optimization algorithms on subspaces revisiting missing data problem in low rank matrix low rank matrix approximation has applications in many fields such as 3d reconstruction from an image sequence and 2d filter design in this paper one issue with low rank matrix approximation is re investigated the missing data problem much effort was devoted to this problem and the wiberg algorithm or the damped newton algorithm were recommended in previous studies however the wiberg or damped newton algorithms do not suit for large especially long matrices because one needs to solve a large linear system in every iteration in this paper we revitalize the usage of the levenberg marquardt algorithm for solving the missing data problem by utilizing the property that low rank approximation is a minimization problem on subspaces in two proposed implementations of the levenberg marquardt algorithm one only needs to solve a much smaller linear system in every iteration especially for long matrices simulations and experiments on real data show the superiority of the proposed algorithms though the proposed algorithms achieve a high success rate in estimating the optimal solution by random initialization as illustrated by real examples it still remains an open issue how to properly do the initialization in a severe situation that is a large amount of data is missing and with high level noise

tracking while zooming using affine transfer and multifocal tensors this paper presents algorithms for tracking unknown objects in the presence of zoom since prior models are unavailable point and line matches in affine views are used to characterize the structure and to transfer a fixation point into new images in a sequence because any affine projection matrix is permitted the intrinsic camera parameters such as focal length may change freely also since the techniques do not require long feature tracks a further desirable property is insensitivity to partial occlusion caused for instance by part of the object falling off the image plane while zooming in if only point matches are available a previous method based on factorization is applied when also incorporating lines the affine trifocal and quadrifocal tensors are used for tracking in monocular and stereo systems respectively methods for computing the tensors minimizing algebraic error are developed in comparison with their projective counterparts the affine tensors offer significant advantages in terms of computation time and convenience of parameterization and the relations between the different tensors are shown to be much simpler successful tracking is demonstrated on several real image sequences

a new characterization of bergman schatten spaces and a duality result abstract let b 0 d l 2 denote the space of all upper triangular matrices a such that lim r 1 1 r 2 a c r b l 2 0 we also denote by b 0 c d l 2 the closed banach subspace of b 0 d l 2 consisting of all upper triangular matrices whose diagonals are compact operators in this paper we give a duality result between b 0 c d l 2 and the bergman schatten spaces l a 1 d l 2 we also give a characterization of the more general bergman schatten spaces l a p d l 2 1 p in terms of taylor coefficients which is similar to that of m mateljevic and m pavlovic m mateljevic m pavlovic l p behaviour of the integral means of analytic functions studia math 77 1984 219 237 for classical bergman spaces

an analysis of linear subspace approaches for computer vision and pattern recognition linear subspace analysis lsa has become rather ubiquitous in a wide range of problems arising in pattern recognition and computer vision the essence of these approaches is that certain structures are intrinsically or approximately low dimensional for example the factorization approach to the problem of structure from motion sfm and principal component analysis pca based approach to face recognition in lsa the singular value decomposition svd is usually the basic mathematical tool however analysis of the performance in the presence of noise has been lacking we present such an analysis here first the denoising capacity of the svd is analysed specifically given a rank r matrix corrupted by noise how much noise remains in the rank r projected version of that corrupted matrix second we study the learning capacity of the lsa based recognition system in a noise corrupted environment specifically lsa systems that attempt to capture a data class as belonging to a rank r column space will be affected by noise in both the training samples measurement noise will mean the learning samples will not produce the true subspace and the test sample which will also have measurement noise on top of the ideal clean sample belonging to the true subspace these results should help one to predict aspects of performance and to design more optimal systems in computer vision particularly in tasks such as sfm and face recognition our analysis agrees with certain observed phenomenon and these observations together with our simulations verify the correctness of our theory

rank conditions on the multiple view matrix geometric relationships governing multiple images of points and lines and associated algorithms have been studied to a large extent separately in multiple view geometry the previous studies led to a characterization based on multilinear constraints which have been extensively used for structure and motion recovery feature matching and image transfer in this paper we present a universal rank condition on the so called multiple view matrix m for arbitrarily combined point and line features across multiple views the condition gives rise to a complete set of constraints among multiple images all previously known multilinear constraints become simple instantiations of the new condition in particular the relationship between bilinear trilinear and quadrilinear constraints can be clearly revealed from this new approach the theory enables us to carry out global geometric analysis for multiple images as well as systematically characterize all degenerate configurations without breaking image sequence into pairwise or triple wise sets of views this global treatment allows us to utilize all incidence conditions governing all features in all images simultaneously for a consistent recovery of motion and structure from multiple views in particular a rank based multiple view factorization algorithm for motion and structure recovery is derived from the rank condition simulation results are presented to validate the multiple view matrix based approach

operators on weighted bergman spaces 0 p leq 1 and applications we describe the boundedness of a linear operator from bp f d c analytic d 1 z 1 z f z da z 1 p for 0 p 1 under some conditions on the weight function into a general banach space x by means of the growth conditions at the boundary of certain fractional derivatives of a single x valued analytic function this in particular allows us to characterize the dual of bp for 0 p 1 and to give a formulation of generalized carleson measures in terms of the inclusion b1 l d we then apply the result to the study of multipliers hankel operators and composition operators acting on bp spaces 1991 math subject class primary 47b38 47b35 secondary 42a45

relative affine structure canonical model for 3d from 2d geometry and applications we propose an affine framework for perspective views captured by a single extremely simple equation based on a viewer centered invariant we call relative affine structure via a number of corollaries of our main results we show that our framework unifies previous work including euclidean projective and affine in a natural and simple way and introduces new extremely simple algorithms for the tasks of reconstruction from multiple views recognition by alignment and certain image coding applications

motion and structure from line correspondences closed form solution uniqueness and optimization this work discusses estimating motion and structure parameters from line correspondences of a rigid scene the authors present a closed form solution to motion and structure parameters from line correspondences through three monocular perspective views the algorithm makes use of redundancy in the data to improve the accuracy of the solutions the uniqueness of the solution is established and necessary and sufficient conditions for degenerate spatial line configurations are given optimization has been employed to further improve the accuracy of the estimates in the presence of noise simulations have shown that the errors of the optimized estimates are close to the theoretical lower error bound

rigid structure from motion from a blind source separation perspective we present an information theoretic approach to define the problem of structure from motion sfm as a blind source separation one given that for almost all practical joint densities of shape points the marginal densities are non gaussian we show how higher order statistics can be used to provide improvements in shape estimates over the methods of factorization via singular value decomposition svd bundle adjustment and bayesian approaches previous techniques have either explicitly or implicitly used only second order statistics in models of shape or noise a further advantage of viewing sfm as a blind source problem is that it easily allows for the inclusion of noise and shape models resulting in maximum likelihood ml or maximum a posteriori map shape and motion estimates a key result is that the blind source separation approach has the ability to recover the motion and shape matrices without the need to explicitly know the motion or shape pdf we demonstrate that it suffices to know whether the pdf is sub or super gaussian i e semi parametric estimation and derive a simple formulation to determine this from the data we provide extensive experimental results on synthetic and real tracked points in order to quantify the improvement obtained from this technique

nearly weakly orthonormal sequences singular value estimates and calderon zygmund operators abstract the direct way to estimate the singular values of a compact operator is to decompose it as a sum of orthogonal rank one pieces however such decompositions can generally not be found in practice we give a variation of the decomposition using nearly weakly orthonormal nwo sequences the nwo decomposition is easier to do in examples but is still strong enough to give good singular value estimates we illustrate this by giving sharp trace ideal estimates for the double layer potential and for the first and higher commutators of multiplication operators and calderon zygmund operators in particular nwo sequences seem well suited for dealing with operators which depend nonlinearly on their symbol we also show that the class of operators which admit nwo expansions is well behaved under composition under conjugation by weights and under conjugation by changes of variable

lipschitz type conditions on homogeneous banach spaces of analytic functions abstract in this paper we deal with banach spaces of analytic functions x defined on the unit disk satisfying that r t f x for any t 0 and f x where r t f z f e i t z we study the space of functions in x such that p r d f x o 1 r 1 r r 1 where d f z n 0 n 1 a n z n and is a continuous and non decreasing weight satisfying certain mild assumptions the space under consideration is shown to coincide with the subspace of functions in x satisfying any of the following conditions a r t f f x o t b p r f f x o 1 r c n f x o 2 n or d f s n f x o n 1 where p r f z f r z s n f z k 0 n a k z k and n f s 2 n f s 2 n 1 f our results extend those known for hardy or bergman spaces and power weights t t

replica symmetry breaking condition exposed by random matrix calculation of landscape complexity we start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random n 1 dimensional gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface this is used to show that for a generic smooth concave confining potentials the condition of the zero temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in n for such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically different behaviour reported in our earlier short communication fyodorov et al in jetp lett 85 261 2007 was due to non analyticity of the hard wall confinement potential finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points in particular we show that in the vicinity of critical confinement the saddle points with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the hessian

affine structure from line correspondences with uncalibrated affine cameras this paper presents a linear algorithm for recovering 3d affine shape and motion from line correspondences with uncalibrated affine cameras the algorithm requires a minimum of seven line correspondences over three views the key idea is the introduction of a one dimensional projective camera this converts 3d affine reconstruction of line directions into 2d projective reconstruction of points in addition a line based factorization method is also proposed to handle redundant views experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm

spaces of analytic functions on the disc where the growth of mp f r depends on a weight abstract we consider spaces of analytic functions depending on a weight p t 0 t 0 1 defined by certain conditions namely 1 1 m p f r o p 1 r 1 r 2 2 m p ft r o p 1 r 1 r 2 3 3 1 0 p 1 r 1 r m p f r dr we study boundary value problems and duality for these spaces depending on the properties of the weight function

recovering the missing components in a large noisy low rank matrix application to sfm in computer vision it is common to require operations on matrices with missing data for example because of occlusion or tracking failures in the structure from motion sfm problem such a problem can be tackled allowing the recovery of the missing values if the matrix should be of low rank when noise free the filling in of missing values is known as imputation imputation can also be applied in the various subspace techniques for face and shape classification online recommender systems and a wide variety of other applications however iterative imputation can lead to the recovery of data that is seriously in error in this paper we provide a method to recover the most reliable imputation in terms of deciding when the inclusion of extra rows or columns containing significant numbers of missing entries is likely to lead to poor recovery of the missing parts although the proposed approach can be equally applied to a wide range of imputation methods this paper addresses only the sfm problem the performance of the proposed method is compared with jacobs and shum s methods for sfm

3d motion recovery via affine epipolar geometry algorithms to perform point based motion estimation under orthographic and scaled orthographic projection abound in the literature a key limitation of many existing algorithms is that they operate on the minimum amount of data required often requiring the selection of a suitable minimal set from the available data to serve as a local coordinate frame such approaches are extremely sensitive to errors and noise in the minimal set and forfeit the advantages of using the full data set furthermore attention is seldom paid to the statistical performance of the algorithms

minimal projective reconstruction including missing data the minimal data necessary for projective reconstruction from image points is well known when each object point is visible in all images we formulate and propose solutions to a family of reconstruction problems for multiple images from minimal data where there are missing points in some of the images the ability to handle the minimal cases with missing data is of great theoretical and practical importance it is unavoidable to use them to bootstrap robust estimation such as ransac and lms algorithms and optimal estimation such as bundle adjustment first we develop a framework to parameterize the multiple view geometry needed to handle the missing data cases then we present a solution to the minimal case of eight points in three images where one different point is missing in each of the three images we prove that there are in general as many as 11 solutions for this minimal case furthermore all minimal cases with missing data for three and four images are catalogued finally we demonstrate the method on both simulated and real images and show that the algorithms presented in the paper can be used for practical problems

projective reconstruction of all visual primitives we present a method to simultaneously reconstruct cameras points lines degenerate quadrics and full rank quadrics from a set of n uncalibrated views of a static scene our method is based on the use of bundle adjustment to refine an initial reconstruction during the initialization stage a novel algorithm was proposed for the conic outline correction in n views constrains derived from this algorithm were also integrated in the cost function of bundle adjustment to help preserve the right topology of the reconstructed quadrics

affine structure and motion from points lines and conics in this paper several new methods for estimating scene structure and camera motion from an image sequence taken by affine cameras are presented all methods can incorporate both point line and conic features in a unified manner the correspondence between features in different images is assumed to be known three new tensor representations are introduced describing the viewing geometry for two and three cameras the centred affine epipoles can be used to constrain the location of corresponding points and conics in two images the third order or alternatively the reduced third order centred affine tensors can be used to constrain the locations of corresponding points lines and conics in three images the reduced third order tensors contain only 12 components compared to the 16 components obtained when reducing the trifocal tensor to affine cameras a new factorization method is presented the novelty lies in the ability to handle not only point features but also line and conic features concurrently another complementary method based on the so called closure constraints is also presented the advantage of this method is the ability to handle missing data in a simple and uniform manner finally experiments performed on both simulated and real data are given including a comparison with other methods

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

density of critical points for a gaussian random function critical points of a scalar quantitiy are either extremal points or saddle points the character of the critical points is determined by the sign distribution of the eigenvalues of the hessian matrix for a two dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words the density of the saddle points and extrema agree this argument breaks down in three dimensions all ratios of the densities of saddle points and extrema larger than one are possible for a homogeneous gaussian random field one finds no longer an equidistribution of signs saddle points are slightly more frequent

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

affine approximation for direct batch recovery of euclidian structure and motion from sparse data we present a batch method for recovering euclidian camera motion from sparse image data the main purpose of the algorithm is to recover the motion parameters using as much of the available information and as few computational steps as possible the algorithm thus places itself in the gap between factorisation schemes which make use of all available information in the initial recovery step and sequential approaches which are able to handle sparseness in the image data euclidian camera matrices are approximated via the affine camera model thus making the recovery direct in the sense that no intermediate projective reconstruction is made using a little known closure constraint the fa closure we are able to formulate the camera coefficients linearly in the entries of the affine fundamental matrices the novelty of the presented work is twofold firstly the presented formulation allows for a particularly good conditioning of the estimation of the initial motion parameters but also for an unprecedented diversity in the choice of possible regularisation terms secondly the new autocalibration scheme presented here is in practice guaranteed to yield a least squares estimate of the calibration parameters as a bi product the affine camera model is rehabilitated as a useful model for most cameras and scene configurations e g wide angle lenses observing a scene at close range experiments on real and synthetic data demonstrate the ability to reconstruct scenes which are very problematic for previous structure from motion techniques due to local ambiguities and error accumulation

carleson measures and multipliers of dirichlet type spaces a function p from 0 1 onto itself is a dirichlet weight if it is increasing p 0 and limr o x p x 0 the corresponding dirichlettype space dp consists of those bounded holomorphic functions on u z e c izl 1 such that if z 12p 1 izi is integrable with respect to lebesgue measure on u we characterize in terms of a carleson type maximal operator the functions in the set of pointwise multipliers of dp m dp g u c gf e dp vf e dp