l_ 1 norm low rank matrix decomposition by neural networks and mollifiers the l 1 norm cost function of the low rank approximation of the matrix with missing entries is not smooth and also cannot be transformed into a standard linear or quadratic programming problem and thus the optimization of this cost function is still not well solved to tackle this problem first a mollifier is used to smooth the cost function high closeness of the smoothed function to the original one can be obtained by tuning the parameters contained in the mollifier next a recurrent neural network is proposed to optimize the mollified function which will converge to a local minimum in addition to boost the speed of the system the mollifying process is implemented by a filtering procedure the influence of two mollifier parameters is theoretically analyzed and experimentally confirmed showing that one of the parameters is critical to computational efficiency and accuracy while the other not a large number of experiments on synthetic data show that the proposed method is competitive to the state of the art methods in particular the experiments on large matrices and a real application in the structure from motion indicate that the memory requirement of the proposed algorithm is mild making it suitable for real applications that often involve large scale matrix decomposition

toward dynamic image mosaic generation with robustness to parallax mosaicing is largely dependent on the quality of registration among the constituent input images parallax and object motion present challenges to image registration leading to artifacts in the result to reduce the impact of these artifacts traditional image mosaicing approaches often impose planar scene constraints or rely on purely rotational camera motion or dense sampling however these requirements are often impractical or fail to address the needs of all applications instead taking advantage of depth cues and a smooth transition criterion we achieve significantly improved mosaicing results for static scenes coping effectively with nontrivial parallax in the input we extend this approach to the synthesis of dynamic video mosaics incorporating foreground background segmentation and a consistent motion perception criterion although further additions are required to cope with unconstrained object motion our algorithm can synthesize a perceptually convincing output conveying the same appearance of motion as seen in the input sequences

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

3d trajectory reconstruction under perspective projection we present an algorithm to reconstruct the 3d trajectory of a moving point from its correspondence in a collection of temporally non coincidental 2d perspective images given the time of capture that produced each image and the relative camera poses at each time instant triangulation based solutions do not apply as multiple views of the point may not exist at each time instant we represent a 3d trajectory using a linear combination of compact trajectory basis vectors such as the discrete cosine transform basis that have been shown to approximate object independence we note that such basis vectors are also coordinate independent which allows us to directly use camera poses estimated from stationary areas in the scene in contrast to nonrigid structure from motion techniques where cameras are simultaneously estimated this reduces the reconstruction optimization to a linear least squares problem allowing us to robustly handle missing data that often occur due to motion blur texture deformation and self occlusion we present an algorithm to determine the number of trajectory basis vectors individually for each trajectory via a cross validation scheme and refine the solution by minimizing the geometric error the relationship between point and camera motion can cause degeneracies to occur we geometrically analyze the problem by studying the relationship of the camera motion point motion and trajectory basis vectors we define the reconstructability of a 3d trajectory under projection and show that the estimate approaches the ground truth when reconstructability approaches infinity this analysis enables us to precisely characterize cases when accurate reconstruction is achievable we present qualitative results for the reconstruction of several real world scenes from a series of 2d projections where high reconstructability can be guaranteed and report quantitative results on motion capture sequences

linear local models for monocular reconstruction of deformable surfaces recovering the 3d shape of a nonrigid surface from a single viewpoint is known to be both ambiguous and challenging resolving the ambiguities typically requires prior knowledge about the most likely deformations that the surface may undergo it often takes the form of a global deformation model that can be learned from training data while effective this approach suffers from the fact that a new model must be learned for each new surface which means acquiring new training data and may be impractical in this paper we replace the global models by linear local models for surface patches which can be assembled to represent arbitrary surface shapes as long as they are made of the same material not only do they eliminate the need to retrain the model for different surface shapes they also let us formulate 3d shape reconstruction from correspondences as either an algebraic problem that can be solved in closed form or a convex optimization problem whose solution can be found using standard numerical packages we present quantitative results on synthetic data as well as qualitative results on real images

spatio temporal union of subspaces for multi body non rigid structure from motion abstract non rigid structure from motion nrsfm has so far been mostly studied for recovering 3d structure of a single non rigid deforming object to handle the real world challenging multiple deforming objects scenarios existing methods either pre segment different objects in the scene or treat multiple non rigid objects as a whole to obtain the 3d non rigid reconstruction however these methods fail to exploit the inherent structure in the problem as the solution of segmentation and the solution of reconstruction could not benefit each other in this paper we propose a unified framework to jointly segment and reconstruct multiple non rigid objects to compactly represent complex multi body non rigid scenes we propose to exploit the structure of the scenes along both temporal direction and spatial direction thus achieving a spatio temporal representation specifically we represent the 3d non rigid deformations as lying in a union of subspaces along the temporal direction and represent the 3d trajectories as lying in the union of subspaces along the spatial direction this spatio temporal representation not only provides competitive 3d reconstruction but also outputs robust segmentation of multiple non rigid objects the resultant optimization problem is solved efficiently using the alternating direction method of multipliers admm extensive experimental results on both synthetic and real multi body nrsfm datasets demonstrate the superior performance of our proposed framework compared with the state of the art methods

fixed point theory for almost convex functions traditionally metric fixed point theory has sought classes of spaces in which a given type of mapping nonexpansive assymptotically or generalized nonexpansive uniformly lipschitz etc from a nonempty weakly compact convex set into itself always has a fixed point in some situations the class of space is determined by the application while there is some degree of freedom in constructing the map to be used with this in mind we seek to relax the conditions on the space by considering more restrictive types of mappings

subspace procrustes analysis procrustes analysis pa has been a popular technique to align and build 2 d statistical models of shapes given a set of 2 d shapes pa is applied to remove rigid transformations later a non rigid 2 d model is computed by modeling the residual e g pca although pa has been widely used it has several limitations for modeling 2 d shapes occluded landmarks and missing data can result in local minima solutions and there is no guarantee that the 2 d shapes provide a uniform sampling of the 3 d space of rotations for the object to address previous issues this paper proposes subspace pa spa given several instances of a 3 d object spa computes the mean and a 2 d subspace that can model rigid and non rigid deformations of the 3 d object we propose a discrete dspa and continuous cspa formulation for spa assuming that 3 d samples of an object are provided dspa extends the traditional pa and produces unbiased 2 d models by uniformly sampling different views of the 3 d object cspa provides a continuous approach to uniformly sample the space of 3 d rotations being more efficient in space and time we illustrate the benefits of spa in two different applications first spa is used to learn 2 d face and body models from 3 d datasets experiments on the facewarehouse and cmu motion capture mocap datasets show the benefits of our 2 d models against the state of the art pa approaches and conventional 3 d models second spa learns an unbiased 2 d model from cmu mocap dataset and it is used to estimate the human pose on the leeds sports dataset

computable elastic distances between shapes we dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other the energy is formally dened from a left invariant riemannian distance on an innite dimensional group acting on the curves which can be explicitly computed the obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed an analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem which can eciently be implemented as illustrated by experiments 1 1 generalities the problem of matching two objects together is very impor tant in computer vision and shape recognition in many cases recognition is based on shapes outlines with the help of some suitably designed distance a general principle is to associate with any pair o1 o 2 of objects to be compared a measure of discrepancy d o1 o 2 the recognition of an observed object o may be done by nding from a dictionary of templates the previously recorded object otemp for which d o otemp is minimal clearly the denition of the distance is the crucial step of the method and much research has been done in this direction we shall not try here to provide a review of the huge literature existing on the subject see for example 17 but rather focus on methods related to deformable templates with which we are directly concerned instead of basing recognition on a nite collection of points of interest primitives taken from the outline of an object corners inflexion points etc which is a popular way of handling the problem our purpose is to base the comparison on the whole outline considered as a plane curve the distance we shall dene incorporates some deformation energy between the curves the approach as we will see turns out to be intrinsic and robust to usual euclidean transformations the method is related to the wide literature on snakes 14 7 21 etc in the way that our distance corresponds to some continuous process of deformation of one curve into another it is also related to papers on elastic matching such as 8 however we provide an elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes taking into account possible invariance to scaling or euclidean transformations in the case they are required from this point of view our results are indebted to the seminal work of grenander on group theory applied to pattern recognition cf 10 and 11 in particular see also 1 2 12 another source of inspiration may come from mathematical physics since we are going to look to the path process of lowest energy which deforms one object into

3d reconstruction of human motion from monocular image sequences this article tackles the problem of estimating non rigid human 3d shape and motion from image sequences taken by uncalibrated cameras similar to other state of the art solutions we factorize 2d observations in camera parameters base poses and mixing coefficients existing methods require sufficient camera motion during the sequence to achieve a correct 3d reconstruction to obtain convincing 3d reconstructions from arbitrary camera motion our method is based on a priorly trained base poses we show that strong periodic assumptions on the coefficients can be used to define an efficient and accurate algorithm for estimating periodic motion such as walking patterns for the extension to non periodic motion we propose a novel regularization term based on temporal bone length constancy in contrast to other works the proposed method does not use a predefined skeleton or anthropometric constraints and can handle arbitrary camera motion we achieve convincing 3d reconstructions even under the influence of noise and occlusions multiple experiments based on a 3d error metric demonstrate the stability of the proposed method compared to other state of the art methods our algorithm shows a significant improvement

cat k spaces weak convergence and fixed points abstract in this paper we show that some of the recent results on fixed point for cat 0 spaces still hold true for cat 1 spaces and so for any cat k space under natural boundedness conditions we also introduce a new notion of convergence in geodesic spaces which is related to the convergence and applied to study some aspects on the geometry of cat 0 spaces at this point two recently posed questions in w a kirk b panyanak a concept of convergence in geodesic spaces nonlinear anal 68 12 2008 3689 3696 are answered in the negative the work finishes with the study of the lifsic characteristic and property p of lim xu to derive fixed point results for uniformly lipschitzian mappings in cat k spaces a conjecture raised in s dhompongsa w a kirk b sims fixed points of uniformly lipschitzian mappings nonlinear anal 65 2006 762 772 on the lifsic characteristic function of cat k spaces is solved in the positive

isometric non rigid shape from motion with riemannian geometry solved in linear time we study isometric non rigid shape from motion iso nrsfm given multiple intrinsically calibrated monocular images we want to reconstruct the time varying 3d shape of a thin shell object undergoing isometric deformations we show that iso nrsfm is solvable from local warps the inter image geometric transformations we propose a new theoretical framework based on the riemmanian manifold to represent the unknown 3d surfaces as embeddings of the camera s retinal plane this allows us to use the manifold s metric tensor and christoffel symbol cs fields these are expressed in terms of the first and second order derivatives of the inverse depth of the 3d surfaces which are the unknowns for iso nrsfm we prove that the metric tensor and the cs are related across images by simple rules depending only on the warps this forms a set of important theoretical results we show that current solvers cannot solve for the first and second order derivatives of the inverse depth simultaneously we thus propose an iterative solution in two steps 1 we solve for the first order derivatives assuming that the second order derivatives are known we initialise the second order derivatives to zero which is an infinitesimal planarity assumption we derive a system of two cubics in two variables for each image pair the sum of squares of these polynomials is independent of the number of images and can be solved globally forming a well posed problem for n geq 3 images 2 we solve for the second order derivatives by initialising the first order derivatives from the previous step we solve a linear system of 4n 4 equations in three variables we iterate until the first order derivatives converge the solution for the first order derivatives gives the surfaces normal fields which we integrate to recover the 3d surfaces the proposed method outperforms existing work in terms of accuracy and computation cost on synthetic and real datasets

a stable analytical framework for isometric shape from template by surface integration shape from template sft reconstructs the shape of a deforming surface from a single image a 3d template and a deformation prior for isometric deformations this is a well posed problem however previous methods which require no initialization break down when the perspective effects are small which happens when the object is small or viewed from larger distances that is they do not handle all projection geometries we propose stable sft methods that accurately reconstruct the 3d shape for all projection geometries we follow the existing approach of using first order differential constraints and obtain local analytical solutions for depth and the first order quantities the depth gradient or the surface normal previous methods use the depth solution directly to obtain the 3d shape we prove that the depth solution is unstable when the projection geometry tends to affine while the solution for the first order quantities remain stable for all projection geometries we therefore propose to solve sft by first estimating the first order quantities either depth gradient or surface normal and integrating them to obtain shape we validate our approach with extensive synthetic and real world experiments and obtain significantly more accurate results compared to previous initialization free methods our approach does not require any optimization which makes it very fast

quasi perspective projection model theory and application to structure and motion factorization from uncalibrated image sequences this paper addresses the problem of factorization based 3d reconstruction from uncalibrated image sequences previous studies on structure and motion factorization are either based on simplified affine assumption or general perspective projection the affine approximation is widely adopted due to its simplicity whereas the extension to perspective model suffers from recovering projective depths to fill the gap between simplicity of affine and accuracy of perspective model we propose a quasi perspective projection model for structure and motion recovery of rigid and nonrigid objects based on factorization framework the novelty and contribution of this paper are as follows firstly under the assumption that the camera is far away from the object with small lateral rotations we prove that the imaging process can be modeled by quasi perspective projection which is more accurate than affine model from both geometrical error analysis and experimental studies secondly we apply the model to establish a framework of rigid and nonrigid factorization under quasi perspective assumption finally we propose an extended cholesky decomposition to recover the rotation part of the euclidean upgrading matrix we also prove that the last column of the upgrading matrix corresponds to a global scale and translation of the camera thus may be set freely the proposed method is validated and evaluated extensively on synthetic and real image sequences and improved results over existing schemes are observed

geodesic estimation for large deformation anatomical shape averaging and interpolation the goal of this research is to promote variational methods for anatomical averaging that operate within the space of the underlying image registration problem this approach is effective when using the large deformation viscous framework where linear averaging is not valid or in the elastic case the theory behind this novel atlas building algorithm is similar to the traditional pairwise registration problem but with single image forces replaced by average forces these group forces drive an average transport ordinary differential equation allowing one to estimate the geodesic that moves an image toward the mean shape configuration this model gives large deformation atlases that are optimal with respect to the shape manifold as defined by the data and the image registration assumptions we use the techniques in the large deformation context here but they also pertain to small deformation atlas construction furthermore a natural inherently inverse consistent image registration is gained for free as is a tool for constant arc length geodesic shape interpolation the geodesic atlas creation algorithm is quantitatively compared to the euclidean anatomical average to elucidate the need for optimized atlases the procedures generate improved average representations of highly variable anatomy from distinct populations

minimal basis subspace representation a unified framework for rigid and non rigid motion segmentation motion segmentation and non rigid structure from motion are two challenging computer vision problems that have attracted numerous research interests while the previous works handle these two problems separately we present a general motion segmentation framework in this paper for solving these two seemingly different problems in a unified manner at the heart of our general motion segmentation framework is a model selection mechanism based on finding the minimal basis subspace representation by seeking the joint sparse representation of the data matrix however such formulation is np hard and we solve the convex proxy instead unlike other compressive sensing related works this convex proxy solution is insufficient for our problem the convex relaxation artefacts and noise yield multiple subspace representations making identification of the exact number of motion subspaces challenging we solve for the right number of subspaces by transforming this problem into a facility location problem with global cost and solve the factor graph formulation using max product belief propagation message passing

a generic framework for tracking using particle filter with dynamic shape prior tracking deforming objects involves estimating the global motion of the object and its local deformations as functions of time tracking algorithms using kalman filters or particle filters pfs have been proposed for tracking such objects but these have limitations due to the lack of dynamic shape information in this paper we propose a novel method based on employing a locally linear embedding in order to incorporate dynamic shape information into the particle filtering framework for tracking highly deformable objects in the presence of noise and clutter the pf also models image statistics such as mean and variance of the given data which can be useful in obtaining proper separation of object and background

procrustean normal distribution for non rigid structure from motion a well defined deformation model can be vital for non rigid structure from motion nrsfm most existing methods restrict the deformation space by assuming a fixed rank or smooth deformation which are not exactly true in the real world and they require the degree of deformation to be predetermined which is impractical meanwhile the errors in rotation estimation can have severe effects on the performance i e these errors can make a rigid motion be misinterpreted as a deformation in this paper we propose an alternative to resolve these issues motivated by an observation that non rigid deformations excluding rigid changes can be concisely represented in a linear subspace without imposing any strong constraints such as smoothness or low rank this observation is embedded in our new prior distribution the procrustean normal distribution pnd which is a shape distribution exclusively for non rigid deformations because of this unique characteristic of the pnd rigid and non rigid changes can be strictly separated which leads to better performance the proposed algorithm em pnd fits a pnd to given 2d observations to solve nrsfm without any user determined parameters the experimental results show that em pnd gives the state of the art performance for the benchmark data sets confirming the adequacy of the new deformation model

complex non rigid 3d shape recovery using a procrustean normal distribution mixture model recovering the 3d shape of a non rigid object is a challenging problem existing methods make the low rank assumption and do not scale well with the increased degree of freedom found in complex non rigid deformations or shape variations moreover in general the degree of freedom of deformation is assumed to be known in advance which limits the applicability of non rigid structure from motion algorithms in a practical situation in this paper we propose a method for handling complex shape variations based on the assumption that complex shape variations can be represented probabilistically by a mixture of primitive shape variations the proposed model is a generative probabilistic model called a procrustean normal distribution mixture model which can model complex shape variations without rank constraints experimental results show that the proposed method significantly outperforms existing methods

nonrigid structure from motion estimating shape and motion with hierarchical priors this paper describes methods for recovering time varying shape and motion of nonrigid 3d objects from uncalibrated 2d point tracks for example given a video recording of a talking person we would like to estimate the 3d shape of the face at each instant and learn a model of facial deformation time varying shape is modeled as a rigid transformation combined with a nonrigid deformation reconstruction is ill posed if arbitrary deformations are allowed and thus additional assumptions about deformations are required we first suggest restricting shapes to lie within a low dimensional subspace and describe estimation algorithms however this restriction alone is insufficient to constrain reconstruction to address these problems we propose a reconstruction method using a probabilistic principal components analysis ppca shape model and an estimation algorithm that simultaneously estimates 3d shape and motion for each instant learns the ppca model parameters and robustly fills in missing data points we then extend the model to represent temporal dynamics in object shape allowing the algorithm to robustly handle severe cases of missing data

a dissipative singular sturm liouville problem with a spectral parameter in the boundary condition abstract in the hilbert space l w 2 a b we consider nonselfadjoint singular sturm liouville boundary value problem with two singular end points a and b in limit circle cases at a and b and with a spectral parameter in the boundary condition the approach is based on the use of the maximal dissipative operator and the spectral analysis of this operator is adequate for boundary value problem we construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations which make it possible to determine the scattering matrix of the dilation we also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding sturm liouville equation on the basis of the results obtained regarding the theory of the characteristic function we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and sturm liouville boundary value problem

the fixed point property and unbounded sets in cat 0 spaces abstract in this work we study the fixed point property for nonexpansive self mappings defined on convex and closed subsets of a cat 0 space we will show that a positive answer to this problem is very much linked with the euclidean geometry of the space while the answer is more likely to be negative if the space is more hyperbolic as a consequence we extend a very well known result of w o ray on hilbert spaces

a concept of convergence in geodesic spaces abstract a cat 0 space is a geodesic space for which each geodesic triangle is at least as thin as its comparison triangle in the euclidean plane a notion of convergence introduced independently several years ago by lim and kuczumow is shown in cat 0 spaces to be very similar to the usual weak convergence in banach spaces in particular many banach space results involving weak convergence have precise analogues in this setting at the same time many questions remain open

a simple prior free method for non rigid structure from motion factorization this paper proposes a simple prior free method for solving the non rigid structure from motion nrsfm factorization problem other than using the fundamental low order linear combination model assumption our method does not assume any extra prior knowledge either about the non rigid structure or about the camera motions yet it works effectively and reliably producing optimal results and not suffering from the inherent basis ambiguity issue which plagued most conventional nrsfm factorization methods our method is very simple to implement which involves solving a very small sdp semi definite programming of fixed size and a nuclear norm minimization problem we also present theoretical analysis on the uniqueness and the relaxation gap of our solutions extensive experiments on both synthetic and real motion capture data assuming following the low order linear combination model are conducted which demonstrate that our method indeed outperforms most of the existing non rigid factorization methods this work offers not only new theoretical insight but also a practical everyday solution to nrsfm

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

the behavior of fixed point free nonexpansive mappings in geodesic spaces abstract dropping the existence of fixed points of a nonexpansive mapping is an interesting and unusual task in metric fixed point theory hyperbolic geometry proved to be very relevant in the study of the behavior of fixed point free nonexpansive mappings in this work we generalize some of the results in that direction in geodesic spaces more precisely we show under which additional assumptions the picard iterative sequence of a mapping defined on a hyperbolic geodesic space tends to a point of the boundary

fixed points of generalized contractive multi valued mappings in a recent paper n mizoguchi and w takahashi gave a positive answer to the conjecture of s reich concerning the existence of fixed points of multi valued mappings that satisfy a certain contractive condition in this paper we provide an alternative and somewhat more straightforward proof for the theorem of mizoguchi and takahashi also the problems associated with fixed points of weakly contractive multi valued mappings are studied finally we make a few comments that improve other results from their paper j math anal appl 141 1989 177 188

the geometry of shape spaces kendall introduced shape spaces which classified n tuples of points in r k and explained their role for statistics the aim of this paper is to study such shape spaces and in particular to explore the corresponding spaces for points on spheres the ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant the geometry of the shape space is critical in determining the type of statistical technique which one should use therefore the major part of this paper is devoted to an examination of the geometry and the statistics is only introduced in a simple case when considering random processes on the shape space

adaptive non rigid registration and structure from motion from image trajectories this paper addresses the problem of registering a known 3d model to a set of 2d deforming image trajectories the proposed approach can adapt to a scenario where the 3d model to register is not an exact description of the measured image data this results in finding a 2d 3d registration given the complexity of having both 2d deforming data and a coarse description of the image observations the method acts in two distinct phases first an affine step computes a factorization for both the 2d image data and the 3d model using a joint subspace decomposition this initial solution is then upgraded by finding the best projection to the image plane complying with the metric constraints given by a scaled orthographic camera both steps are computed efficiently in closed form with the additional feature of being robust to degenerate motions which may possibly affect the 2d image data i e lack of relevant rigid motion a further extension of the approach allows to compute the full 3d deformations of the shape given the first initial rigid registration this step results in solving a non rigid structure from motion nrsfm problem using the 3d known shape as a prior experimental results show the robustness of the method in registration tasks such as pose estimation and 3d reconstruction when degenerate image motion is present

a fast approach to deformable surface 3d tracking deformable surface 3d tracking is a severely under constrained problem and great efforts have been made to solve it a recent state of the art approach solves this problem by formulating it as a second order cone programming socp problem however one drawback of this approach is that it is time consuming in this paper we propose an effective method for 3d deformable surface tracking first we formulate the deformable surface tracking problem as a linear programming lp problem then we solve the lp problem with an algorithm which converges superlinearly rather than bisection algorithm whose convergence speed is linear our experimental studies on synthetic and real data have demonstrated the proposed method can not only reliably recover 3d structures of surfaces but also run faster than the state of the art method

combining local physical and global statistical models for sequential deformable shape from motion in this paper we simultaneously estimate camera pose and non rigid 3d shape from a monocular video using a sequential solution that combines local and global representations we model the object as an ensemble of particles each ruled by the linear equation of the newton s second law of motion this dynamic model is incorporated into a bundle adjustment framework in combination with simple regularization components that ensure temporal and spatial consistency the resulting approach allows to sequentially estimate shape and camera poses while progressively learning a global low rank model of the shape that is fed back into the optimization scheme introducing thus global constraints the overall combination of local physical and global statistical constraints yields a solution that is both efficient and robust to several artifacts such as noisy and missing data or sudden camera motions without requiring any training data at all validation is done in a variety of real application domains including articulated and non rigid motion both for continuous and discontinuous shapes our on line methodology yields significantly more accurate reconstructions than competing sequential approaches being even comparable to the more computationally demanding batch methods

spectral analysis for discontinuous non self adjoint singular dirac operators with eigenparameter dependent boundary condition abstract in this paper a discontinuous non self adjoint dissipative dirac operator with eigenparameter dependent boundary condition and with two singular endpoints is studied the interface conditions are imposed on the discontinuous point firstly we pass the considered problem to a maximal dissipative operator l h by using operator theoretic formulation the self adjoint dilation t h of l h in the space h is constructed furthermore the incoming and outgoing representations of t h and functional model are also constructed hence in light of lax phillips theory we derive the scattering matrix using the equivalence between scattering matrix and characteristic function completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved

convex optimization for nonrigid stereo reconstruction we present a method for recovering 3 d nonrigid structure from an image pair taken with a stereo rig more specifically we dedicate to recover shapes of nearly inextensible deformable surfaces in our approach we represent the surface as a 3 d triangulated mesh and formulate the reconstruction problem as an optimization problem consisting of data terms and shape terms the data terms are model to image keypoint correspondences which can be formulated as second order cone programming socp constraints using l norm the shape terms are designed to retaining original lengths of mesh edges which are typically nonconvex constraints we will show that this optimization problem can be turned into a sequence of socp feasibility problems in which the nonconvex constraints are approximated as a set of convex constraints thanks to the efficient socp solver the reconstruction problem can then be solved reliably and efficiently as opposed to previous methods ours neither involves smoothness constraints nor need an initial estimation which enables us to recover shapes of surfaces with smooth sharp and other complex deformations from a single image pair the robustness and accuracy of our approach are evaluated quantitatively on synthetic data and qualitatively on real data

kronecker markov prior for dynamic 3d reconstruction recovering dynamic 3d structures from 2d image observations is highly under constrained because of projection and missing data motivating the use of strong priors to constrain shape deformation in this paper we empirically show that the spatiotemporal covariance of natural deformations is dominated by a kronecker pattern we demonstrate that this pattern arises as the limit of a spatiotemporal autoregressive process and derive a kronecker markov random field as a prior distribution over dynamic structures this distribution unifies shape and trajectory models of prior art and has the individual models as its marginals the key assumption of the kronecker mrf is that the spatiotemporal covariance is separable into the product of a temporal and a shape covariance and can therefore be modeled using the matrix normal distribution analysis on motion capture data validates that this distribution is an accurate approximation with significantly fewer free parameters using the trace norm we present a convex method to estimate missing data from a single sequence when the marginal shape distribution is unknown the kronecker markov distribution fit to a single sequence outperforms state of the art methods at inferring missing 3d data and additionally provides covariance estimates of the uncertainty

geodesic rays the lion man game and the fixed point property abstract this paper focuses on the relation among the existence of different types of curves such as directional ones quasi geodesic or geodesic rays the approximate fixed point property for nonexpansive mappings and a discrete lion and man game our main result holds in the setting of cat 0 spaces that are additionally gromov hyperbolic

monocular template based reconstruction of inextensible surfaces we present a monocular 3d reconstruction algorithm for inextensible deformable surfaces it uses point correspondences between a single image of the deformed surface taken by a camera with known intrinsic parameters and a template the main assumption we make is that the surface shape as seen in the template is known since the surface is inextensible its deformations are isometric to the template we exploit the distance preservation constraints to recover the 3d surface shape as seen in the image though the distance preservation constraints have already been investigated in the literature we propose a new way to handle them spatial smoothness priors are easily incorporated as well as temporal smoothness priors in the case of reconstruction from a video the reconstruction can be used for 3d augmented reality purposes thanks to a fast implementation we report results on synthetic and real data some of them are compared to stereo based 3d reconstructions to demonstrate the efficiency of our method

convolutional sparse coding for trajectory reconstruction trajectory basis non rigid structure from motion nrsfm refers to the process of reconstructing the 3d trajectory of each point of a non rigid object from just their 2d projected trajectories reconstruction relies on two factors i the condition of the composed camera trajectory basis matrix and ii whether the trajectory basis has enough degrees of freedom to model the 3d point trajectory these two factors are inherently conflicting employing a trajectory basis with small capacity has the positive characteristic of reducing the likelihood of an ill conditioned system when composed with the camera during reconstruction however this has the negative characteristic of increasing the likelihood that the basis will not be able to fully model the object s true 3d point trajectories in this paper we draw upon a well known result centering around the reduced isometry property rip condition for sparse signal reconstruction rip allow us to relax the requirement that the full trajectory basis composed with the camera matrix must be well conditioned further we propose a strategy for learning an over complete basis using convolutional sparse coding from naturally occurring point trajectory corpora to increase the likelihood that the rip condition holds for a broad class of point trajectories and camera motions finally we propose an ell 1 inspired objective for trajectory reconstruction that is able to adaptively select the smallest sub matrix from an over complete trajectory basis that balances i and ii we present more practical 3d reconstruction results compared to current state of the art in trajectory basis nrsfm

averaged alternating reflections in geodesic spaces abstract we study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context we show that weak convergence results from hilbert spaces find natural counterparts in spaces of constant curvature moreover in this particular setting one obtains strong convergence

moment approach to the boundary exact controllability of an active constrained layer beam abstract using the moment approach we consider the boundary exact controllability of an active constrained layer acl beam consisting of three layers which is modeled as a rayleigh beam coupled with two wave equations we convert the controllability problem of the acl beam into a corresponding moment problem which can be solvable in a hilbert space l 2 then we conclude that the acl beam is exactly controllable when the control time is greater than the maximum value among of the optical lengths of the two waves and the square root of the moment of inertia of the rayleigh beam the well posedness and asymptotic spectral expressions of the acl beam are also presented

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

boundary eigenvalue problems for differential equations nη λpη with λ polynomial boundary conditions abstract the present paper deals with the spectral properties of boundary eigenvalue problems for differential equations of the form n p on a compact interval with boundary conditions which depend on the spectral parameter polynomially here n as well as p are regular differential operators of order n and p respectively with n p 0 the main results concern the completeness minimality and riesz basis properties of the corresponding eigenfunctions and associated functions they are obtained after a suitable linearization of the problem and by means of a detailed asymptotic analysis of the green s function the function spaces where the above properties hold are described by independent boundary conditions an application to a problem from elasticity theory is given

kernel density estimation and intrinsic alignment for shape priors in level set segmentation in this paper we make two contributions to the field of level set based image segmentation firstly we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups the invariance is obtained by an intrinsic registration of the evolving level set function in contrast to existing approaches to invariance in the level set framework this closed form solution removes the need to iteratively optimize explicit pose parameters the resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object s pose secondly based on these invariant shape dissimilarity measures we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes this prior constitutes an extension of kernel density estimators to the level set domain in contrast to the commonly employed gaussian distribution such nonparametric density estimators are suited to model aribtrary distributions we demonstrate the advantages of this multi modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence and on the segmentation of the left ventricle in cardiac ultrasound images we give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes

schwarps locally projective image warps based on 2d schwarzian derivatives image warps or just warps capture the geometric deformation existing between two images of a deforming surface the current approach to enforce a warp s smoothness is to penalize its second order partial derivatives bookstein in ieee trans pattern anal mach intell 11 567 585 1989 rueckert et al in ieee trans med imaging 18 712 721 1999 because this favors locally affine warps this fails to capture the local projective component of the image deformation this may have a negative impact on applications such as image registration and deformable 3d reconstruction we propose a novel penalty designed to smooth the warp while capturing the deformation s local projective structure our penalty is based on equivalents to the schwarzian derivatives which are projective differential invariants exactly preserved by homographies we propose a methodology to derive a set of partial differential equations with only homographies as solutions we call this system the schwarzian equations and we explicitly derive them for 2d functions using differential properties of homographies we name as schwarp a warp which is estimated by penalizing the residual of schwarzian equations experimental evaluation shows that schwarps outperform existing warps in modeling and extrapolation power and lead to better results in three deformable reconstruction methods namely shape reconstruction in shape from template camera calibration in shape from template and non rigid structure from motion

deformotion deforming motion shape average and the joint registration and approximation of structures in images what does it mean for a deforming object to be moving how can we separate the overall motion a finite dimensional group action from the more general deformation a diffeomorphism in this paper we propose a definition of motion for a deforming object and introduce a notion of shape average as the entity that separates the motion from the deformation our definition allows us to derive novel and efficient algorithms to register non identical shapes using region based methods and to simultaneously approximate and align structures in greyscale images we also extend the notion of shape average to that of a moving average in order to track moving and deforming objects through time the algorithms we propose extend prior work on landmark based matching to smooth curves and involve the numerical integration of partial differential equations which we address within the framework of level set methods

stochastic exploration of ambiguities for nonrigid shape recovery recovering the 3d shape of deformable surfaces from single images is known to be a highly ambiguous problem because many different shapes may have very similar projections this is commonly addressed by restricting the set of possible shapes to linear combinations of deformation modes and by imposing additional geometric constraints unfortunately because image measurements are noisy such constraints do not always guarantee that the correct shape will be recovered to overcome this limitation we introduce a stochastic sampling approach to efficiently explore the set of solutions of an objective function based on point correspondences this allows us to propose a small set of ambiguous candidate 3d shapes and then use additional image information to choose the best one as a proof of concept we use either motion or shading cues to this end and show that we can handle a complex objective function without having to solve a difficult nonlinear minimization problem the advantages of our method are demonstrated on a variety of problems including both real and synthetic data

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

what shape are dolphins building 3d morphable models from 2d images 3d morphable models are low dimensional parameterizations of 3d object classes which provide a powerful means of associating 3d geometry to 2d images however morphable models are currently generated from 3d scans so for general object classes such as animals they are economically and practically infeasible we show that given a small amount of user interaction little more than that required to build a conventional morphable model there is enough information in a collection of 2d pictures of certain object classes to generate a full 3d morphable model even in the absence of surface texture the key restriction is that the object class should not be strongly articulated and that a very rough rigid model should be provided as an initial estimate of the mean shape the model representation is a linear combination of subdivision surfaces which we fit to image silhouettes and any identifiable key points using a novel combined continuous discrete optimization strategy results are demonstrated on several natural object classes and show that models of rather high quality can be obtained from this limited information

a closed form solution to non rigid shape and motion recovery recovery of three dimensional 3d shape and motion of non static scenes from a monocular video sequence is important for applications like robot navigation and human computer interaction if every point in the scene randomly moves it is impossible to recover the non rigid shapes in practice many non rigid objects e g the human face under various expressions deform with certain structures their shapes can be regarded as a weighted combination of certain shape bases shape and motion recovery under such situations has attracted much interest previous work on this problem bregler c hertzmann a and biermann h 2000 in proc int conf computer vision and pattern recognition brand m 2001 in proc int conf computer vision and pattern recognition torresani l yang d alexander g and bregler c 2001 in proc int conf computer vision and pattern recognition utilized only orthonormality constraints on the camera rotations rotation constraints this paper proves that using only the rotation constraints results in ambiguous and invalid solutions the ambiguity arises from the fact that the shape bases are not unique an arbitrary linear transformation of the bases produces another set of eligible bases to eliminate the ambiguity we propose a set of novel constraints basis constraints which uniquely determine the shape bases we prove that under the weak perspective projection model enforcing both the basis and the rotation constraints leads to a closed form solution to the problem of non rigid shape and motion recovery the accuracy and robustness of our closed form solution is evaluated quantitatively on synthetic data and qualitatively on real video sequences

optimal metric projections for deformable and articulated structure from motion this paper describes novel algorithms for recovering the 3d shape and motion of deformable and articulated objects purely from uncalibrated 2d image measurements using a factorisation approach most approaches to deformable and articulated structure from motion require to upgrade an initial affine solution to euclidean space by imposing metric constraints on the motion matrix while in the case of rigid structure the metric upgrade step is simple since the constraints can be formulated as linear deformability in the shape introduces non linearities in this paper we propose an alternating bilinear approach to solve for non rigid 3d shape and motion associated with a globally optimal projection step of the motion matrices onto the manifold of metric constraints our novel optimal projection step combines into a single optimisation the computation of the orthographic projection matrix and the configuration weights that give the closest motion matrix that satisfies the correct block structure with the additional constraint that the projection matrix is guaranteed to have orthonormal rows i e its transpose lies on the stiefel manifold this constraint turns out to be non convex the key contribution of this work is to introduce an efficient convex relaxation for the non convex projection step efficient in the sense that for both the cases of deformable and articulated motion the proposed relaxations turned out to be exact i e tight in all our numerical experiments the convex relaxations are semi definite sdp or second order cone socp programs which can be readily tackled by popular solvers an important advantage of these new algorithms is their ability to handle missing data which becomes crucial when dealing with real video sequences with self occlusions we show successful results of our algorithms on synthetic and real sequences of both deformable and articulated data we also show comparative results with state of the art algorithms which reveal that our new methods outperform existing ones

geodesic active contours and level sets for the detection and tracking of moving objects this paper presents a new variational framework for detecting and tracking multiple moving objects in image sequences motion detection is performed using a statistical framework for which the observed interframe difference density function is approximated using a mixture model this model is composed of two components namely the static background and the mobile moving objects one both components are zero mean and obey laplacian or gaussian law this statistical framework is used to provide the motion detection boundaries additionally the original frame is used to provide the moving object boundaries then the detection and the tracking problem are addressed in a common framework that employs a geodesic active contour objective function this function is minimized using a gradient descent method a new approach named hermes is proposed which exploits aspects from the well known front propagation algorithms and compares favorably to them very promising experimental results are provided using real video sequences

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

fixed points for multivalued mappings defined on unbounded sets in banach spaces abstract we give some fixed points theorems for multivalued nonexpansive and pseudocontractive mappings defined on closed convex unbounded subsets of a banach space some of such results extend to the multivoque case a few of the theorems already known for the univoque case while some other results improve a few of the theorems already known for the multivoque case

shape from template we study a problem that we call shape from template which is the problem of reconstructing the shape of a deformable surface from a single image and a 3d template current methods in the literature address the case of isometric deformations and relax the isometry constraint to the convex inextensibility constraint solved using the so called maximum depth heuristic we call these methods zeroth order since they use image point locations the zeroth order differential structure to solve the shape inference problem from a perspective image we propose a novel class of methods that we call first order the key idea is to use both image point locations and their first order differential structure the latter can be easily extracted from a warp between the template and the input image we give a unified problem formulation as a system of pdes for isometric and conformal surfaces that we solve analytically this has important consequences first it gives the first analytical algorithms to solve this type of reconstruction problems second it gives the first algorithms to solve for the exact constraints third it allows us to study the well posedness of this type of reconstruction we establish that isometric surfaces can be reconstructed unambiguously and that conformal surfaces can be reconstructed up to a few discrete ambiguities and a global scale in the latter case the candidate solution surfaces are obtained analytically experimental results on simulated and real data show that our isometric methods generally perform as well as or outperform state of the art approaches in terms of reconstruction accuracy while our conformal methods largely outperform all isometric methods for extensible deformations

sequential non rigid structure from motion using physical priors we propose a new approach to simultaneously recover camera pose and 3d shape of non rigid and potentially extensible surfaces from a monocular image sequence for this purpose we make use of the extended kalman filter based simultaneous localization and mapping ekf slam formulation a bayesian optimization framework traditionally used in mobile robotics for estimating camera pose and reconstructing rigid scenarios in order to extend the problem to a deformable domain we represent the object s surface mechanics by means of navier s equations which are solved using a finite element method fem with these main ingredients we can further model the material s stretching allowing us to go a step further than most of current techniques typically constrained to surfaces undergoing isometric deformations we extensively validate our approach in both real and synthetic experiments and demonstrate its advantages with respect to competing methods more specifically we show that besides simultaneously retrieving camera pose and non rigid shape our approach is adequate for both isometric and extensible surfaces does not require neither batch processing all the frames nor tracking points over the whole sequence and runs at several frames per second

procrustean regression a flexible alignment based framework for nonrigid structure estimation non rigid structure from motion nrsfm is a fundamental problem of computer vision recently it has been shown that incorporating shape alignment in nrsfm can improve the performance significantly compared with the other algorithms which do not consider shape alignment however realizing this idea was at a cost of a heavy complicated process which limits its usefulness and possible extensions in this paper we propose a novel regression framework for nrsfm of which the variables 3d shapes are regularized based on their aligned shapes we show that this can be casted into an unconstrained problem or a problem with simple bound constraints which can be efficiently solved by existing solvers this framework can be easily integrated with numerous existing models and assumptions such as orthographic or perspective camera models occlusion low rank assumption smooth deformations and so on which makes it more practical for various real situations the experimental results show that the proposed method gives competitive result to the state of the art methods for orthographic projection with much less time complexity and memory requirement and outperforms the existing methods for perspective projection

monocular 3d tracking of deformable surfaces using sequential second order cone programming reconstructing structures of deformable objects from monocular image sequences is important for applications like visual servoing and augmented reality in this paper we propose a method to recover 3d shapes of deformable surfaces using sequential second order cone programming socp the key of our approach is to represent the surface as a triangulated mesh and introduce two sets of constraints one for model to image keypoint correspondences which are socp constraints another for retaining the original lengths of the mesh edges which are non convex constraints in the process of tracking the surface structure is iteratively updated by solving sequential socp feasibility problems in which the non convex constraints are replaced by a set of convex constraints over a local convex region the shape constraints used in our approach is more generic than previous methods that enables us to reliably recover surface shapes with smooth sharp and other complex deformations the capability and efficiency of our approach are evaluated quantitatively with synthetic image sequences and qualitatively with real image sequences

a linearization for a class of λ nonlinear boundary eigenvalue problems abstract in this paper a new linearization of boundary eigenvalue problems for systems a0 a1 of n first order differential equations with polynomial boundary conditions is proposed the linearized problem is again a boundary eigenvalue problem for a system y a0y a1y of first order differential equations of dimension n n where n is the total polynomial degree of the boundary conditions as a particular case we consider systems of first order differential equations induced by nth order differential equations n p and we give an application to the orr sommerfeld equation with quadratic boundary conditions

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

the riemannian structure of euclidean shape spaces a novel environment for statistics the riemannian metric structure of the shape space sum k m for k labelled points in mathbb r m was given by kendall for the atypically simple situations in which m 1 or 2 and k geq 2 here we deal with the general case m geq 1 k geq 2 by using the properties of riemannian submersions and warped products as studied by o neill the approach is via the associated size and shape space that is the warped product of the shape space and the half line mathbb r carrying size the warping function being equal to the square of the size when combined with parallel studies by le of the corresponding global geodesic geometry the results obtained here determine the environment in which shape statistical calculations have to be acted out finally three different applications are discussed that illustrate the theory and its use in practice

exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects abstract we study the boundary stabilization of an active constrained layer acl beam consisting of a stiff layer a viscoelastic layer and a piezoelectric layer the piezoelectric layer is actuated by a voltage source without magnetic effects the system is modeled as a rayleigh beam coupled with two wave equations by using an asymptotic technique we present the asymptotic expressions for the eigenpairs of the system we show that the generalized eigenfunctions form a riesz basis in the state space and hence the spectrum determined growth condition holds finally the exponential stability of the closed loop system is established

deform pf mt particle filter with mode tracker for tracking nonaffine contour deformations we propose algorithms for tracking the boundary contour of a deforming object from an image sequence when the nonaffine local deformation over consecutive frames is large and there is overlapping clutter occlusions low contrast or outlier imagery when the object is arbitrarily deforming each or at least most contour points can move independently contour deformation then forms an infinite in practice very large dimensional space direct application of particle filters pf for large dimensional problems is impractically expensive however in most real problems at any given time most of the contour deformation occurs in a small number of dimensions effective basis space while the residual deformation in the rest of the state space residual space is small this property enables us to apply the particle filtering with mode tracking pf mt idea that was proposed for such large dimensional problems in recent work since most contour deformation is low spatial frequency we propose to use the space of deformation at a subsampled set of locations as the effective basis space the resulting algorithm is called deform pf mt it requires significant modifications compared to the original pf mt because the space of contours is a non euclidean infinite dimensional space

bilinear modeling via augmented lagrange multipliers balm this paper presents a unified approach to solve different bilinear factorization problems in computer vision in the presence of missing data in the measurements the problem is formulated as a constrained optimization where one of the factors must lie on a specific manifold to achieve this we introduce an equivalent reformulation of the bilinear factorization problem that decouples the core bilinear aspect from the manifold specificity we then tackle the resulting constrained optimization problem via augmented lagrange multipliers the strength and the novelty of our approach is that this framework can seamlessly handle different computer vision problems the algorithm is such that only a projector onto the manifold constraint is needed we present experiments and results for some popular factorization problems in computer vision such as rigid non rigid and articulated structure from motion photometric stereo and 2d 3d non rigid registration

trajectory space a dual representation for nonrigid structure from motion existing approaches to nonrigid structure from motion assume that the instantaneous 3d shape of a deforming object is a linear combination of basis shapes these bases are object dependent and therefore have to be estimated anew for each video sequence in contrast we propose a dual approach to describe the evolving 3d structure in trajectory space by a linear combination of basis trajectories we describe the dual relationship between the two approaches showing that they both have equal power for representing 3d structure we further show that the temporal smoothness in 3d trajectories alone can be used for recovering nonrigid structure from a moving camera the principal advantage of expressing deforming 3d structure in trajectory space is that we can define an object independent basis this results in a significant reduction in unknowns and corresponding stability in estimation we propose the use of the discrete cosine transform dct as the object independent basis and empirically demonstrate that it approaches principal component analysis pca for natural motions we report the performance of the proposed method quantitatively using motion capture data and qualitatively on several video sequences exhibiting nonrigid motions including piecewise rigid motion partially nonrigid motion such as a facial expressions and highly nonrigid motion such as a person walking or dancing

shapes shocks and deformations i the components of two dimensional shape and the reaction diffusion space we undertake to develop a general theory of two dimensional shape by elucidating several principles which any such theory should meet the principles are organized around two basic intuitions first if a boundary were changed only slightly then in general its shape would change only slightly this leads us to propose an operational theory of shape based on incremental contour deformations the second intuition is that not all contours are shapes but rather only those that can enclose physical material a theory of contour deformation is derived from these principles based on abstract conservation principles and hamilton jacobi theory these principles are based on the work of sethian 1985a c the osher sethian 1988 level set formulation the classical shock theory of lax 1971 1973 as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of gage hamilton grayson 1986 1989 the result is a characterization of the computational elements of shape deformations parts bends and seeds which show where to place the components of a shape the theory unifies many of the diverse aspects of shapes and leads to a space of shapes the reaction diffusion space which places shapes within a neighborhood of similar ones such similarity relationships underlie descriptions suitable for recognition

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

continuous generalized procrustes analysis two dimensional shape models have been successfully applied to solve many problems in computer vision such as object tracking recognition and segmentation typically 2d shape models are learned from a discrete set of image landmarks corresponding to projection of 3d points of an object after applying generalized procustes analysis gpa to remove 2d rigid transformations however the standard gpa process suffers from three main limitations firstly the 2d training samples do not necessarily cover a uniform sampling of all the 3d transformations of an object this can bias the estimate of the shape model secondly it can be computationally expensive to learn the shape model by sampling 3d transformations thirdly standard gpa methods use only one reference shape which can might be insufficient to capture large structural variability of some objects to address these drawbacks this paper proposes continuous generalized procrustes analysis cgpa cgpa uses a continuous formulation that avoids the need to generate 2d projections from all the rigid 3d transformations it builds an efficient in space and time non biased 2d shape model from a set of 3d model of objects a major challenge in cgpa is the need to integrate over the space of 3d rotations especially when the rotations are parameterized with euler angles to address this problem we introduce the use of the haar measure finally we extended cgpa to incorporate several reference shapes experimental results on synthetic and real experiments show the benefits of cgpa over gpa highlightscontinuous formulation of the generalized procrustes analysis cgpa avoids the need to generate 2d projections from all 3d rigid transformations cgpa builds an efficient non biased 2d shape model from 3d objects cgpa uses the haar measure to integrate over the space of 3d rotations experimental results building 2d shape models for different public datasets

a framework for image segmentation using shape models and kernel space shape priors segmentation involves separating an object from the background in a given image the use of image information alone often leads to poor segmentation results due to the presence of noise clutter or occlusion the introduction of shape priors in the geometric active contour gac framework has proved to be an effective way to ameliorate some of these problems in this work we propose a novel segmentation method combining image information with prior shape knowledge using level sets following the work of leventon et al we propose to revisit the use of pca to introduce prior knowledge about shapes in a more robust manner we utilize kernel pca kpca and show that this method outperforms linear pca by allowing only those shapes that are close enough to the training data in our segmentation framework shape knowledge and image information are encoded into two energy functionals entirely described in terms of shapes this consistent description permits to fully take advantage of the kernel pca methodology and leads to promising segmentation results in particular our shape driven segmentation technique allows for the simultaneous encoding of multiple types of shapes and offers a convincing level of robustness with respect to noise occlusions or smearing

template based monocular 3d shape recovery using laplacian meshes we show that by extending the laplacian formalism which was first introduced in the graphics community to regularize 3d meshes we can turn the monocular 3d shape reconstruction of a deformable surface given correspondences with a reference image into a much better posed problem this allows us to quickly and reliably eliminate outliers by simply solving a linear least squares problem this yields an initial 3d shape estimate which is not necessarily accurate but whose 2d projections are the initial shape is then refined by a constrained optimization problem to output the final surface reconstruction our approach allows us to reduce the dimensionality of the surface reconstruction problem without sacrificing accuracy thus allowing for real time implementations

exponential stabilization of laminated beams with structural damping and boundary feedback controls we study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two layered objects by using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs furthermore we prove that there is a sequence of generalized eigenfunctions that forms a riesz basis in the state hilbert space and hence the spectrum determined growth condition holds furthermore exponential stability of the closed loop system can be deduced from the eigenvalue expressions in particular the semigroup generated by the system operator is a c 0 group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis

the tps direct transport a new method for transporting deformations in the size and shape space modern shape analysis allows the fine comparison of shape changes occurring between different objects very often the classic machineries of generalized procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks however if size and shape data are structured in different groups thus constituting different morphological trajectories a data centering is needed if one wants to compare solely the deformation representing the trajectories to do that inter individual variation must be filtered out this maneuver is rarely applied in studies using simulated or real data a geometrical procedure named parallel transport that can be based on various connection types is necessary to perform such kind of data centering usually the levi civita connection is used for interpolation of curves in a riemannian space it can also be used to transport a deformation we demonstrate that this procedure does not preserve some important characters of the deformation even in the affine case we propose a novel procedure called tps direct transport which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools we recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study

monocular 3 d tracking of inextensible deformable surfaces under l_2 norm we present a method for recovering the 3 d shape of an inextensible deformable surface from a monocular image sequence state of the art methods on this problem utilize l norm of reprojection residual vectors and formulate the tracking problem as a second order cone programming socp problem instead of using l which is sensitive to outliers we use l 2 norm of reprojection errors generally using l 2 leads a nonconvex optimization problem which is difficult to minimize instead of solving the nonconvex problem directly we design an iterative l 2 norm approximation process to approximate the nonconvex objective function in which only a linear system needs to be solved at each iteration furthermore we introduce a shape regularization term into this iterative process in order to keep the inextensibility of the recovered mesh compared with previous methods ours performs more robust to image noises outliers and large interframe motions with high computational efficiency the robustness and accuracy of our approach are evaluated quantitatively on synthetic data and qualitatively on real data

planar structure from motion with affine camera models closed form solutions ambiguities and degeneracy analysis planar structure from motion sfm is the problem of reconstructing a planar object or surface from a set of 2d images using motion information the problem is well understood with the perspective camera model and can be solved with homography decomposition hd however when the structure is small and or viewed far from the camera the perspective effects diminish and in the limit the projections become affine in these situations hd fails because the problem itself becomes ill posed we propose a stable alternative using affine camera models these have been used extensively to reconstruct non planar structures however a general accurate and closed form method for planar structures has been missing the problem is fundamentally different with planar structures because the types of affine camera models one can use are more restricted and it is inherently more ambiguous and non linear we provide a closed form method for the orthographic camera model that solves the general problem three or more views with three or more correspondences and missing correspondences and returns all metric structure solutions and corresponding camera poses the method does not require initialisation and optimises an objective function that is very similar to the reprojection error in fact there is no clear benefit in refining its solutions with bundle adjustment which is a remarkable result we also present a new theoretical analysis that deepens our understanding of the problem the main result is the necessary and sufficient geometric conditions for the problem to be degenerate with the orthographic camera we also show there can exist up to two solutions for metric structure with four or more views previously it was assumed to be unique and we give the necessary and sufficient geometric conditions for disambiguation other theoretical results include showing that in the case of three images the optimal reconstruction with respect to reprojection error can usually be found in closed form and additional prior knowledge needed to solve with non orthographic affine cameras