ambiguity in structure from motion sphere versus plane if 3d rigid motion can be correctly estimated from image sequences the structure of the scene can be correctly derived using the equations for image formation however an error in the estimation of 3d motion will result in the computation of a distorted version of the scene structure of computational interest are these regions in space where the distortions are such that the depths become negative because in order for the scene to be visible it has to lie in front of the image and thus the corresponding depth estimates have to be positive the stability analysis for the structure from motion problem presented in this paper investigates the optimal relationship between the errors in the estimated translational and rotational parameters of a rigid motion that results in the estimation of a minimum number of negative depth values the input used is the value of the flow along some direction which is more general than optic flow or correspondence for a planar retina it is shown that the optimal configuration is achieved when the projections of the translational and rotational errors on the image plane are perpendicular furthermore the projection of the actual and the estimated translation lie on a line through the center for a spherical retina given a rotational error the optimal translation is the correct one given a translational error the optimal rotational negative deptherror depends both in direction and value on the actual and estimated translation as well as the scene in view the proofs besides illuminating the confounding of translation and rotation in structure from motion have an important application to ecological optics the same analysis provides a computational explanation of why it is easier to estimate self motion in the case of a spherical retina and why shape can be estimated easily in the case of a planar retina thus suggesting that nature s design of compound eyes or panoramic vision for flying systems and camera type eyes for primates and other systems that perform manipulation is optimal

generalization of selberg s frac 3 16 theorem and affine sieve an analogue of the well known frac 3 16 lower bound for the first eigenvalue of the laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non elementary subgroup l of sl 2 z the proof in the case that the hausdorff of the limit set of l is bigger than frac 1 2 is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting in the case that delta is less than frac 1 2 we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function these spectral gaps are then applied to sieving problems on orbits of such groups

three dimensional ego motion estimation from motion fields observed with multiple cameras abstract in this paper we present a robust method to estimate the three dimensional ego motion of an observer moving in a static environment this method combines the optical flow fields observed with multiple cameras to avoid the ambiguity of 3 d motion recovery due to small field of view and small depth variation in the field of view two residual functions are proposed to estimate the ego motion for different situations in the non degenerate case both the direction and the scale of the three dimensional rotation and translation can be obtained in the degenerate case rotation can still be obtained but translation can only be obtained up to a scale factor both the number of cameras and the camera placement affect the accuracy of the estimated ego motion we compare different camera configurations through simulation some results of real world experiments are also given to demonstrate the benefits of our method

correspondence free structure from motion we present a novel approach for the estimation of 3d motion directly from two images using the radon transform the feasibility of any camera motion is computed by integrating over all feature pairs that satisfy the epipolar constraint this integration is equivalent to taking the inner product of a similarity function on feature pairs with a dirac function embedding the epipolar constraint the maxima in this five dimensional motion space will correspond to compatible rigid motions the main novelty is in the realization that the radon transform is a filtering operator if we assume that the similarity and dirac functions are defined on spheres and the epipolar constraint is a group action of rotations on spheres then the radon transform is a correlation integral we propose a new algorithm to compute this integral from the spherical fourier transform of the similarity and dirac functions generating the similarity function now becomes a preprocessing step which reduces the complexity of the radon computation by a factor equal to the number of feature pairs processed the strength of the algorithm is in avoiding a commitment to correspondences thus being robust to erroneous feature detection outliers and multiple motions

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

extracting structure from optical flow using the fast error search technique in this paper we present a globally optimal and computationally efficient technique for estimating the focus of expansion foe of an optical flow field using fast partial search for each candidate location on a discrete sampling of the image area we generate a linear system of equations for determining the remaining unknowns viz rotation and inverse depth we compute the least squares error of the system without actually solving the equations to generate an error surface that describes the goodness of fit across the hypotheses using fourier techniques we prove that given an n n flow field the foe and subsequently rotation and structure can be estimated in cal o n 2 log n operations since the resulting system is linear bounded perturbations in the data lead to bounded errors we support the theoretical development and proof of our technique with experiments on synthetic and real data through a series of experiments on synthetic data we prove the correctness robustness and operating envelope of our algorithm we demonstrate the utility of our technique by applying it for detecting obstacles from a monocular sequence of images

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

correspondence free determination of the affine fundamental matrix fundamental matrix estimation is a central problem in computer vision and forms the basis of tasks such as stereo imaging and structure from motion existing algorithms typically analyze the relative geometries of matched feature points identified in both projected views automated feature matching is itself a challenging problem results typically have a large number of false matches traditional fundamental matrix estimation methods are very sensitive to matching errors which led naturally to the application of robust statistical estimation techniques to the problem in this work an entirely novel approach is proposed to the fundamental matrix estimation problem instead of analyzing the geometry of matched feature points the problem is recast in the frequency domain through the use of integral projection showing how this is a reasonable model for orthographic cameras the problem now reduces to one of identifying matching lines in the frequency domain which most importantly requires no feature matching or correspondence information experimental results on both real and synthetic data are presented that demonstrate the algorithm is a practical technique for fundamental matrix estimation the behavior of the proposed algorithm is additionally characterized with respect to input noise feature counts and other parameters of interest

computation of the generalized inverse of a polynomial matrix and applications abstract the computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its laurent expansion of a nonregular polynomial matrix the proposed algorithm constitutes a generalization of the algorithm proposed by fragulis et al for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis

self calibration from image derivatives this study investigates the problem of estimating camera calibration parameters from image motion fields induced by a rigidly moving camera with unknown parameters where the image formation is modeled with a linear pinhole camera model the equations obtained show the flow to be separated into a component due to the translation and the calibration parameters and a component due to the rotation and the calibration parameters a set of parameters encoding the latter component is linearly related to the flow and from these parameters the calibration can be determined however as for discrete motion in general it is not possible to decouple image measurements obtained from only two frames into translational and rotational components geometrically the ambiguity takes the form of a part of the rotational component being parallel to the translational component and thus the scene can be reconstructed only up to a projective transformation in general for full calibration at least four successive image frames are necessary with the 3d rotation changing between the measurements the geometric analysis gives rise to a direct self calibration method that avoids computation of optical flow or point correspondences and uses only normal flow measurements new constraints on the smoothness of the surfaces in view are formulated to relate structure and motion directly to image derivatives and on the basis of these constraints the transformation of the viewing geometry between consecutive images is estimated the calibration parameters are then estimated from the rotational components of several flow fields as the proposed technique neither requires a special set up nor needs exact correspondence it is potentially useful for the calibration of active vision systems which have to acquire knowledge about their intrinsic parameters while they perform other tasks or as a tool for analyzing image sequences in large video databases

epipolar plane image analysis an approach to determining structure from motion we present a technique for building a three dimensional description of a static scene from a dense sequence of images these images are taken in such rapid succession that they form a solid block of data in which the temporal continuity from image to image is approximately equal to the spatial continuity in an individual image the technique utilizes knowledge of the camera motion to form and analyze slices of this solid these slices directly encode not only the three dimensional positions of objects but also such spatiotemporal events as the occlusion of one object by another for straight line camera motions these slices have a simple linear structure that makes them easier to analyze the analysis computes the three dimensional positions of object features marks occlusion boundaries on the objects and builds a three dimensional map of free space in our article we first describe the application of this technique to a simple camera motion and then show how projective duality is used to extend the analysis to a wider class of camera motions and object types that include curved and moving objects

model based brightness constraints on direct estimation of structure and motion we describe a direct method for estimating structure and motion from image intensities of multiple views we extend the direct methods of horn and weldon 1988 to three views adding the third view enables us to solve for motion and compute a dense depth map of the scene directly from image spatio temporal derivatives in a linear manner without first having to find point correspondences or compute optical flow we describe the advantages and limitations of this method which are then verified with experiments using real images

feature matching constrained by cross ratio invariance abstract the main aim of this work is to propose a new technique to solve the well known feature correspondence problem for motion estimation the problem is formulated as an optimization process whose energy function includes constraints based on projective invariance of cross ratio of five coplanar points starting from some approximated correspondences estimated by radiometric similarity for features with high directional variance optimal matches are obtained through an optimization technique the new contribution of this work consists of a matching process refining the raw measurements based on an energy function minimization technique converging to an optimal solution for most of the features by taking advantage of some good initial guess and in the use of cross ratio as geometrical invariant constraint to detect and correct the mismatches due to wrong radiometric similarity measures though the method is based on geometrical invariance of coplanar points it is not required that all features have to be coplanar or to preprocess the images to detect the planar regions experimental results are presented for real and synthetic images and the performance of the novel approach is evaluated on different image sequences and compared to well known techniques

structure from motion using line correspondences a theory is presented for the computation of three dimensional motion and structure from dynamic imagery using only line correspondences the traditional approach of corresponding microfeatures interesting points highlights corners high curvature points etc is reviewed and its shortcomings are discussed then a theory is presented that describes a closed form solution to the motion and structure determination problem from line correspondences in three views the theory is compared with previous ones that are based on nonlinear equations and iterative methods

discretization of riemannian manifolds applied to the hodge laplacian for kappa geq 0 and r 0 0 let bbb m n kappa r 0 be the set of all connected compact n dimensional riemannian manifolds such that k g leq kappa and inj m g geq r 0 we study the relation between the k rm th positive eigenvalue of the hodge laplacian on differential forms and the k rm th positive eigenvalue of the combinatorial laplacian associated to an open cover acting on v c ech cochains we show that for a fixed sufficiently small varepsilon 0 there exist positive constants c 1 and c 2 depending only on n kappa and varepsilon such that for any m in bbb m n kappa r 0 and for any varepsilon discretization x of m we have c 1 lambda k p x leq lambda k p m leq c 2 lambda k p x for any k leq k k depends on x moreover we find a lower bound for the spectrum of the combinatorial laplacian and a lower bound for the spectrum of the hodge laplacian

on the analysis of discrete linear time invariant singular systems the discrete singular equation over an interval can represent a two point boundary value problem or it can be considered as a dynamical relation developing forward in time a theory is provided by giving analytic solutions and discussing system properties in both cases that encompasses both interpretations the singular system fundamental matrix is used to provide analytic solutions to a time invariant discrete singular equation defined over an interval the two distinct cases in which the singular relation is interpreted as a two point boundary value problem or as a forward dynamical system on the interval are both considered also considered is the case in which the singular relation is considered as a backward dynamical system and the relationship between the index of nilpotence and the laurent series coefficients resulting from the solutions of certain state space equations is shown reachability and observability are discussed and the point is made that these properties are different depending on how the singular relation is interpreted

3 d motion estimation from motion field abstract several experiments suggest that the first stage of motion perception is the measurement of visual motion the result of this stage is called the motion field which assigns a velocity vector to each point in the image plane the second stage involves interpreting the motion field in terms of objects and motion in the three dimensional world recovering 3 d motion of the object from the motion field has been difficult owing to the nonlinear system of equations involved and the sensitivity of the system to noise the need for the stability of the system is essential as only the optical flow field can be recovered from a sequence of images which is at best a crude approximation to the motion field we define two sets of basic parameters which can be recovered from the motion field by solving a linear system of equations the relationship between the basic parameters and the motion parameter being one to one and linear we obtain a closed form solution for the 3 d motion parameter by solving a system of linear equations only we prove the correctness completeness and robustness of the approach and in that sense the problem of recovering the motion parameter from the motion field may be said to be solved we present the results of extensive experimentation with real and simulated image sequences

rotation recovery from spherical images without correspondences this paper addresses the problem of rotation estimation directly from images defined on the sphere and without correspondence the method is particularly useful for the alignment of large rotations and has potential impact on 3d shape alignment the foundation of the method lies in the fact that the spherical harmonic coefficients undergo a unitary mapping when the original image is rotated the correlation between two images is a function of rotations and we show that it has an so 3 fourier transform equal to the pointwise product of spherical harmonic coefficients of the original images the resolution of the rotation space depends on the bandwidth we choose for the harmonic expansion and the rotation estimate is found through a direct search in this 3d discretized space a refinement of the rotation estimate can be obtained from the conservation of harmonic coefficients in the rotational shift theorem a novel decoupling of the shift theorem with respect to the euler angles is presented and exploited in an iterative scheme to refine the initial rotation estimates experiments show the suitability of the method for large rotations and the dependence of the method on bandwidth and the choice of the spherical harmonic coefficients

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

on the cayley hamilton theorem for two dimensional systems an alternate proof of the 2 d cayley hamilton theorem is supplied together with an extension to derived polynomials

reconstruction of consistent shape from inconsistent data optimization of 2 raise0 5ex hbox scriptstyle 1 kern 0 1em kern 0 15em lower0 25ex hbox scriptstyle 2 d sketches although the 3d orientations of edges and surfaces are theoretically sufficient for reconstructing the 3d object shape this does not mean that the 3d object shape can actually be reconstructed specifying the edge and surface orientations is often overspecification and inconsistency may result if image data contain errors we propose a scheme of optimization to construct a consistent polyhedron shape from inconsistent data our optimization is achieved by solving a set of linear equations no searchers and iterations are necessary this technique is first applied to the problem of shape from motion and then to the 3d recovery based on the rectangularity hypothesis and the parallelism hypothesis we also present a strategy of heuristic reasoning on rectangularity and parallelism

optical flow constraints on deformable models with applications to face tracking optical flow provides a constraint on the motion of a deformable model we derive and solve a dynamic system incorporating flow as a hard constraint producing a model based least squares optical flow solution our solution also ensures the constraint remains satisfied when combined with edge information which helps combat tracking error accumulation constraint enforcement can be relaxed using a kalman filter which permits controlled constraint violations based on the noise present in the optical flow information and enables optical flow and edge information to be combined more robustly and efficiently we apply this framework to the estimation of face shape and motion using a 3d deformable face model this model uses a small number of parameters to describe a rich variety of face shapes and facial expressions we present experiments in extracting the shape and motion of a face from image sequences which validate the accuracy of the method they also demonstrate that our treatment of optical flow as a hard constraint as well as our use of a kalman filter to reconcile these constraints with the uncertainty in the optical flow are vital for improving the performance of our system

directions of motion fields are hardly ever ambiguous if instead of the full motion field we consider only the direction of the motion field due to a rigid motion what can we say about the three dimensional motion information contained in it this paper provides a geometric analysis of this question based solely on the constraint that the depth of the surfaces in view is positive the motivation behind this analysis is to provide a theoretical foundation for image constraints employing only the sign of flow in various directions and justify their utilization for addressing 3d dynamic vision problems it is shown that considering as the imaging surface the whole sphere independently of the scene in view two different rigid motions cannot give rise to the same directional motion field if we restrict the image to half of a sphere or an infinitely large image plane two different rigid motions with instantaneous translational and rotational velocities t 1 1 and t 2 2 cannot give rise to the same directional motion field unless the plane through t 1 and t 2 is perpendicular to the plane through 1 and 2 i e t 1 t 2 1 2 e 0 in addition in order to give practical significance to these uniqueness results for the case of a limited field of view we also characterize the locations on the image where the motion vectors due to the different motions must have different directions if 1 2 t 1 t 2 e 0 and certain additional constraints are met then the two rigid motions could produce motion fields with the same direction for this to happen the depth of each corresponding surface has to be within a certain range defined by a second and a third order surface similar more restrictive constraints are obtained for the case of multiple motions consequently directions of motion fields are hardly ever ambiguous a byproduct of the analysis is that full motion fields are never ambiguous with a half sphere as the imaging surface

random construction of riemann surfaces we develop a new approach for the study of typical riemann surfaces with high genus the method that we use is the construction of random riemann surfaces from oriented cubic graphs this construction enables us to get a control over the global geometry properties of compact riemann surfaces we use the theory of random regular graphs to show that almost all such surfaces have large first eigenvalues and large cheeger constants moreover a closer analysis of the probability space of oriented cubic graphs shows that on a typical surface there is a large embedded hyperbolic ball

motion estimation based on multiple local constraints and nonlinear smoothing abstract a new algorithm for estimating motion from image sequences is presented initial motion estimates are determined based on a least squares solution to a set of independent linear constraints on the motion at a pixel these initial estimates are then improved by a nonlinear smoothing operation the results of this algorithm are compared with those obtained by the horn schunck algorithm on a number of image sequences

coarse non amenability and covers with small eigenvalues given a closed riemannian manifold m and a virtual epimorphism pi 1 m twoheadrightarrow mathbb f 2 of the fundamental group onto a free group of rank 2 we construct a tower of finite sheeted regular covers left m n right n 0 infty of m such that 1 mn 0 as n this is the first example of such a tower which is not obtainable up to uniform quasi isometry or even up to uniform coarse equivalence by the previously known methods where 1 m is supposed to surject onto an amenable group

critical surface pairs and triplets in this paper we study situations where a perspective image motion field can have multiple interpretations we show that the ambiguity in interpretation of the three dimensional instantaneous motion of curved surfaces is at most three fold and can occur only in the case of certain hyperboloids of one sheet saddle surfaces and hyperbolic paraboloids we also present analytical expressions for the relationship among multiple interpretations

a general aperture problem for direct estimation of 3 d motion parameters the necessary and sufficient conditions that an object should satisfy so that motion can be uniquely determined by a direct method are discussed this direct method based on the temporal spatial gradient scheme can estimate the three dimensional 3 motion parameters of a rigid moving object from an image sequence by utilizing depth information of the object it is shown that the 3 d motion cannot be uniquely determined for only eight kinds of objects with special geometric structure and surface pattern

the feasibility of motion and structure from noisy time varying image velocity information this research addresses the problem of noise sensitivity inherent in motion and structure algorithms the motion and structure paradigm is a two step process first we measure image velocities and perhaps their spatial and temporal derivatives are obtained from time varying image intensity data and second we use these data to compute the motion of a moving monocular observer in a stationary environment under perspective projection relative to a single 3 d planar surface the first contribution of this article is an algorithm that uses time varying image velocity information to compute the observer s translation and rotation and the normalized surface gradient of the 3 d planar surface the use of time varying image velocity information is an important tool in obtaining a more robust motion and structure calculation the second contribution of this article is an extensive error analysis of the motion and structure problem any motion and structure algorithm that uses image velocity information as its input should exhibit error sensitivity behavior compatible with the results reported here we perform an average and worst case error analysis for four types of image velocity information full and normal image velocities and full and normal sets of image velocity and its derivatives these derivatives are simply the coefficients of a truncated taylor series expansion about some point in space and time the main issues we address here are just how sensitive is a motion and structure computation in the presence of noisy input or alternately how accurate must our image velocity information be how much and what type of input data is needed and under what circumstances is motion and structure feasible that is when can we be sure that a motion and structure computation will produce usable results we base our answers on a numerical error analysis we conduct for a large number of motions

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

recursive estimation of time varying motion and structure parameters we present a computational framework for recovering both first order motion parameters observer direction of translation and observer rotation second order motion parameters observer rotational acceleration and relative depth maps from time varying optical flow we recover translation speed and acceleration in units which are scaled relative to the distance to the object our assumption is that the observer rotational motion is no more than second order in other words observer motion is either constant or has at most constant acceleration we examine the effect of noise on the solution of the motion and structure parameters this ensemble of unknowns comprises a solution to the classical structure and motion from optic flow problem our complete framework utilizes a method for interpreting the bilinear image velocity equation by solving simple systems of linear equations since our noise analysis yields uncertainty measures for each parameter a kalman filter is employed to incrementally integrate new measurements as they become available as each additional frame in the sequence is processed we conclude by analysing this reduction of uncertainty over time as the system converges to a stable solution for both synthetic and real image sequences

optical flow using spatiotemporal filters a model is presented consonant with current views regarding the neurophysiology and psychophysics of motion perception that combines the outputs of a set of spatiotemporal motion energy filters to estimate image velocity a parallel implementation computes a distributed representation of image velocity a measure of image flow uncertainty is formulated preliminary results indicate that this uncertainty measure may be used to recognize ambiguity due to the aperture problem the model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for some patterns that have large differences in contrast at different spatial orientations

isoperimetric control of the spectrum of a compact hypersurface upper bounds for the eigenvalues of the laplace beltrami operator on a hypersurface bounding a domain in some ambient riemannian manifold are given in terms of the isoperimetric ratio of the domain these results are applied to the extrinsic geometry of isometric embeddings

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

the general state space model for a two dimensional linear digital system the general state space model for a 2 d linear digital system is presented a new definition of state transition matrix is given based on the definition it is easy to calculate the state transition matrix for any linear digital system the general response formula for a system follows simply from the definition a new definition of the characteristic function of a system and a theorem parallel to the cayley hamilton theorem are also given the presented results apply to any linear causal system

computation of the transfer function matrix and its laurent expansion of generalized two dimensional systems an algorithm is developed for the computation of the transfer function matrix of a two dimensional system which is given in its generalized form the algorithm is a recursion in terms of the original system matrix and does not require the inversion of a two variable polynomial matrix an algorithm for the evaluation of the laurent expansion of the inverse of a two variable polynomial matrix is also presented

transfer function matrix of 2 d systems a formula is given which allows the determination of the transfer function matrix of linear multivariable two dimensional 2 d systems directly in terms of the state transition matrix and the characteristic equation

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

computation of the inverse of a polynomial matrix and evaluation of its laurent expansion abstract an algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil se a due to mertzios 1984 in the case of general systems described in differential operator form recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix as well as for the coefficients of the characteristic polynomial of the polynomial matrix a simple method is also presented that allows the evaluation of the laurent expansion for the inverse of a polynomial matrix of any order specifically an arma model is given for the computation of the coefficient matrices of the laurent expansion in terms of the coefficient matrices of the polynomial matrix without inverting the latter the laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular or generalized systems

the singular general model of 2d systems and its solution a general singular model for 2d linear discrete systems with constant coefficients is defined necessary and sufficient conditions for the existence of a solution to the model are established the considerations of the author s previous work 1986 can be extended to singular models of 2d linear systems with variable coefficients

robust and direct estimation of 3 d motion and scene depth from stereo image sequences abstract in this paper we propose a robust algorithm that estimates 3 d motion parameters and scene depth with calibrated stereo image sequences in the direct method based on the brightness change constraint equation we show that the previous linear or least squares methods are prone to error about the structured noises which are dependent on the contents of image such as intensity edges depth discontinuities and occluded areas we also show that the proposed algorithm is robust to those error distributions with respect to the hierarchical motion estimation framework an effective method for determining the number of iterative refinements which affects the accuracy of 3 d motion estimation is also suggested by using the proposed robust estimation framework computer simulation with a synthetic image sequence and experiments with various real image sequences show that the proposed algorithm has improved the estimation accuracy compared with existing least squares methods

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

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

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

point matching under large image deformations and illumination changes to solve the general point correspondence problem in which the underlying transformation between image patches is represented by a homography a solution based on extensive use of first order differential techniques is proposed we integrate in a single robust m estimation framework the traditional optical flow method and matching of local color distributions these distributions are computed with spatially oriented kernels in the 5d joint spatial color space the estimation process is initiated at the third level of a gaussian pyramid uses only local information and the illumination changes between the two images are also taken into account subpixel matching accuracy is achieved under large projective distortions significantly exceeding the performance of any of the two components alone as an application the correspondence algorithm is employed in oriented tracking of objects

transfer function matrix of singular systems a formula is given which allows the determination of the transfer function matrix of linear multivariable singular systems i e systems of the form e dot x ax bu with e singular directly in terms of the matrices of the state space description and the characteristic polynomial without inverting a polynomial matrix an alternative closed form expression for the transfer function matrix in terms of the pencil a se is also given

on unified optical flow field abstract a new approach to motion analysis from a sequence of stereo images has been developed recently c q shu and y q shi a new approach to motion analysis from a sequence of stereo images technical report no 18 electronic imaging laboratory department of electrical and computer engineering new jersey institute of technology newark nj 1990 in which the optical flow determined by b k p horn and b g schunck determining optical flow artif intell 17 185 203 1981 is extended to a spatial image sequence resulting in a unified optical flow field uoff a set of equations is established to characterize the uoff another set of equations is derived from which 3d motion can be recovered based on the uoff there is no need for feature correspondence in the approach the recovered 3d motion is for a whole continuous field it is therefore a purely optical flow approach in this paper the new concept of the uoff is presented in a more general and compact manner this paper discusses two main aspects about the uoff firstly the brightness function of an image is considered as not only a function of time but also a function of sensor s spatial position the concept of imaging space is presented as an accurate description of the set of all possible brightness functions secondly the brightness invariance is recognized not only for the time variation but also for the space variation so that the brightness invariance equations for both time and space domains are established it is noted that the optical flow discussed in horn and schunck 1981 and the optical flow for a spatial image sequence developed in shu and shi 1990 are both special cases within the framework of the uoff

ends of locally symmetric spaces with maximal bottom spectrum let x be a symmetric space of non compact type and gnx a locally sym metric space then the bottom spectrum l1 gnx satisfies the inequality l1 gnx e l1 x we show that if equality l1 gnx l1 x holds then gnx has either one end which is necessarily of infinite volume or two ends one of infinite volume and another of finite vol ume in the latter case gnx is isometric to r 1 n endowed with a multi warped metric where n is compact

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

concurrent 3 d motion segmentation and 3 d interpretation of temporal sequences of monocular images the purpose of this study is to investigate a variational method for joint multiregion three dimensional 3 d motion segmentation and 3 d interpretation of temporal sequences of monocular images interpretation consists of dense recovery of 3 d structure and motion from the image sequence spatiotemporal variations due to short range image motion the method is direct insomuch as it does not require prior computation of image motion it allows movement of both viewing system and multiple independently moving objects the problem is formulated following a variational statement with a functional containing three terms one term measures the conformity of the interpretation within each region of 3 d motion segmentation to the image sequence spatiotemporal variations the second term is of regularization of depth the assumption that environmental objects are rigid accounts automatically for the regularity of 3 d motion within each region of segmentation the third and last term is for the regularity of segmentation boundaries minimization of the functional follows the corresponding euler lagrange equations this results in iterated concurrent computation of 3 d motion segmentation by curve evolution depth by gradient descent and 3 d motion by least squares within each region of segmentation curve evolution is implemented via level sets for topology independence and numerical stability this algorithm and its implementation are verified on synthetic and real image sequences viewers presented with anaglyphs of stereoscopic images constructed from the algorithm s output reported a strong perception of depth

depth from relative normal flows abstract most of the depth from image flow algorithms has to rely on either good initial guesses or some assumptions about the object surfaces to achieve solutions that agree with the physical world waxman and sinha point out that those restrictions can be relaxed if depth is computed from a relative image flow field since image flow determination is relatively much more difficult than normal flow determination it is of interest to develop an algorithm to recover depth from normal flows in this paper we have shown that similar results can be obtained from relative normal flow fields as from relative image flow fields we have implemented a normal flow estimation algorithm and applied our algorithm to recover depth from intensity images

optical flow 3d segmentation and interpretation a variational method with active curve evolution and level sets this study investigates a variational active curve evolution method for dense three dimensional 3d segmentation and interpretation of optical flow in an image sequence of a scene containing moving rigid objects viewed by a possibly moving camera this method jointly performs 3d motion segmentation 3d interpretation recovery of 3d structure and motion and optical flow estimation the objective functional contains two data terms for each segmentation region one based on the motion only equation which relates the essential parameters of 3d rigid body motion to optical flow and the other on the horn and schunck optical flow constraint it also contains two regularization terms for each region one for optical flow the other for the region boundary the necessary conditions for a minimum of the functional result in concurrent 3d motion segmentation by active curve evolution via level sets and linear estimation of each region essential parameters and optical flow subsequently the screw of 3d motion and regularized relative depth are recovered analytically for each region from the estimated essential parameters and optical flow examples are provided which verify the method and its implementation

the accuracy of the computation of optical flow and of the recovery of motion parameters the accuracy and the dependence on parameters of a general scheme for the analysis of time varying image sequences are discussed the approach is able to produce vector fields from which it is possible to recover 3 d motion parameters such as time to collision and angular velocity the numerical stability of the computed optical flow and the dependence of the recovery of 3 d motion parameters on spatial and temporal filtering is investigated by considering optical flows computed on subsampled images or along single scanlines it is also possible to recover 3 d motion parameters from reduced optical flows an adequate estimate of time to collision can be obtained from sequences of images with spatial resolution reduced to 128 128 pixels or from sequences of single scanlines passing near the focus of expansion the use of kalman filtering increases the accuracy and the robustness of the estimation of motion parameters the proposed approach seems to be able to provide not only a theoretical background but also practical tools that are adequate for the analysis of time varying image sequences

further remarks on the cayley hamilton theorem and leverrier s method for the matrix pencil se a some results for matrix pencils are extended to the singular case se a a singular leverrier s relation cayley hamilton theorem and newton s formula are given a finite series expansion for se a 1 is given in terms of the generalized tschirnhausen polynomials

leverrier s algorithm for singular systems an algorithm is presented which allows the computation of the transfer function of a singular system from its state space description without inverting a polynomial matrix this algorithm is an extension of the leverrier algorithm for the more general case of singular systems and it reduces the computational cost

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

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

on a constraint equation for the estimation of displacement rates in image sequences the commonly used constraint equation delta g sup t u g sub t 0 for the estimation of optical flow can only be justified by assumptions that are in general far too restrictive for image sequences of real world scenes b g schunck 1985 86 recently argued that a constraint equation for the estimation of what he called image flow has to include a term containing the divergence of this image flow without presenting however a stringent derivation based on perspective projection from 3 d scene space the present author derives a constraint equation based on a combination of perspective projection and notions from differential geometry in addition he demonstrates the quantitive effects of taking into account radiometric considerations based on the use of lambertian reflection properties and isotropic illumination in scene space

motion fields are hardly ever ambiguous there has been much concern with ambiguity in the recovery of motion and structure from time varying images i show here that the class of surfaces leading to ambiguous motion fields is extremely restricted only certain hyperboloids of one sheet and some degenerate forms qualify furthermore the viewer must be on the surface for it to lead to a potentially ambiguous motion field thus the motion field over an appreciable image region almost always uniquely defines the instantaneous translational and rotational velocities as well as the shape of the surface up to a scale factor

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

a theory of self calibration of a moving camera there is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement the epipolar transformation imposes two algebraic constraints on the camera calibration if two epipolar transformations arising from different camera displacements are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four the curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces or by a curve of degree six in the dual of the image plane the curve in the dual plane has one singular point of order three and three singular points of order two if three epipolar transformations are available then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration the two curves have a common singular point of order three experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation the real intersections of the two dual curves are found by locating the zeros of a function defined on the interval 0 2 the intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations

on the consistency of instantaneous rigid motion estimation instantaneous camera motion estimation is an important research topic in computer vision although in theory more than five points uniquely determine the solution in an ideal situation in practice one can usually obtain better estimates by using more image velocity measurements because of the noise present in the velocity measurements however the usefulness of using a large number of observations has never been analyzed in detail in this paper we formulate this problem in the statistical estimation framework we show that under certain noise models consistency of motion estimation can be established that is arbitrarily accurate estimates of motion parameters are possible with more and more observations this claim does not simply follow from the general consistency result for maximum likelihood estimates some experiments will be provided to verify our theory our analysis and experiments also indicate that many previously proposed algorithms are inconsistent under even very simple noise models

adjusting shape parameters using model based optical flow residuals we present a method for estimating the shape of a deformable model using the least squares residuals from a model based optical flow computation this method is built on top of an estimation framework using optical flow and image features where optical flow affects only the motion parameters of the model using the results of this computation our new method adjusts all of the parameters so that the residuals from the flow computation are minimized we present face tracking experiments that demonstrate that this method obtains a better estimate of shape compared to related frameworks

estimation of the velocity field of two dimensional deformable motion abstract for a deformable motion the velocities of neighboring points are different in general the traditional smoothness constraints which minimize the spatial derivatives of the velocity are not applicable a new way is proposed to integrate the motion fundamental constraint and the deformation linear constraint an iterative algorithm based on the maximum a posteriori map estimation criterion is proposed to estimate the velocity field of a two dimensional 2d deformable motion based on the statistical approach the relative weights of the two constraints used in the estimation process can be represented by a single parameter methods to avoid the critical problem of determining the region size for which the linear constraint holds are discussed the constraint propagation property is also used to overcome this problem by assuming that the flow field is analytic a promising application for optical flow techniques is to visualize the 2d heart motion quantitatively the 2d echocardiographic images taken by the short axis view are used in the experiments to study the performance of the proposed algorithm the results of using the method proposed by horn and schunck are also given for comparison

determining shape and motion from monocular camera determining the spatial motion of a moving camera from a video is a classical problem in computer vision the difficulty of this problem is that the flow pattern directly observable in the video is generally not the complete flow pattern induced by the motion but only the partial information of it which is known as the normal flow in this paper we present a direct method which neither requires the establishment of feature correspondences nor the recovery of optical flow between two image frames but we directly utilize all observable normal flow data to recover the camera motion we propose a two stage iterative algorithm to search the solution in the motion space in a coarse to fine framework the first stage involves the use of the direction part of the normal flow each of these normal flow data can provide a constrained solution space to the direction of motion the intersection of the motion solutions from all the available normal flow data can reduce the motion ambiguity to a certain extent we then use the globality of the rotational magnitude to all image positions to constrain the motion parameters further once the camera motion is determined the depth map of the imaged scene up to an arbitrary scale can be recovered experimental results on synthetic data and real images are provided to reveal the performance of the proposed method highlightswe determine camera motion and shape directly from normal flows our method does not require optical flows or feature correspondences two constraints are developed to determine camera motion experimental results are comparable to the state of the art methods

direct recovering of nth order surface structure using unified optical flow field abstract there are two different approaches for estimation of structure and or motion of objects from image sequences in the computer vision community today one is the optical flow approach and the other is the feature correspondence approach there are many difficulties and limitations encountered with the feature correspondence method while the optical flow method requires a substantial amount of extra calculations if the optical flow is to be computed as an intermediate step direct methods have been developed that use the optical flow approach but avoid computing the full optical flow field as an intermediate step for recovering structure and motion the unified optical flow field uoff theory was recently established it is an extension of the optical flow formulations to the temporal spatial domain in this paper a direct method is developed to reconstruct the curved surface structure characterized by an n degree polynomial equation based on the uoff it is a direct method since it also directly recovers surface structure without computing the quantities in the uoff however it is different from the direct method in that it is based on the theoretical framework of the unified spatial and temporal optical field instead of the optical flow field determined hence the structure is recovered from a given pair of stereo images instead of from a monocular image sequence two computer simulations a sphere which is a second order surface and an unbounded shaped surface which is a third order one are presented the fairly good results demonstrate the effectiveness of our direct method using the uoff some numerical consideration and error analysis are also included compared with the existing techniques of the direct method where only the first order surface can be recovered the new method has made significant progress in the recovery of curved structure

on the generalized cayley hamilton theorem the cayley hamilton theorem is extended to the case where two a and b n times n matrices are involved results similar to the regular case are presented the results are useful for problems that concern the analysis as well as the synthesis of singular systems for the definition of a function of two matrices f a b and in various other problems in linear algebra