3 d motion estimation using a sequence of noisy stereo images models estimation and uniqueness results a kinematic model based approach for the estimation of 3 d motion parameters from a sequence of noisy stereo images is discussed the approach is based on representing the constant acceleration translational motion and constant precession rotational motion in the form of a bilinear state space model using standard rectilinear states for translation and quaternions for rotation closed form solutions of the state transition equations are obtained to propagate the quaternions the measurements are noisy perturbations of 3 d feature points represented in an inertial coordinate system it is assumed that the 3 d feature points are extracted from the stereo images and matched over the frames owing to the nonlinearity in the state model nonlinear filters are designed for the estimation of motion parameters simulation results are included the cramer rao performance bounds for motion parameter estimates are computed a constructive proof for the uniqueness of motion parameters is given it is shown that with uniform sampling in time three noncollinear feature points in five consecutive binocular image pairs contain all the spatial and temporal information both nondegenerate and degenerate motions are analyzed a deterministic algorithm to recover motion parameters from a stereo image sequence is summarized from the constructive proof

nonlinear extraction of independent components of natural images using radial gaussianization we consider the problem of efficiently encoding a signal by transforming it to a new representation whose components are statistically independent a widely studied linear solution known as independent component analysis ica exists for the case when the signal is generated as a linear transformation of independent nongaussian sources here we examine a complementary case in which the source is nongaussian and elliptically symmetric in this case no invertible linear transform suffices to decompose the signal into independent components but we show that a simple nonlinear transformation which we call radial gaussianization rg is able to remove all dependencies we then examine this methodology in the context of natural image statistics we first show that distributions of spatially proximal bandpass filter responses are better described as elliptical than as linearly transformed independent sources consistent with this we demonstrate that the reduction in dependency achieved by applying rg to either nearby pairs or blocks of bandpass filter responses is significantly greater than that achieved by ica finally we show that the rg transformation may be closely approximated by divisive normalization which has been used to model the nonlinear response properties of visual neurons

a posteriori error control for distributed elliptic optimal control problems with control constraints discretized by hp finite elements abstract a distributed elliptic control problem with control constraints is considered which is formulated as a three field problem and consists of two variational equations for the state and the co state variables as well as of a variational inequality for the control variable two discretization approaches with h p finite elements are discussed in the first discretization approach all variables state co state and control are discretized a semi smooth newton method is introduced for solving the resulting algebraic system in the second discretization approach only the state and the co state variables are discretized whereas the control is determined by projection a simple fixed point scheme is presented for the iterative solution of this approach the main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates which enables h p adaptive mesh refinements in particular the estimates can be applied to the iteration solutions so that they can be used as a stopping criterion of the iterative solution schemes in several numerical experiments the order of convergence of the adaptive discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied

estimating three dimensional motion parameters of a rigid planar patch ii singular value decomposition we show that the three dimensional 3 d motion parameters of a rigid planar patch can be determined by computing the singular value decomposition svd of a 3 3 matrix containing the eight so called pure parameters furthermore aside from a scale factor for the translation parameters the number of solutions is either one or two depending on the multiplicity of the singular values of the matrix

emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces olshausen and field 1996 applied the principle of independence maximization by sparse coding to extract features from natural images this leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency thus resembling gabor functions and simple cell receptive fields in this article we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features similar to those found in complex cells this new kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces instead of the independence of simple linear filter outputs the norms of the projections on such independent feature subspaces then indicate the values of invariant features

pointwise a posteriori error estimates for elliptic problems on highly graded meshes pointwise a posteriori error estimates are derived for linear second order elliptic problems over general polygonal domains in 2d the analysis carries over regardless of convexity accounting even for slit domains and applies to highly graded unstructured meshes as well a key ingredient is a new asymptotic a priori estimate for regularized green s functions the estimators lead always to upper bounds for the error in the maximum norm along with lower bounds under very mild regularity and nondegeneracy assumptions the effect of both point and line singularities is examined three popular local estimators for the energy norm are shown to be equivalent when suitably interpreted to those introduced here

l asymptotic behavior for a finite element approximation in parabolic quasi variational inequalities related to impulse control problem abstract in this paper the parabolic quasi variational inequalities are transformed into a noncoercive elliptic quasi variational inequalities a new iterative discrete algorithm is proposed to show the existence and uniqueness and a simple proof to asymptotic behavior in uniform norm is also given using the theta time scheme combined with a finite element spatial approximation the proposed approach stands on a discrete l stability property with respect to the right hand side and obstacle defined as an impulse control problem

residual error estimators for coulomb friction this paper is concerned with residual error estimators for finite element approximations of coulomb frictional contact problems a recent uniqueness result by renard in siam j math anal 38 2006 pp 452 467 for the continuous problem allows us to perform an a posteriori error analysis we propose study and implement numerically two residual error estimators associated with two finite element discretizations in both cases the estimators permit us to obtain upper and lower bounds of the discretization error

estimation of object motion parameters from noisy images an approach is presented for the estimation of object motion parameters based on a sequence of noisy images the problem considered is that of a rigid body undergoing unknown rotational and translational motion the measurement data consists of a sequence of noisy image coordinates of two or more object correspondence points by modeling the object dynamics as a function of time estimates of the model parameters including motion parameters can be extracted from the data using recursive and or batch techniques this permits a desired degree of smoothing to be achieved through the use of an arbitrarily large number of images some assumptions regarding object structure are presently made results are presented for a recursive estimation procedure the case considered here is that of a sequence of one dimensional images of a two dimensional object thus the object moves in one transverse dimension and in depth preserving the fundamental ambiguity of the central projection image model loss of depth information an iterated extended kalman filter is used for the recursive solution noise levels of 5 10 percent of the object image size are used approximate cramer rao lower bounds are derived for the model parameter estimates as a function of object trajectory and noise level this approach may be of use in situations where it is difficult to resolve large numbers of object match points but relatively long sequences of images 10 to 20 or more are available

what is the goal of sensory coding a number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy in this paper two strategies are contrasted both strategies take advantage of the redundancy in the environment to produce more effective representations the first is described as a compact coding scheme a compact code performs a transform that allows the input to be represented with a reduced number of vectors cells with minimal rms error this approach has recently become popular in the neural network literature and is related to a process called principal components analysis pca a number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex however in this paper it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway in contrast it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of sparse distributed coding in a sparse distributed code all cells in the code have an equal response probability across the class of images but have a low response probability for any single image in such a code the dimensionality is not reduced rather the redundancy of the input is transformed into the redundancy of the firing pattern of cells it is proposed that the signature for a sparse code is found in the fourth moment of the response distribution i e the kurtosis in measurements with 55 calibrated natural scenes the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex codes resembling wavelet transforms are proposed to be effective because the response histograms of such codes are sparse i e show high kurtosis when presented with natural scenes it is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image it is suggested that natural scenes to a first approximation can be considered as a sum of self similar local functions the inverse of a wavelet possible reasons for why sensory systems would evolve toward sparse coding are presented

a posteriori error estimators for regularized total variation of characteristic functions we consider a nonuniformly elliptic double obstacle problem arising from convexifying and regularizing a minimization of a functional with total variation in the set of characteristic functions we derive a posteriori estimators for the discretization error with linear finite elements which are uniform in the regularization incorporate computable and local information on the conditioning vanish in the intersection of discrete and exact contact sets and are not affected by possible nonuniqueness moreover we integrate these estimators in an adaptive algorithm and illustrate their properties by various numerical experiments

estimating three dimensional motion parameters of a rigid planar patch iii finite point correspondences and the three view problem two results are presented in this paper first it is shown that in estimating three dimensional motion of a rigid planar patch the eight pure parameters used in 1 and 2 are uniquely determined from the image correspondences of four points no three collinear and can be estimated by solving a set of linear equations the second result concerns the three view problem it is proved that given four image point correspondences in three perspective views of a planar patch undergoing general three dimensional rigid body motion the number of solutions for the motion parameters is one as opposed to two 2 when only two perspective views are given

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

motion estimation using a complex valued wavelet transform this paper describes a new motion estimation algorithm that is potentially useful for both computer vision and video compression applications it is hierarchical in structure using a separable two dimensional 2 d discrete wavelet transform dwt on each frame to efficiently construct a multiresolution pyramid of subimages the dwt is based on a complex valued pair of four tap fir filters with gabor like characteristics the resulting complex dwt cdwt effectively implements an analysis by an ensemble of gabor like filters with a variety of orientations and scales the phase difference between the subband coefficients of each frame at a given subpel bears a predictable relation to a local translation in the region of the reference frame subtended by that subpel that relation is used to estimate the displacement field at the coarsest scale of the multiresolution pyramid each estimate is accompanied by a directional confidence measure in the form of the parameters of a quadratic matching surface the initial estimate field is progressively refined by a coarse to fine strategy in which finer scale information is appropriately incorporated at each stage the accuracy efficiency and robustness of the new algorithm are demonstrated in comparison testing against hierarchical implementations of intensity gradient based and fractional precision block matching motion estimators

slow feature analysis unsupervised learning of invariances invariant features of temporally varying signals are useful for analysis and classification slow feature analysis sfa is a new method for learning invariant or slowly varying features from a vectorial input signal it is based on a nonlinear expansion of the input signal and application of principal component analysis to this expanded signal and its time derivative it is guaranteed to find the optimal solution within a family of functions directly and can learn to extract a large number of decorrelated features which are ordered by their degree of invariance sfa can be applied hierarchically to process high dimensional input signals and extract complex features sfa is applied first to complex cell tuning properties based on simple cell output including disparity and motion then more complicated input output functions are learned by repeated application of sfa finally a hierarchical network of sfa modules is presented as a simple model of the visual system the same unstructured network can learn translation size rotation contrast or to a lesser degree illumination invariance for one dimensional objects depending on only the training stimulus surprisingly only a few training objects suffice to achieve good generalization to new objects the generated representation is suitable for object recognition performance degrades if the network is trained to learn multiple invariances simultaneously

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

motion estimation with more than two frames the development of a general motion estimation system based on using three or more frames is reported the relevant portion of the sequence is treated as a whole thus simplifying the process of estimating the motion parameters and making possible the computation of the motion parameters in terms of the natural center of motion a general description of this motion estimation method is presented with the derivation of motion equations for three cases three points in three frames two points in four frames and one point in five frames results are given for real and synthetic images for the case of one point in five and six frames on the average the program converged quickly to the right answer for noiseless data for noisy data the answers were reasonable and their accuracy was directly a function of the amount of noise in the disparity vector

bilinear sparse coding for invariant vision recent algorithms for sparse coding and independent component analysis ica have demonstrated how localized features can be learned from natural images however these approaches do not take image transformations into account we describe an unsupervised algorithm for learning both localized features and their transformations directly from images using a sparse bilinear generative model we show that from an arbitrary set of natural images the algorithm produces oriented basis filters that can simultaneously represent features in an image and their transformations the learned generative model can be used to translate features to different locations thereby reducing the need to learn the same feature at multiple locations a limitation of previous approaches to sparse coding and ica our results suggest that by explicitly modeling the interaction between local image features and their transformations the sparse bilinear approach can provide a basis for achieving transformation invariant vision

maximum norm error estimators for three dimensional elliptic problems in this paper we define an a posteriori error estimator for finite element approximations of 3 d elliptic problems we prove that the estimator is equivalent up to logarithmic factors of the meshsize to the maximum norm of the error the results are valid for an arbitrary polyhedral domain and rather general meshes we also obtain analogous results for the nonconforming method of crouzeix raviart finally we present some numerical results comparing adaptive procedures based on controlling the error in different norms

experiments on estimating egomotion and structure parameters using long monocular image sequences this paper presents a simple but robust model based approach to estimating the kinematics of a moving camera and the structure of the objects in a stationary environment using long noisy monocular image sequences both batch and recursive algorithms are presented and the problem due to occlusion is addressed the approach is based on representing the constant translational velocity and constant angular velocity of the camera motion using nine rectilinear motion parameters which are 3 d vectors of the position of the rotation center linear and angular velocities the structure parameters are 3 d coordinates of the salient feature points in the inertial coordinate system due to redundancies in parameterization the total number of independent parameters to be estimated is 3m 7 wherem is the number of feature points the image plane coordinates of these feature points in each frame are first detected and matched over the frames these noisy image coordinates serve as the input to our algorithms due to the nonlinear nature of perspective projection a nonlinear least squares method is formulated for the batch algorithm and a conjugate gradient method is then applied to find the solution a recursive method using an iterated extended kalman filter iekf for incremental estimation of motion and structure is also presented since the plant model is simple in our formulation closed form solutions for the state and covariance transition equations are easily derived experimental results for simulated imagery as well as several real image sequences are included

some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive image frames this paper describes the results of some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive tv images and discusses several factors which affect the accuracy of the results these factors include the sizes of the motion parameters themselves the accuracy of the raw data and the number of point correspondences in addition we address two related topics determining corner positions to subpixel accuracy and matching point patterns with different scales

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

learning invariance from transformation sequences the visual system can reliably identify objects even when the retinal image is transformed considerably by commonly occurring changes in the environment a local learning rule is proposed which allows a network to learn to generalize across such transformations during the learning phase the network is exposed to temporal sequences of patterns undergoing the transformation an application of the algorithm is presented in which the network learns invariance to shift in retinal position such a principle may be involved in the development of the characteristic shift invariance property of complex cells in the primary visual cortex and also in the development of more complicated invariance properties of neurons in higher visual areas

3 d motion estimation understanding and prediction from noisy image sequences this paper presents an approach to understanding general 3 d motion of a rigid body from image sequences based on dynamics a locally constant angular momentum lcam model is introduced the model is local in the sense that it is applied to a limited number of image frames at a time specifically the model constrains the motion over a local frame subsequence to be a superposition of precession and translation thus the instantaneous rotation axis of the object is allowed to change through the subsequence the trajectory of the rotation center is approximated by a vector polynomial the parameters of the model evolve in time so that they can adapt to long term changes in motion characteristics the nature and parameters of short term motion can be estimated continuously with the goal of understanding motion through the image sequence the estimation algorithm presented in this paper is linear i e the algorithm consists of solving simultaneous linear equations based on the assumption that the motion is smooth object positions and motion in the near future can be predicted and short missing subsequences can be recovered noise smoothing is achieved by overdetermination and a leastsquares criterion the framework is flexible in the sense that it allows both overdetermination in number of feature points and the number of image frames

soft mixer assignment in a hierarchical generative model of natural scene statistics gaussian scale mixture models offer a top down description of signal generation that captures key bottom up statistical characteristics of filter responses to images however the pattern of dependence among the filters for this class of models is prespecified we propose a novel extension to the gaussian scale mixturemodel that learns the pattern of dependence from observed inputs and thereby induces a hierarchical representation of these inputs specifically we propose that inputs are generated by gaussian variables modeling local filter structure multiplied by a mixer variable that is assigned probabilistically to each input from a set of possible mixers we demonstrate inference of both components of the generative model for synthesized data and for different classes of natural images such as a generic ensemble and faces for natural images the mixer variable assignments show invariances resembling those of complex cells in visual cortex the statistics of the gaussian components of the model are in accord with the outputs of divisive normalization models we also show how our model helps interrelate a wide range of models of image statistics and cortical processing

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

positivity preserving finite element approximation we consider finite element operators defined on rough functions in a bounded polyhedron in rn insisting on preserving positivity in the approximations we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not we give impossibility results for approximation of general functions to more than first order accuracy at extreme points of we also give impossibility results about invariance of positive operators on finite element functions this is in striking contrast to the well studied case without positivity

a hierarchical bayesian model for learning nonlinear statistical regularities in nonstationary natural signals capturing statistical regularities in complex high dimensional data is an important problem in machine learning and signal processing models such as principal component analysis pca and independent component analysis ica make few assumptions about the structure in the data and have good scaling properties but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary for many natural complex signals the latent variables often exhibit residual dependencies as well as nonstationary statistics here we present a hierarchical bayesian model that is able to capture higher order nonlinear structure and represent nonstationary data distributions the model is a generalization of ica in which the basis function coefficients are no longer assumed to be independent instead the dependencies in their magnitudes are captured by a set of density components each density component describes a common pattern of deviation from the marginal density of the pattern ensemble in different combinations they can describe nonstationary distributions adapting the model to image or audio data yields a nonlinear distributed code for higher order statistical regularities that reflect more abstract invariant properties of the signal

obstacle problems with cohesion a hemivariational inequality approach and its efficient numerical solution motivated by an obstacle problem for a membrane subject to cohesion forces constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered the associated first order optimality system leads to a hemivariational inequality which can also be interpreted as a special complementarity problem in function space besides an analytical investigation of first order optimality a primal dual active set solver is introduced it is associated to a limit case of a semismooth newton method for a regularized version of the underlying problem class for the numerical algorithms studied in this paper global as well as local convergence properties are derived and verified numerically

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

estimating three dimensional motion parameters of a rigid planar patch we present a new direct method of estimating the three dimensional motion parameters of a rigid planar patch from two time sequential perspective views image frames first a set of eight pure parameters are defined these parameters can be determined uniquely from the two given image frames by solving a set of linear equations then the actual motion parameters are determined from these pure parameters by a method which requires the solution of a sixth order polynomial of one variable only and there exists a certain efficient algorithm for solving a sixth order polynomial aside from a scale factor for the translation parameters the number of real solutions never exceeds two in the special case of three dimensional translation the motion parameters can be expressed directly as some simple functions of the eight pure parameters thus only a few arithmetic operations are needed

phase coupling estimation from multivariate phase statistics coupled oscillators are prevalent throughout the physical world dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real world systems and are compelling models of neural systems however these formulations usually deal with the forward problem simulating a system from known coupling parameters here we provide a solution to the inverse problem determining the coupling parameters from measurements starting from the dynamic equations of a system of symmetrically coupled phase oscillators given by a nonlinear langevin equation we derive the corresponding equilibrium distribution this formulation leads us to the maximum entropy distribution that captures pairwise phase relationships to solve the inverse problem for this distribution we derive a closed form solution for estimating the phase coupling parameters from observed phase statistics through simulations we show that the algorithm performs well in high dimensions d 100 and in cases with limited data as few as 100 samples per dimension in addition we derive a regularized solution to the estimation and show that the resulting procedure improves performance when only a limited amount of data is available because the distribution serves as the unique maximum entropy solution for pairwise phase statistics phase coupling estimation can be broadly applied in any situation where phase measurements are made under the physical interpretation the model may be used for inferring coupling relationships within cortical networks

estimating the kinematics and structure of a rigid object from a sequence of monocular images the problem considered involves the use of a sequence of noisy monocular images of a three dimensional moving object to estimate both its structure and kinematics the object is assumed to be rigid and its motion is assumed to be smooth a set of object match points is assumed to be available consisting of fixed features on the object the image plane coordinates of which have been extracted from successive images in the sequence structure is defined as the 3 d positions of these object feature points relative to each other rotational motion occurs about the origin of an object centered coordinate system while translational motion is that of the origin of this coordinate system in this work which is a continuation of the research done by the authors and reported previously ibid vol pami 8 p 90 9 jan 1986 results of an experiment with real imagery are presented involving estimation of 28 unknown translational rotational and structural parameters based on 12 images with seven feature points

learning intermediate level representations of form and motion from natural movies we present a model of intermediate level visual representation that is based on learning invariances from movies of the natural environment the model is composed of two stages of processing an early feature representation layer and a second layer in which invariances are explicitly represented invariances are learned as the result of factoring apart the temporally stable and dynamic components embedded in the early feature representation the structure contained in these components is made explicit in the activities of second layer units that capture invariances in both form and motion when trained on natural movies the first layer produces a factorization or separation of image content into a temporally persistent part representing local edge structure and a dynamic part representing local motion structure consistent with known response properties in early visual cortex area v1 this factorization linearizes statistical dependencies among the first layer units making them learnable by the second layer the second layer units are split into two populations according to the factorization in the first layer the form selective units receive their input from the temporally persistent part local edge structure and after training result in a diverse set of higher order shape features consisting of extended contours multiscale edges textures and texture boundaries the motion selective units receive their input from the dynamic part local motion structure and after training result in a representation of image translation over different spatial scales and directions in addition to more complex deformations these representations provide a rich description of dynamic natural images and testable hypotheses regarding intermediate level representation in visual cortex

tracking a dynamic set of feature points we address the problems of tracking a set of feature points over a long sequence of monocular images as well as how to include and track new feature points detected in successive frames due to the 3 d movement of the camera different parts of the images exhibit different image motion tracking discrete features can therefore be decomposed into several independent and local problems accordingly we propose a localized feature tracking algorithm the trajectory of each feature point is described by a 2 d kinematic model then to track a feature point an interframe motion estimation scheme is designed to obtain the estimates of interframe motion parameters subsequently using the estimates of motion parameters corresponding points are identified to subpixel accuracy afterwards the temporal information is processed to facilitate the tracking scheme since different feature points are tracked independently the algorithm is able to handle the image motion arising from general 3 d camera movements on the other hand in addition to tracking feature points detected at the beginning an efficient way to dynamically include new points extracted in subsequent frames is devised so that the information in a sequence is preserved experimental results for several image sequences are also reported

an a posteriori error analysis of an elliptic optimal control problem in measure space abstract we propose an a posteriori error estimator for an optimal control problem with sparsity promoting cost the control variable lies in the space of regular borel measures we consider a solution technique that relies on the discretization of the control variable as a linear combination of dirac measures the proposed a posteriori error estimator can be decomposed into the sum of two contributions an error estimator in the maximum norm for the discretization of the adjoint equation and an estimator in the l 2 norm that accounts for the approximation of the state equation we prove that the designed error estimator is locally efficient and we explore its reliability properties the analysis is valid for two and three dimensional domains we illustrate the theory with numerical examples

three dimensional motion computation and object segmentation in a long sequence of stereo frames we address the problem of computing the three dimensional motions of objects in a long sequence of stereo frames our approach is bottom up and consists of two levels the first level deals with the tracking of 3d tokens from frame to frame and the estimation of their kinematics the processing is completely parallel for each token the second level groups tokens into objects based on their kinematic parameters controls the processing at the low level to cope with problems such as occlusion disappearance and appearance of tokens and provides information to other components of the system we have implemented this approach using 3d line segments obtained from stereo as the tokens we use classical kinematics and derive closed form solutions for some special but useful cases of motions the motion computation problem is then formulated as a tracking problem in order to apply the extended kalman filter the tracking is performed in a prediction matching update loop in which multiple matches can be handled tokens are labeled by a number called its support of existence which measures their adequation to the measurements if this number goes beyond a threshold the token disappears the individual line segments can be grouped into rigid objects according to the similarity of their kinematic parameters experiments using synthetic and real data have been carried out and the results found to be quite good

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

hierarchical error estimates for the energy functional in obstacle problems we present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems the main result is that the error in the energy minimum is up to oscillation terms equivalent to an appropriate hierarchical estimator the proof does not invoke any saturation assumption we even show that small oscillation implies a related saturation assumption in addition we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators finally we illustrate our theoretical results by numerical computations

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

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

finding trajectories of feature points in a monocular image sequence identifying the same physical point in more than one image the correspondence problem is vital in motion analysis most research for establishing correspondence uses only two frames of a sequence to solve this problem by using a sequence of frames it is possible to exploit the fact that due to inertia the motion of an object cannot change instantaneously by using smoothness of motion it is possible to solve the correspondence problem for arbitrary motion of several nonrigid objects in a scene we formulate the correspondence problem as an optimization problem and propose an iterative algorithm to find trajectories of points in a monocular image sequence a modified form of this algorithm is useful in case of occlusion also we demonstrate the efficacy of this approach considering synthetic laboratory and real scenes

convergence analysis of a conforming adaptive finite element method for an obstacle problem the adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual based a posteriori error estimator each cycle of the adaptive loop consists of the steps solve estimate mark and refine the techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of galerkin orthogonality we establish r linear convergence of the part of the energy above its minimal value if there is appropriate control of the data oscillations surprisingly the adaptive mesh refinement algorithm is the same as in the unconstrained case of a linear pde in fact there is no modification near the discrete free boundary necessary for r linear convergence the arguments are presented for a model obstacle problem with an affine obstacle and homogeneous dirichlet boundary conditions the proof of the discrete local efficiency is more involved than in the unconstrained case numerical results are given to illustrate the performance of the error estimator

direct passive navigation in this correspondence we show how to recover the motion of an observer relative to a planar surface from image brightness derivatives we do not compute the optical flow as an intermediate step only the spatial and temporal brightness gradients at a minimum of eight points we first present two iterative schemes for solving nine nonlinear equations in terms of the motion and surface parameters that are derived from a least squares fomulation an initial pass over the relevant image region is used to accumulate a number of moments of the image brightness derivatives all of the quantities used in the iteration are efficiently computed from these totals without the need to refer back to the image we then show that either of two possible solutions can be obtained in closed form we first solve a linear matrix equation for the elements of a 3 3 matrix the eigenvalue decomposition of the symmetric part of the matrix is then used to compute the motion parameters and the plane orientation a new compact notation allows us to show easily that there are at most two planar solutions

efficient and reliable a posteriori error estimators for elliptic obstacle problems a posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems these estimators yield global upper and local lower bounds for the discretization error here discretization error means the sum of two contributions the distance between continuous and discrete solution in the energy norm and some quantity that is related to the distance of continuous and discrete contact set moreover the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution

estimation of illuminant direction albedo and shape from shading a robust approach to the recovery of shape from shading information is presented assuming uniform albedo and lambertian surface for the imaging model two methods for estimating the azimuth of the illuminant are presented one is based on local estimates on smooth patches and the other method uses shading information along image contours the elevation of the illuminant and surface albedo are estimated from image statistics taking into consideration the effect of self shadowing with the estimated reflectance map parameters the authors then compute the surface shape using a procedure that implements the smoothness constraint by requiring the gradients of reconstructed density to be close to the gradients of the input image the algorithm is data driven stable updates the surface slope and height maps simultaneously and significantly reduces the residual errors in irradiance and integrability terms a hierarchical implementation of the algorithm is presented typical results on synthetic and images are given to illustrate the usefulness of the approach

an efficient and reliable residual type a posteriori error estimator for the signorini problem we derive a new a posteriori error estimator for the signorini problem it generalizes the standard residual type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non linearity remarkably these additional contact related terms vanish in the case of so called full contact we prove reliability and efficiency for two and three dimensional simplicial meshes moreover we address the case of non discrete gap functions numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two and three dimensional case for simplicial as well as for unstructured mixed meshes

a posteriori error estimations of residual type for signorini s problem this paper presents an a posteriori error analysis for the linear finite element approximation of the signorini problem in two space dimensions a posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained we perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error

a posteriori error estimates in the maximum norm for parabolic problems we derive a posteriori error estimates in the l infty 0 t l infty omega norm for approximations of solutions to linear parabolic equations using the elliptic reconstruction technique introduced by makridakis and nochetto and heat kernel estimates for linear parabolic problems we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations we then establish a posteriori bounds for a fully discrete backward euler finite element approximation the elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators

the generic bilinear calibration estimation problem we identify a very general recurring pattern in a number of well known problems in biological and machine vision many problems are of a peculiar double sided nature one attempts to estimate certain properties of the environment using a certain type of equipment and simultaneously one attempts to calibrate the same equipment on the structure of the environment at first sight this appears the kind of the chicken and the egg problem that might well prove to be insoluble however due to basic constraints that universally apply e g the world is only three dimensional a solution up to a certain class of ambiguity transformations often exists the more complicated the problem is the less important the remaining ambiguity will be at least in a relative sense many well known problems are special in that they can be cast in bilinear form sometimes after transformation or the introduction of dummy variables instances include photometric stereo photometric estimations e g of lightness local differential image operators a variety of photogrammetric problems etc it turns out that many of these problems and together these make up a large fraction of the generic problems in machine vision today can be cast in a simple universal framework this framework enables one to handle arbitrarily large that is not minimal consistent configurations noisy thus inconsistent date sets automatically the level at which prior information either of a deterministic or a statistical nature is used assumptions such as constant albedo rigidity uniform distributions etc is clearly separated as an additional typically nonlinear stage

maximum norm a posteriori error estimates for singularly perturbed elliptic reaction diffusion problems residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polyhedral domains standard finite element approximations are considered the error constants are independent of the diameters of mesh elements and the small perturbation parameter in our analysis we employ sharp bounds on the green s function of the linearized differential operator numerical results are presented that support our theoretical findings

topographic independent component analysis in ordinary independent component analysis the components are assumed to be completely independent and they do not necessarily have any meaningful order relationships in practice however the estimated independent components are often not at all independent we propose that this residual dependence structure could be used to define a topographic order for the components in particular a distance between two components could be defined using their higher order correlations and this distance could be used to create a topographic representation thus we obtain a linear decomposition into approximately independent components where the dependence of two components is approximated by the proximity of the components in the topographic representation