consider two broad classes of motions i generic motion features such as motion discontinuities and moving barss and ii non rigid object specific motions such as the motion of human mouths for motion features we construct a basis of steerable flow fields that approximate the motion features for object specific motions we construct basis flow fields from example motions using principal component analysis in both cases the model coefficients can be estimated directly from spatiotemporal image derivatives with a robust multi resolution scheme finally we show how these model coefficients can be use to detect and recognize specific motions such as occlusion boundaries and facial expressions

low rank optimization with trace norm penalty the paper addresses the problem of low rank trace norm minimization we propose an algorithm that alternates between fixed rank optimization and rank one updates the fixed rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable the search space is nonlinear but is equipped with a riemannian structure that leads to efficient computations we present a second order trust region algorithm with a guaranteed quadratic rate of convergence overall the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix to compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor corrector approach that outperforms the naive warm restart approach on the fixed rank quotient manifold the performance of the proposed algorithm is illustrated on p

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

descent algorithms on oblique manifold for source adaptive ica contrast a riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis ica that employs a mutual information based source adaptive contrast function despite the extensive development of manifold techniques catering to the orthonormality constraint only a limited number of works have been dedicated to oblique manifold ob algorithms to intrinsically handle the normality constraint which has been empirically shown to be superior to other riemannian and euclidean approaches imposing the normality constraint implicitly in line with the ica definition essentially guarantees a substantial improvement in the solution accuracy by way of increased degrees of freedom while searching for an optimal unmixing ica matrix in contrast with the orthonormality constraint designs of the steepest descent conjugate gradient with hager zhang or a hybrid update parameter quasi newton and cost effective quasi newton methods intended for ob are presented in this paper their performance is validated using natural images and systematically compared with the popular state of the art approaches in order to assess the performance effects of the choice of algorithm and the use of a riemannian rather than euclidean framework we surmount the computational challenge associated with the direct estimation of the source densities using the improved fast gauss transform in the evaluation of the contrast function and its gradient the proposed ob schemes may find applications in the offline image signal analysis wherein on one hand the computational overhead can be tolerated and on the other the solution quality holds paramount interest

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

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

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

robust optical flow computation based on least median of squares regression an optical flow estimation technique is presented which is based on the least median of squares lmeds robust regression algorithm enabling more accurate flow estimates to be computed in the vicinity of motion discontinuities the flow is computed in a blockwise fashion using an affine model through the use of overlapping blocks coupled with a block shifting strategy redundancy is introduced into the computation of the flow this eliminates blocking effects common in most other techniques based on blockwise processing and also allows flow to be accurately computed in regions containing three distinct motions a multiresolution version of the technique is also presented again based on lmeds regression which enables image sequences containing large motions to be effectively handled an extensive set of quantitative comparisons with a wide range of previously published methods are carried out using synthetic realistic computer generated images of natural scenes with known flow and natural images both angular and absolute flow errors are calculated for those sequences with known optical flow displaced frame difference error used extensively in video compression is used for those natural scenes with unknown flow in all of the sequences tested a comparison with those methods that result in a dense flow field greater than 80 spatial coverage show that the lmeds technique produces the least error irrespective of the error measure used

low rank optimization on the cone of positive semidefinite matrices we propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices this algorithm relies on the factorization x yy t where the number of columns of y fixes an upper bound on the rank of the positive semidefinite matrix x it is thus very effective for solving problems that have a low rank solution the factorization x yy t leads to a reformulation of the original problem as an optimization on a particular quotient manifold the present paper discusses the geometry of that manifold and derives a second order optimization method with guaranteed quadratic convergence it furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem in contrast to existing methods the proposed algorithm converges monotonically to the sought solution its numerical efficiency is evaluated on two applications the maximal cut of a graph and the problem of sparse principal component analysis

estimating piecewise smooth optical flow with global matching and graduated optimization this paper presents a new method for estimating piecewise smooth optical flow we propose a global optimization formulation with three frame matching and local variation and develop an efficient technique to minimize the resultant global energy this technique takes advantage of local gradient global gradient and global matching methods and alleviates their limitations experiments on various synthetic and real data show that this method achieves highly competitive accuracy

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

computing optical flow with physical models of brightness variation although most optical flow techniques presume brightness constancy it is well known that this constraint is often violated producing poor estimates of image motion this paper describes a generalized formulation of optical flow estimation based on models of brightness variations that are caused by time dependent physical processes these include changing surface orientation with respect to a directional illuminant motion of the illuminant and physical models of heat transport in infrared images with these models we simultaneously estimate the 2d image motion and the relevant physical parameters of the brightness change model the estimation problem is formulated using total least squares with confidence bounds on the parameters experiments in four domains with both synthetic and natural inputs show how this formulation produces superior estimates of the 2d image motion

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

optical flow estimation and the interaction between measurement errors at adjacent pixel positions in order to estimate both components of optical flow as well as their first spatio temporal derivatives it is postulated that the optical flow constraint equation ofce is valid in a spatio temporal neighborhood of pixels so far it has been tacitly assumed that the partial derivatives of the gray value distribution which are required for this approach at the pixel positions involved are independent from each other it is shown how dropping this assumption affects the estimation procedure based on well established approaches of estimation theory the insight gained thereby is used to develop an approach towards merging image regions based on the compatibility of optical flow estimates obtained within the regions considered for merger

error analysis of robust optical flow estimation by least median of squares methods for the varying illumination model the apparent pixel motion in an image sequence called optical flow is a useful primitive for automatic scene analysis and various other applications of computer vision in general however the optical flow estimation suffers from two significant problems the problem of illumination that varies with time and the problem of motion discontinuities induced by objects moving with respect to either other objects or with respect to the background various integrated approaches for solving these two problems simultaneously have been proposed of these those that are based on the lmeds least median of squares appear to be the most robust the goal of this paper is to carry out an error analysis of two different lmeds based approaches one based on the standard lmeds regression and the other using a modification thereof as proposed by us recently while it is to be expected that the estimation accuracy of any approach would decrease with increasing levels of noise for lmeds like methods it is not always clear as to how much of that decrease in performance can be attributed to the fact that only a small number of randomly selected samples is used for forming temporary solutions to answer this question our study here includes a baseline implementation in which all of the image data is used for forming motion estimates we then compare the estimation errors of the two lmeds based methods with the baseline implementation our error analysis demonstrates that for the case of gaussian noise our modified lmeds approach yields better estimates at moderate levels of noise but is outperformed by the standard lmeds method as the level of noise increases for the case of salt and pepper noise the modified lmeds method consistently performs better than the standard lmeds method

intrinsic representation of tangent vectors and vector transports on matrix manifolds the quasi newton methods on riemannian manifolds proposed thus far do not appear to lend themselves to satisfactory convergence analyses unless they resort to an isometric vector transport this prompts us to propose a computationally tractable isometric vector transport on the stiefel manifold of orthonormal p frames in mathbb r n rn specifically it requires o np 2 o np2 flops which is considerably less expensive than existing alternatives in the frequently encountered case where n gg p n p we then build on this result to also propose computationally tractable isometric vector transports on other manifolds namely the grassmann manifold the fixed rank manifold and the positive semidefinite fixed rank manifold in the process we also propose a convenient way to represent tangent vectors to these manifolds as elements of mathbb r d rd where d is the dimension of the manifold we call this an intrinsic representation as opposed to extrinsic representations as elements of mathbb r w rw where w is the dimension of the embedding space finally we demonstrate the performance of the proposed isometric vector transport in the context of a riemannian quasi newton method applied to minimizing the brockett cost function

multimodal estimation of discontinuous optical flow using markov random fields the estimation of dense velocity fields from image sequences is basically an ill posed problem primarily because the data only partially constrain the solution it is rendered especially difficult by the presence of motion boundaries and occlusion regions which are not taken into account by standard regularization approaches in this paper the authors present a multimodal approach to the problem of motion estimation in which the computation of visual motion is based on several complementary constraints it is shown that multiple constraints can provide more accurate flow estimation in a wide range of circumstances the theoretical framework relies on bayesian estimation associated with global statistical models namely markov random fields the constraints introduced here aim to address the following issues optical flow estimation while preserving motion boundaries processing of occlusion regions fusion between gradient and feature based motion constraint equations deterministic relaxation algorithms are used to merge information and to provide a solution to the maximum a posteriori estimation of the unknown dense motion field the algorithm is well suited to a multiresolution implementation which brings an appreciable speed up as well as a significant improvement of estimation when large displacements are present in the scene experiments on synthetic and real world image sequences are reported

fast robust correlation a new fast statistically robust exhaustive translational image matching technique is presented fast robust correlation existing methods are either slow or non robust or rely on optimization fast robust correlation works by expressing a robust matching surface as a series of correlations speed is obtained by computing correlations in the frequency domain computational cost is analyzed and the method is shown to be fast speed is comparable to conventional correlation and for large images thousands of times faster than direct robust matching three experiments demonstrate the advantage of the technique over standard correlation

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

robust regression methods for computer vision a review regression analysis fitting a model to noisy data is a basic technique in computer vision robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance we review several robust estimation techniques and describe in detail the least median of squares lmeds method the method yields the correct result even when half of the data is severely corrupted its efficiency in the presence of gaussian noise can be improved by complementing it with a weighted least squares based procedure the high time complexity of the lmeds algorithm can be reduced by a monte carlo type speed up technique we discuss the relationship of lmeds with the ransac paradigm and its limitations in the presence of noise corrupting all the data and we compare its performance with the class of robust m estimators references to published applications of robust techniques in computer vision are also given

computing least median of squares regression lines and guided topological sweep abstract given a set of data points pi xi yi for 1 i n the least median of squares regression line is a line y ax b for which the median of the squared residuals is a minimum over all choices of a and b an algorithm is described that computes such a line in o n 2 time and o n memory space thus improving previous upper bounds on the problem this algorithm is an application of a general method built on top of the topological sweep of line arrangements

multidimensional orientation estimation with applications to texture analysis and optical flow the problem of detection of orientation in finite dimensional euclidean spaces is solved in the least squares sense the theory is developed for the case when such orientation computations are necessary at all local neighborhoods of the n dimensional euclidean space detection of orientation is shown to correspond to fitting an axis or a plane to the fourier transform of an n dimensional structure the solution of this problem is related to the solution of a well known matrix eigenvalue problem the computations can be performed in the spatial domain without actually doing a fourier transformation along with the orientation estimate a certainty measure based on the error of the fit is proposed two applications in image analysis are considered texture segmentation and optical flow the theory is verified by experiments which confirm accurate orientation estimates and reliable certainty measures in the presence of noise the comparative results indicate that the theory produces algorithms computing robust texture features as well as optical flow