packing stability for symplectic 4 manifolds abstractthe packing stability in symplectic geometry was rst noticed bybiran bir97 the symplectic obstructions to embed several balls into amanifold disappear when their size is small enough this phenomenonis known to hold for all closed manifolds with rational symplectic class see bir99 for the 4 dimensional case and bh11 bh13 for higherdimensions as well as for all ellipsoids bh13 in this note we show that packing stability holds for all closed andseveral open symplectic 4 manifolds 1 introduction in bir97 bir99 biran discovered the packing stability phenomenon insome symplectic 4 manifolds the symplectic obstructions to pack nidenti cal balls disappear when nbecomes large enough he later generalized thisresult to every closed symplectic 4 manifold with rational symplectic class h 2 m q in this paper we generalize these results in several directions by applyingthe singular in ation technique developed in ops13b mo13 we generalizebiran s packing stability to all symplectic 4 manifolds and also to a class ofopensymplectic 4 manifolds includingellipsoids anddomainswe call pseudo balls for these manifolds we establish not just packing stability but strongpacking stability which we de ne now the de nition is in the spirit ofbiran s rst results on the subject bir97 throughout b will denote anopen ball of capacity see de nition 1 3 and we will write u

vector sparse representation of color image using quaternion matrix analysis traditional sparse image models treat color image pixel as a scalar which represents color channels separately or concatenate color channels as a monochrome image in this paper we propose a vector sparse representation model for color images using quaternion matrix analysis as a new tool for color image representation its potential applications in several image processing tasks are presented including color image reconstruction denoising inpainting and super resolution the proposed model represents the color image as a quaternion matrix where a quaternion based dictionary learning algorithm is presented using the k quaternion singular value decomposition qsvd generalized k means clustering for qsvd method it conducts the sparse basis selection in quaternion space which uniformly transforms the channel images to an orthogonal color space in this new color space it is significant that the inherent color structures can be completely preserved during vector reconstruction moreover the proposed sparse model is more efficient comparing with the current sparse models for image restoration tasks due to lower redundancy between the atoms of different color channels the experimental results demonstrate that the proposed sparse image model avoids the hue bias issue successfully and shows its potential as a general and powerful tool in color image analysis and processing domain

hofer s question on intermediate symplectic capacities roughly twenty five years ago hofer asked can the cylinder b 2 1 times mathbb r 2 n 1 be symplectically embedded into b 2 n 1 r times mathbb r 2 for some r 0 we show that this is the case if r geq sqrt 2 n 1 2 n 2 2 we deduce that there are no intermediate capacities between 1 capacities first constructed by gromov in 1985 and n capacities answering another question of hofer in 2008 guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property

quaternion collaborative and sparse representation with application to color face recognition collaborative representation based classification crc and sparse rc src have recently achieved great success in face recognition fr previous crc and src are originally designed in the real setting for grayscale image based fr they separately represent the color channels of a query color image and ignore the structural correlation information among the color channels to remedy this limitation in this paper we propose two novel rc methods for color fr namely quaternion crc qcrc and quaternion src qsrc using quaternion ell 1 minimization by modeling each color image as a quaternionic signal they naturally preserve the color structures of both query and gallery color images while uniformly coding the query channel images in a holistic manner despite the empirical success of crc and src on fr a few theoretical results are developed to guarantee their effectiveness another purpose of this paper is to establish the theoretical guarantee for qcrc and qsrc under mild conditions comparisons with competing methods on benchmark real world databases consistently show the superiority of the proposed methods for both color fr and reconstruction

singularities of j holomorphic curves let v j be an almost complex manifold recall that a map from a riemann surface to v is j holomorphic if its derivative is complex linear a j holomorphic curve or j curve for short is the image of such a map nowhere locally constant the study of singularities of j holomorphic curves began with the statement of m gromov g about the positivity of intersections in the case when v is of dimension 4 the homological intersection of two compact j curves is a sum of positive local contributions and a similar property holds for the self intersection of one curve adjunction formula this property is crucial for applications of the theory to symplectic topology in dimension 4 and contact topology in dimension 3 gromov mcduff eliashberg hofer the positivity of intersections was proved by d mcduff mcd1 see also chapter vi in al then in mcd2 she proved that in any dimension a germ of j holomorphic map c 0 v v0 is topologically equivalent to a standard holomorphic germ from c 0 to cn 0 this was then strengthened to a c 1equivalence by m micallef and b white mw as a special case of a much more general result on singularities of generalized minimal surfaces they proved also that it applies to reducible germs

a stability property of symplectic packing we prove that for any closed symplectic 4 manifold m with h 2 m q there exists a number n 0 such that for every n n 0 m admits full symplectic packing by n equal balls we also indicate how to compute this n 0 our approach is based on donaldson s symplectic submanifold theorem and on tools from the framework of taubes theory of gromov invariants

restricted p isometry properties of nonconvex matrix recovery recently a nonconvex relaxation of low rank matrix recovery lmr called the schatten p quasi norm minimization 0 p 1 was introduced instead of the previous nuclear norm minimization in order to approximate the problem of lmr closer in this paper we introduce a notion of the restricted p isometry constants 0 p 1 and derive a p rip condition for exact reconstruction of lmr via schatten p quasi norm minimization in particular we determine how many random gaussian measurements are needed for the p rip condition to hold with high probability which gives a theoretical result that it needs fewer measurements with small p for exact recovery via schatten p quasi norm minimization than when p 1

low rank matrix recovery via modified schatten p norm minimization with convergence guarantees in recent years low rank matrix recovery problems have attracted much attention in computer vision and machine learning the corresponding rank minimization problems are both combinational and np hard in general which are mainly solved by both nuclear norm and schatten p 0 norm based optimization algorithms however inspired by weighted nuclear norm and schatten p norm as the relaxations of rank function the main merits of this work firstly provide a modified schatten p norm in the affine matrix rank minimization problem denoted as the modified schatten p norm minimization ms p nm secondly its surrogate function is constructed and the equivalence relationship with the ms p nm is further achieved thirdly the iterative singular value thresholding algorithm isvta is devised to optimize it and its accelerated version i e aisvta is also obtained to reduce the number of iterations through the well known nesterov s acceleration strategy most importantly the convergence guarantees and their relationship with objective function stationary point and variable sequence generated by the proposed algorithms are established under some specific assumptions e g kurdyka ojasiewicz k property finally numerical experiments demonstrate the effectiveness of the proposed algorithms in the matrix completion problem for image inpainting and recommender systems it should be noted that the accelerated algorithm has a much faster convergence speed and a very close recovery precision when comparing with the proposed non accelerated one

bilinear factor matrix norm minimization for robust pca algorithms and applications the heavy tailed distributions of corrupted outliers and singular values of all channels in low level vision have proven effective priors for many applications such as background modeling photometric stereo and image alignment and they can be well modeled by a hyper laplacian however the use of such distributions generally leads to challenging non convex non smooth and non lipschitz problems and makes existing algorithms very slow for large scale applications together with the analytic solutions to ell p norm minimization with two specific values of p i e p 1 2 and p 2 3 we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis we first define the double nuclear norm and frobenius nuclear hybrid norm penalties and then prove that they are in essence the schatten 1 2 and 2 3 quasi norms respectively which lead to much more tractable and scalable lipschitz optimization problems our experimental analysis shows that both our methods yield more accurate solutions than original schatten quasi norm minimization even when the number of observations is very limited finally we apply our penalties to various low level vision problems e g text removal moving object detection image alignment and inpainting and show that our methods usually outperform the state of the art methods

almost contact 5 manifolds are contact the existence of a contact structure is proved in any homotopy class of almost contact structures on a closed 5 dimensional manifold

factorization with uncertainty factorization using singular value decomposition svd is often used for recovering 3d shape and motion from feature correspondences across multiple views svd is powerful at finding the global solution to the associated least square error minimization problem however this is the correct error to minimize only when the x and y positional errors in the features are uncorrelated and identically distributed but this is rarely the case in real data uncertainty in feature position depends on the underlying spatial intensity structure in the image which has strong directionality to it hence the proper measure to minimize is covariance weighted squared error or the mahalanobis distance in this paper we describe a new approach to covariance weighted factorization which can factor noisy feature correspondences with high degree of directional uncertainty into structure and motion our approach is based on transforming the raw data into a covariance weighted data space where the components of noise in the different directions are uncorrelated and identically distributed applying svd to the transformed data now minimizes a meaningful objective function in this new data space this is followed by a linear but suboptimal second step to recover the shape and motion in the original data space we empirically show that our algorithm gives very good results for varying degrees of directional uncertainty in particular we show that unlike other svd based factorization algorithms our method does not degrade with increase in directionality of uncertainty even in the extreme when only normal flow data is available it thus provides a unified approach for treating corner like points together with points along linear structures in the image

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

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

depth estimation of face images using the nonlinear least squares model in this paper we propose an efficient algorithm to reconstruct the 3d structure of a human face from one or more of its 2d images with different poses in our algorithm the nonlinear least squares model is first employed to estimate the depth values of facial feature points and the pose of the 2d face image concerned by means of the similarity transform furthermore different optimization schemes are presented with regard to the accuracy levels and the training time required our algorithm also embeds the symmetrical property of the human face into the optimization procedure in order to alleviate the sensitivities arising from changes in pose in addition the regularization term based on linear correlation is added in the objective function to improve the estimation accuracy of the 3d structure further a model integration method is proposed to improve the depth estimation accuracy when multiple nonfrontal view face images are available experimental results on the 2d and 3d databases demonstrate the feasibility and efficiency of the proposed methods

symplectic manifolds with contact type boundaries an example of a 4 dimensional symplectic manifold with disconnected boundary of contact type is constructed a collection of other results about symplectic manifolds with contact type boundaries are derived using the theory ofj holomorphic spheres in particular the following theorem of eliashberg floer mcduff is proved if a neighbourhood of the boundary of v is symplectomorphic to a neighbourhood ofs2n 1 in standard euclidean space and if vanishes on all 2 spheres inv thenv is diffeomorphic to the ballb2n

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

weighted homogeneous singularities and rational homology disk smoothings we classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings the nonexistence of rational homology disk smoothings is shown by symplectic geometric methods while the existence is verified via smoothings of negative weight

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

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

quaternion based bilinear factor matrix norm minimization for color image inpainting as a new color image representation tool quaternion has achieved excellent results in the color image processing because it treats the color image as a whole rather than as a separate color space component thus it can make full use of the high correlation among rgb channels recently low rank quaternion matrix completion lrqmc methods have proven very useful for color image inpainting in this article we propose three novel lrqmc methods based on three quaternion based bilinear factor qbf matrix norm minimization models specifically we define quaternion double frobenius norm q dfn quaternion double nuclear norm q dnn and quaternion frobenius nuclear norm q fnn and then show their relationship with quaternion based matrix schatten p q schatten p norm for certain p values the proposed methods can avoid computing quaternion singular value decompositions qsvd for large quaternion matrices and thus can effectively reduce the calculation time compared with existing lrqmc methods furthermore we also analyze the convergence of the algorithms and introduce a rank estimation method for the quaternion matrix the experimental results demonstrate the superior performance of the proposed methods over some state of the art low rank quaternion matrix completion methods

computing smooth time trajectories for camera and deformable shape in structure from motion with occlusion we address the classical computer vision problems of rigid and nonrigid structure from motion sfm with occlusion we assume that the columns of the input observation matrix w describe smooth 2d point trajectories over time we then derive a family of efficient methods that estimate the column space of w using compact parameterizations in the discrete cosine transform dct domain our methods tolerate high percentages of missing data and incorporate new models for the smooth time trajectories of 2d points affine and weak perspective cameras and 3d deformable shape we solve a rigid sfm problem by estimating the smooth time trajectory of a single camera moving around the structure of interest by considering a weak perspective camera model from the outset we directly compute euclidean 3d shape reconstructions without requiring postprocessing steps such as euclidean upgrade and bundle adjustment our results on real sfm data sets with high percentages of missing data compared positively to those in the literature in nonrigid sfm we propose a novel 3d shape trajectory approach that solves for the deformable structure as the smooth time trajectory of a single point in a linear shape space a key result shows that compared to state of the art algorithms our nonrigid sfm method can better model complex articulated deformation with higher frequency dct components while still maintaining the low rank factorization constraint finally we also offer an approach for nonrigid sfm when w is presented with missing data

centered complexity one hamiltonian torus actions we consider symplectic manifolds with hamiltonian torus actions which are almost but not quite completely integrable the dimension of the torus is one less than half the dimension of the manifold we provide a complete set of invariants for such spaces when they are centered and the moment map is proper in particular this classifies the preimages under the moment map of all sufficiently small open sets which is an important step towards global classification as an application we construct a full packing of each of the grassmannians gr 2 r5 and gr 2 r6 by two equal symplectic balls

sparse representation for color image restoration sparse representations of signals have drawn considerable interest in recent years the assumption that natural signals such as images admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data in particular the design of well adapted dictionaries for images has been a major challenge the k svd has been recently proposed for this task and shown to perform very well for various grayscale image processing tasks in this paper we address the problem of learning dictionaries for color images and extend the k svd based grayscale image denoising algorithm that appears in this work puts forward ways for handling nonhomogeneous noise and missing information paving the way to state of the art results in applications such as color image denoising demosaicing and inpainting as demonstrated in this paper

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

fast and accurate matrix completion via truncated nuclear norm regularization recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications such as image inpainting and recommender systems many existing approaches formulate this problem as a general low rank matrix approximation problem since the rank operator is nonconvex and discontinuous most of the recent theoretical studies use the nuclear norm as a convex relaxation one major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized and thus the rank may not be well approximated in practice in this paper we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm which is given by the nuclear norm subtracted by the sum of the largest few singular values in addition we develop a novel matrix completion algorithm by minimizing the truncated nuclear norm we further develop three efficient iterative procedures tnnr admm tnnr apgl and tnnr admmap to solve the optimization problem tnnr admm utilizes the alternating direction method of multipliers admm while tnnr agpl applies the accelerated proximal gradient line search method apgl for the final optimization for tnnr admmap we make use of an adaptive penalty according to a novel update rule for admm to achieve a faster convergence rate our empirical study shows encouraging results of the proposed algorithms in comparison to the state of the art matrix completion algorithms on both synthetic and real visual datasets

the theory of quaternion matrix derivatives a systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced the proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables this rectifies a mathematical shortcut in the existing methods which incorrectly use the traditional product rule we also show that within the proposed framework the derivatives of quaternion matrix functions can be calculated directly without using quaternion differentials or resorting to the isomorphism with real vectors illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications

low rank matrix fitting based on subspace perturbation analysis with applications to structure from motion the task of finding a low rank r matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering the problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements the former problem can be solved by concatenating a set of r column matrices that share a common single r dimensional solution space unfortunately the number of possible submatrices is generally very large and hence the results obtained with one set of r column matrices will generally be different from that captured by a different set ideally we would like to find that solution that is least affected by noise this requires that we determine which of the r column matrices i e which of the original feature points are less influenced by the unknown noise term this paper presents a criterion to successfully carry out such a selection our key result is to formally prove that the more distinct the r vectors of the r column matrices are the less they are swayed by noise this key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r column matrices it is shown how this criterion can be effectively used to recover the noise free matrix of rank r finally we derive the affine and projective structure from motion sfm algorithms using the proposed criterion extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art

convergence analysis of alternating direction method of multipliers for a family of nonconvex problems the alternating direction method of multipliers admm is widely used to solve large scale linearly constrained optimization problems convex or nonconvex in many engineering fields however there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex in this paper we analyze the convergence of the admm for solving certain nonconvex consensus and sharing problems we show that the classical admm converges to the set of stationary solutions provided that the penalty parameter in the augmented lagrangian is chosen to be sufficiently large for the sharing problems we show that the admm is convergent regardless of the number of variable blocks our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many admm variants involving proximal update rules and various flexible block selection rules

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

irreversibility from staircases in symplectic embeddings abstract we present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of hamiltonian systems of many degrees of freedom we use recent flexibility and rigidity results of symplectic embeddings quantified via the stabilized fibonacci and pell staircases to encode the underlying breadth of the possible initial conditions which alongside the multitude of degrees of freedom of the underlying system give rise to time irreversibility

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

weighted nuclear norm minimization and its applications to low level vision as a convex relaxation of the rank minimization model the nuclear norm minimization nnm problem has been attracting significant research interest in recent years the standard nnm regularizes each singular value equally composing an easily calculated convex norm however this restricts its capability and flexibility in dealing with many practical problems where the singular values have clear physical meanings and should be treated differently in this paper we study the weighted nuclear norm minimization wnnm problem which adaptively assigns weights on different singular values as the key step of solving general wnnm models the theoretical properties of the weighted nuclear norm proximal wnnp operator are investigated albeit nonconvex we prove that wnnp is equivalent to a standard quadratic programming problem with linear constrains which facilitates solving the original problem with off the shelf convex optimization solvers in particular when the weights are sorted in a non descending order its optimal solution can be easily obtained in closed form with wnnp the solving strategies for multiple extensions of wnnm including robust pca and matrix completion can be readily constructed under the alternating direction method of multipliers paradigm furthermore inspired by the reweighted sparse coding scheme we present an automatic weight setting method which greatly facilitates the practical implementation of wnnm the proposed wnnm methods achieve state of the art performance in typical low level vision tasks including image denoising background subtraction and image inpainting

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

topology of symplectomorphism groups of s 2 s 2 in dimension 4 due to non existence of adequate tools very little is known about the topology of groups of diffeomorphisms for example it is unknown if the group of compactly supported diffeomorphisms of r is connected the situation is much better if one wants to study groups of symplectomorphisms this is due to the existence of powerful tools going by the name of pseudo holomorphic curve techniques and introduced in symplectic geometry by m gromov in his seminal paper of 1985 5 gromov proved in that paper among several other remarkable results the contractibility of the group of compactly supported symplectomorphisms of r with its standard symplectic form dx dy dx dy gromov also studied the following example letm be the symplectic manifold s s 1 0 0 where 0 r and 0 is a standard area form on s with total area equal to 1 denote by g the group of symplectomorphisms of m that act as the identity on h2 s s z gromov proved in 5 that g0 is homotopy equivalent to its subgroup of standard isometries so 3 so 3 he also showed why that would not be true for g with 0 and in 1987 d mcduff 10 constructed explicitly an element of infinite order in h1 g 0 in this paper we will give a more detailed description of g for 0 1 in particular we will prove the following on leave from instituto superior tecnico lisbon portugal partially supported by jnict programas ciencia and praxis fundacao luso americana para o desenvolvimento and a sloan doctoral dissertation fellowship current address school of mathematics institute for advanced study princeton nj 08540 usa e mail abreu math ias edu

a quaternion framework for color image smoothing and segmentation in this paper we present feature detail preserving models for color image smoothing and segmentation using the hamiltonian quaternion framework first we introduce a novel quaternionic gabor filter qgf which can combine the color channels and the orientations in the image plane we show that these filters are optimally localized both in the spatial and frequency domains and provide a good approximation to quaternionic quadrature filters using the qgfs we extract the local orientation information in the color images second in order to model this derived orientation information we propose continuous mixtures of appropriate exponential basis functions and derive analytic expressions for these models these analytic expressions take the form of spatially varying kernels which when convolved with a color image or the signed distance function of an evolving contour placed in the color image yield a detail preserving smoothing and segmentation respectively several examples on widely used image databases are shown to depict the performance of our algorithms

weighted schatten p norm minimization for image denoising and background subtraction low rank matrix approximation lrma which aims to recover the underlying low rank matrix from its degraded observation has a wide range of applications in computer vision the latest lrma methods resort to using the nuclear norm minimization nnm as a convex relaxation of the nonconvex rank minimization however nnm tends to over shrink the rank components and treats the different rank components equally limiting its flexibility in practical applications we propose a more flexible model namely the weighted schatten p norm minimization wsnm to generalize the nnm to the schatten p norm minimization with weights assigned to different singular values the proposed wsnm not only gives better approximation to the original low rank assumption but also considers the importance of different rank components we analyze the solution of wsnm and prove that under certain weights permutation wsnm can be equivalently transformed into independent non convex l p norm subproblems whose global optimum can be efficiently solved by generalized iterated shrinkage algorithm we apply wsnm to typical low level vision problems e g image denoising and background subtraction extensive experimental results show both qualitatively and quantitatively that the proposed wsnm can more effectively remove noise and model the complex and dynamic scenes compared with state of the art methods

filling by holomorphic curves in symplectic 4 manifolds we develop a general framework for embedded immersed jholomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4 dimensional symplectic manifolds in particular a deformation theory and an intersection theory for j holomorphic curves with boundary are developed bishop s local filling theorem is extended to almost complex manifolds existence and uniqueness of global fillings are given complete proofs then they are extended to the situation with nontrivial jholomorphic spheres culminating in the construction of singular fillings

isotopy of symplectic balls gromov s radius and the structure of ruled symplectic 4 manifolds there are various criteria enriques castelnuovo mori in algebraic geometry of complex surfaces that characterize rational or ruled surfaces up to birationality mori s criterion in particular states that a minimal compact algebraic surface is biholomorphic to cp 2 or to a ruled rational or irrational surface exactly when it contains a curve of strictly positive chern index this criterion has been partly generalized to symplectic 4 manifold by mcduff who showed the following any symplectic 4 manifold which contains an embedded symplectic sphere or immersed with positive normal double points of chern index larger than i is symplectomorphic to a manifold obtained by blowing up a finite number of symplectic balls either on p2 with its standard kahler structure or on a symplectic s2 fibration over a closed oriented real surface we need the answer to the following two problems in order to complete the classification 1 the classification of irrational symplectic s2 fibrations is not complete the statement of theorem 1 3 of 1 9 is not correct in the irrational case in particular it does not fully take into account the kahler structures coming from the projectivization of non split holomorphic rank 2 bundles the sharpest results on this problem are those of the recent article 1 11 when the area of the base of a symplectic s2 fibration v co s o over a surface of genus g is sufficiently large with respect to the area of the fiber the cohomology class of a symplectic form determines its isotopy class and therefore determines the symplectic type of the manifold see theorem 1 2 below actually under this cohomological hypothesis the symplectic ruled manifold v o may be obtained by cutting and pasting from a rational ruled

a characterization of the tight 3 sphere ii recall that every closed oriented connected 3 manifold m admits a contact form we shall give conditions on a periodic orbit p0 of the reeb vector field defined by that allows the construction of an open book decomposition of m p0 into embedded planes asymptotic to p0 such that p0 is the binding orbit of the decomposition it follows that m is the tight 3 sphere as a by product a new dynamical criterion for the tightness of a contact structure is derived in the proof the results extend earlier results 1999 john wiley sons inc