eliminating constraint drift in the numerical simulation of constrained dynamical systems by means of the udwadia kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift the basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion in order to avoid the drift typically experienced in constrained multibody simulations the embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense the formulation allows one to use a standard ode solver avoiding the need for iterative constraint stabilization the equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities the proposed method takes into account the finite word length of the computational environment and also accommodates possibly inconsistent initial conditions

2 1 2 d visual servoing with respect to unknown objects through a new estimation scheme of camera displacement classical visual servoing techniques need a strong a priori knowledge of the shape and the dimensions of the observed objects in this paper we present how the 2 1 2 d visual servoing scheme we have recently developed can be used with unknown objects characterized by a set of points our scheme is based on the estimation of the camera displacement from two views given by the current and desired images since vision based robotics tasks generally necessitate to be performed at video rate we focus only on linear algorithms classical linear methods are based on the computation of the essential matrix in this paper we propose a different method based on the estimation of the homography matrix related to a virtual plane attached to the object we show that our method provides a more stable estimation when the epipolar geometry degenerates this is particularly important in visual servoing to obtain a stable control law especially near the convergence of the system finally experimental results confirm the improvement in the stability robustness and behaviour of our scheme with respect to classical methods

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

polynomial methods for structure from motion the authors analyze the limitations of structure from motion sfm methods presented in the literature and propose the use of a polynomial system of equations with the unit quaternions representing rotation to recover sfm under perspective projection the authors combine the equations by the method of resultants with the maxima symbolic algebra system reducing the system to a single polynomial its real roots are then found with sturm sequences since this system has multiple solutions a hypothesize and verify scheme is used to eliminate incorrect ones the scheme diminishes the sensitivity of using polynomial equations the authors examine the effect of different rotation axes and angles on sfm accuracy and compare the performance of the algorithm to a few earlier approaches generally it is found that a large amount of motion is the most important factor in getting good sfm accuracy

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

motion parameter estimation from global flow field data presented are two methods for the determination of the parameters of motion of a sensor given the vector flow field induced by an imaging system governed by a perspective transformation of a rigid scene both algorithms integrate global data to determine motion parameters the first the flow circulation algorithm determines the rotational parameters the second the foe search algorithm determines the translational parameters of the motion independently of the first algorithm several methods for determining when the function has the appropriate form are suggested one method involves filtering the function by a collection of circular surround zero mean receptive fields the other methods project the function onto a linear space of quadratic polynomials and measures the distance between the two functions the error function for the first two methods is a quadratic polynomial of the candidate position yielding a very rapid search strategy

blended implicit methods for the numerical solution of dae problems recently a new approach for solving the discrete problems generated by the application of block implicit methods for the numerical solution of initial value problems for odes has been devised l brugnano blended block bvms b 3vms a family of economical implicit methods for odes j comput appl math 116 2000 41 62 l brugnano c magherini blended implementation of block implicit methods for odes appl numer math 42 2002 29 45 l brugnano d trigiante block implicit methods for odes in d trigiante ed recent trends in numerical analysis nova science publishers new york 2001 pp 81 105 this approach is based on the so called blended implementation of the methods giving corresponding blended implicit methods the latter have been implemented in the computational code bim l brugnano c magherini the bim code for the numerical solution of odes j comput appl math 164 165 2004 145 158 blended implicit methods are here extended to handle the numerical solution of dae problems resulting in a straightforward generalization of the basic approach

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

three dimensional modeling from two dimensional video this paper presents the surface based factorization method to recover three dimensional 3 d structure i e the 3 d shape and 3 d motion of a rigid object from a two dimensional 2 d video sequence the main ingredients of our approach are as follows 1 we describe the unknown shape of the 3 d rigid object by polynomial patches 2 projections of these patches in the image plane move according to parametric 2 d motion models 3 we recover the parameters describing the 3 d shape and 3 d motion from the 2 d motion parameters by factorizing a matrix that is rank 1 in a noiseless situation our method is simultaneously an extension and a simplification of the original factorization method of tomasi and kanade 1992 we track regions where the 2 d motion in the image plane is described by a single set of parameters avoiding the need to track a large number of pointwise features in general a difficult task then our method estimates the parameters describing the 3 d structure by factoring a rank 1 matrix not rank 3 as in tomasi and kanade this allows the use of fast iterative algorithms to compute the 3 d structure that best fits the data experimental results with real life video sequences illustrate the good performance of our approach

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

motion analysis of a homogeneously deformable object using subset correspondences abstract analysis of generalized motion parameters involves the estimation of translation rotation and the deformation parameters for a non rigid object most motion analysis methods require a preprocessing step of establishing the feature point correspondence between subsequent time frames such a preprocessing of data is computationally demanding and very often leads to erroneous results we explore the possibility of relaxing the above requirements for a homogeneously deformable object a computationally efficient method is proposed here that needs only subset correspondences between 3d feature points on the object at two different time instants instead of having explicit point correspondence the deformation parameters are obtained utilizing the changes in the euclidean distance for individual feature points belonging to different subsets under the proposed model the object may now be transformed into a rigid one utilizing the estimated deformation parameters the rotation parameters are estimated by considering the invariance properties of the principal moments simulation experiments are carried out to evaluate the performance of the proposed method the applicability of the method is also tested on a real data set

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

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

a diffusion mechanism for obstacle detection from size change information a mechanism fur the visual detection of obstacles is presented a new immediacy measure representing the imminence of collision between an object and a moving observer is defined a diffusion process on the image domain whose initial condition is determined by the motion field normal to the object s boundary is shown to converge asymptotically to the immediacy measure a network of locally connected cells derived from a finite difference approximation of the diffusion equation estimates the immediacy measure from normal velocity and boundary information provided by a motion measurement and segmentation stage the algorithm s performance on real image sequences is demonstrated

some properties of the e matrix in two view motion estimation in the eight point linear algorithm for determining 3d motion structure from two perspective views using point correspondences the e matrix plays a central role the e matrix is defined as a skew symmetrical matrix containing the translation components postmultiplied by a rotation matrix the authors show that a necessary and sufficient condition for a 3 3 matrix to be so decomposable is that one of its singular values is zero and the other two are equal several other forms of this property are presented some applications are briefly described

a correlation relaxation labeling framework for computing optical flow template matching from a new perspective optical flow estimation is discussed based on a model for time varying images more general than that implied by horn and schunk 1981 the emphasis is on applications where low contrast imagery nonrigid or evolving object patterns movement as well as large interframe displacements are encountered template matching is identified as having advantages over point correspondence and the gradient based approach in dealing with such applications the two fundamental uncertainties in feature matching whether template matching or feature point correspondences are discussed correlation template matching procedures are established based on likelihood measurement a method for determining optical flow is developed by combining template matching and relaxation labeling a number of candidate displacements for each template and their respective likelihood measures are determined then relaxation labeling is employed to iteratively update each candidate s likelihood by requiring smoothness within a motion field real cloud images from satellites are used to test the method

analysis of video image sequences using point and line correspondences abstract in this research we investigate the problem of analyzing time varying imagery using a feature based approach we assume a scenario where the imaged objects remain stationary while the camera moves the goal is to compute the structure of the imaged objects and the motion of the camera from a sequence of video images the proposed method exploits the principle of the invariance of rigid configurations during motion using the rigidity constraints we specify equations based on distance and angular invariance to compute the structural parameters of the imaged 3d objects independently of the camera s motion once the structural parameters are recovered the motion parameters can be computed the strength of our approach is in the decomposition of the computations of structure and motion and in the use of point and line correspondences simultaneously the main advantage of using both points and lines is that our approach is not limited to objects whose images have only a particular type of feature in abundance in this research we consider the special case of four points and one line which are the minimum feature sets required for computing the structure and motion parameters we then consider additional features in an overdetermined system of equations to improve the reliability and accuracy of the computation we present computer simulation results as well as results from real image sequences to demonstrate the algorithm s validity

power series with restricted coefficients and a root on a given ray we consider bounds on the smallest possible root with a specified argument o of a power series f z 1 n 1 a i z i with coefficients a i in the interval g g we describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when o 2 is rational when g 2 2 3 we show that the smallest disc containing two roots has radius g 1 1 coinciding with the smallest double real root possible for such a series it is clear from our computations that the behaviour is more complicated for smaller g we give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument we conclude by briefly discussing variants of the beta numbers where the defining integer sequence is generated by taking the nearest integer rather than the integer part we show that the conjugates a of these pseudobeta numbers either lie inside the unit circle or their reciprocals must be roots of 1 2 1 2 power series in particular we obtain the sharp inequality 3 2

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

motion detection using background constraints abstract we introduce a technique for detecting moving objects from a moving camera using the background constraint the camera is mounted on a platform that can both rotate and translate motion is detected by computing a mapping that corresponds pixels in successive images whenever possible assuming the background is planar provides knowledge that is used to recover projected local velocity u v background compensation can be used to eliminate the effects of rotation and translation on the mapping function to increase the robustness of this technique false motion caused by inaccuracies in sensor readings are eliminated this is achieved by using a morphological filter which consists of two successive operations erosion and dilation performed on the motion detection image experimental results with real images are presented which show the robustness of the proposed algorithm in detecting independently moving objects from a mobile platform

on the problem of correspondence in range data and some inelastic uses for elastic nets in this work the authors propose a novel method to obtain correspondence between range data across image frames using neural like mechanisms the method is computationally efficient and tolerant of noise and missing points elastic nets which evolved out of research into mechanisms to establish ordered neural projections between structures of similar geometry are used to cast correspondence as an optimization problem this formulation is then used to obtain approximations to the motion parameters under the assumption of rigidity inelasticity these parameter scan be used to recover correspondence experimental results are presented to establish the veracity of the scheme and the method is compared to earlier attempts in this direction

direct computation of qualitative 3 d shape and motion invariants structure from motion often refers to the computation of three dimensional structure from a matched sequence of images however a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition given matched images it is shown that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line segments in one image using this result local surface patches can be classified as convex concave cylindrical hyperbolic saddle point or planar at the same time the translational component of the optical flow from which the focus of expansion can be computed is obtained

interpretation of image flow rigid curved surfaces in motion a new method is described for interpreting image flow or optical flow in a small field of view produced by a rigidly moving curved surface the equations relating the shape and motion of the surface to the image flow are formulated these equations are solved to obtainexplicit analytic expressions for the motion orientation and curvatures of the surface in terms of the spatial derivatives up to second order of the image flow we state and prove somenew theoretical results concerning the existence of multiple interpretations numerical examples are given for some interesting cases where multiple solutions exist the solution method described here is simpler and more direct than previous methods the method and the representation described here are part of a unified approach for the interpretation of image motion in a variety of cases e g planar curved surfaces constant accelerated motion etc thus the representation and the method of analysis adopted here have some advantages in comparison with previous approaches

on kineopsis and computation of structure and motion this correspondence deals with the computation of structure and motion of rigid objects in space from image positions and optical flow a test for rigid motion of objects in space is introduced which yields a new formulation of the problem assuming a central projection model for the viewing system it is shown that image positions and optical flow at four points can achieve this perception

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

a comment on analysis of video image sequences using point and line correspondences abstract in this paper we would like to contest the results of y f wang n karandikar and j k aggarwal pattern recognition 24 1065 1085 1991 raising two fundamental claims a line does not contribute anything to recognition of motion parameters from two images four traceable points are not sufficient to recover motion parameters from two perspective projections to be constructive however we show that four traceable points are sufficient to recover motion parameters from two frames under orthogonal projection and that five points are sufficient to simplify the solution of the two frame problem under orthogonal projection to solving a linear equation system

second order bounds for linear recurrences with negative coefficients this paper introduces a generalization of fibonacci and pell polynomials in order to obtain optimal second order bounds for general linear recurrences with negative coefficients an important aspect of the derived bounds is that they are applicable and easily computable the results imply bounds on all entries in inverses of triangular matrices as well as on coefficients of reciprocals of power series

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

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

numerical solutions of chemical differential algebraic equations in this paper the solution for a chemical differential algebraic equation dae can be expanded up to arbitrary order using maple computer algebra systems first we calculate power series of the given equations system then transform it into pade series form which give an arbitrary order for solving chemical dae numerically

optimal motion and structure estimation the causes of existing linear algorithms exhibiting various high sensitivities to noise are analyzed it is shown that even a small pixel level perturbation may override the epipolar information that is essential for the linear algorithms to distinguish different motions this analysis indicates the need for optimal estimation in the presence of noise methods are introduced for optimal motion and structure estimation under two situations of noise distribution known and unknown computationally the optimal estimation amounts to minimizing a nonlinear function for the correct convergence of this nonlinear minimization a two step approach is used the first step is using a linear algorithm to give a preliminary estimate for the parameters the second step is minimizing the optimal objective function starting from that preliminary estimate as an initial guess a remarkable accuracy improvement has been achieved by this two step approach over using the linear algorithm alone

pose determination from line to plane correspondences existence condition and closed form solutions a class of pose determination problems in which the sensory data are lines and the corresponding reference data are planes is discussed the lines considered are different from edge lines in that they are not the intersection of boundary faces of the object the author describes a polynomial approach that does not require a priori knowledge about the object location closed form solutions for orthogonal parallel and coplanar feature configurations of critical importance in real applications are derived findings concerning the necessary and sufficient conditions under which the line to plane pose determination problem can be solved are described

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

vehicle type motion estimation from multi frame images a model for vehicle type motion which assumes that the motion is a rotation around an axis through the vehicle center followed by a forward translation along the main axis of the vehicle is proposed when the rotation and the amplitude of translation are constant this type of motion is shown to be equivalent to a constant camera centered motion this indicates that a constant motion in the conventional camera centered model which is commonly considered to be artificial can in fact be a reasonable model in real life it is shown that a constant vehicle type motion can be interpreted as a constant screw motion a linear algorithm for estimating constant vehicle type motion is presented and experiments using real scene images are included as an extension vehicle type motion with constant rotation and constantly accelerated translation is discussed

estimating motion and structure from correspondences of line segments between two perspective images presents an algorithm for determining 3d motion and structure from correspondences of line segments between two perspective images to the author s knowledge this paper is the first investigation of use of line segments in motion and structure from motion classical methods use their geometric abstraction namely straight lines but then three images are necessary for the motion and structure determination process in this paper the author shows that it is possible to recover motion from two views when using line segments the assumption used is that two matched line segments contain the projection of a common part of the corresponding line segment in space i e they overlap indeed this is what the author uses to match line segments between different views this assumption constrains the possible motion between two views to an open set in motion parameter space a heuristic consisting of maximizing the overlap leads to a unique solution both synthetic and real data have been used to test the proposed algorithm and excellent results have been obtained with real data containing a relatively large set of line segments

paraperspective affine it is shown that the set of all paraperspective images with arbitrary reference point and the set of affine images of a 3 d object are identical consequently all uncalibrated paraperspective images of an object can be constructed from a 3 d model of the object by applying an affine transformation to the model and every affine image of the object represents some uncalibrated paraperspective image of the object it follows that the paraperspective images of an object can be expressed as linear combinations of any two non degenerate images of the object when the image position of the reference point is given the parameters of the affine transformation and likewise the coefficients of the linear combinations satisfy two quadratic constraints conversely when the values of parameters are given the image position of the reference point is determined by solving a bi quadratic equation

least squares completions for nonlinear differential algebraic equations a method has been proposed for numerically solving lower dimensional nonlinear higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit runge kutta may not be appropriate this method is based on solving nonlinear dae derivative arrays using nonlinear singular least squares methods the theoretical foundations generality and limitations of this approach remain to be determined this paper carefully examines several key aspects of this approach the emphasis is on general results rather than specific results based on the structure of various applications

robust shape recovery from occluding contours using a linear smoother recovering the shape of an object from two views fails at occluding contours of smooth objects because the extremal contours are view dependent for three or more views shape recovery is possible and several algorithms have recently been developed for this purpose we present a new approach to the multiframe stereo problem that does not depend on differential measurements in the image which may be noise sensitive instead we use a linear smoother to optimally combine all of the measurements available at the contours and other edges in all of the images this allows us to extract a robust and reasonably dense estimate of surface shape and to integrate shape information from both surface markings and occluding contours results are presented which empirically show that in the presence of noise smoothing over more than three views reduces the error even when the epipolar curve is nonplanar

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

state space realizations of linear differential algebraic equation systems with control dependent state space this note addresses the derivation of state space realizations for the feedback control of linear high index differential algebraic equation systems that are not controllable at infinity in particular a class of systems is considered for which the underlying algebraic constraints involve the control inputs and thus a state space realization cannot be derived independently of the feedback controller the proposed methodology involves the design of a dynamic state feedback compensator such that the underlying algebraic constraints in the resulting modified system are independent of the new inputs a state space realization of the feedback modified system is then derived that can be used as the basis for controller synthesis

a multi frame approach to visual motion perception one of the main issues in the area of motion estimation given the correspondences of some features in a sequence of images is sensitivity to error in the input the main way to attack the problem as with several other problems in science and engineering is redundancy in the data up to now all the algorithms developed either used two frames or depended on assumptions about the motion or the shape of the scene we present in this paper an algorithm based on multiple frames that employs only the rigidity assumption is simple and mathematically elegant and experimentally proves to be a major improvement over the two frame algorithms the algorithm does minimization of the squared error which we prove equivalent to an eigenvalue minimization problem one of the side effects of this mean square method is that the algorithm can have a very descriptive physical interpretation in terms of the loaded spring model

what can be seen in a noisy optical flow field projected by a moving planar patch in 3d space abstract in this paper we would like to propose a brand new interpretation to the so called structure from motion sfm problem the optical flow field projected by a moving rigid planar patch in 3d space is our main consideration instead of just obtaining an explicit 3d motion pose solution like the old approaches did before we focus our attention on analyzing its error sensitivity uncertainty and ambiguity from another point of view our new method can handle the above error analysis easily as known well before the optical flow field projected by a 3d moving planar patch can be completely expressed by eight coefficients two for second order four for first order and two for zeroth order based on these flow coefficients easily determined by a linear regression method or other similar approaches the error sensitivity of 3d estimates can be analyzed quantitatively and qualitatively in a coarse to fine way the concepts of camera fixation and singular value decomposition svd play important roles in our analysis there are three goals for our experiments 1 to prove the correctness of the algorithms simulated image 2 to show the tendency of error sensitivity when the 3d poses of the target planar patch are varied in a controlled manner simulated image 3 to show that our analysis is workable in the real world application real world image

rigid body segmentation and shape description from dense optical flow under weak perspective we present an algorithm for identifying and tracking independently moving rigid objects from optical flow some previous attempts at segmentation via optical flow have focused on finding discontinuities in the flow field while discontinuities do indicate a change in scene depth they do not in general signal a boundary between two separate objects the proposed method uses the fact that each independently moving object has a unique epipolar constraint associated with its motion thus motion discontinuities based on self occlusion can be distinguished from those due to separate objects the use of epipolar geometry allows for the determination of individual motion parameters for each object as well as the recovery of relative depth for each point on the object the algorithm assumes an affine camera where perspective effects are limited to changes in overall scale no camera calibration parameters are required a kalman filter based approach is used for tracking motion parameters with time

anti hadamard matrices abstract an anti hadamard matrix may be loosely defined as a real 0 1 matrix which is invertible but only just let a be an invertible 0 1 matrix with eigenvalues i singular values i and inverse b bij we are interested in the four closely related problems of finding n mina i i n mina i i n maxa i j bij and n maxa ijb2ij then a is an anti hadamard matrix if it attains n we show that n n are between 2n 1 n 4 n 2 and c n 2 274 n where c is a constant c 2 274 n n 2 n 4 n 2 and c 5 172 n n 4n 2 n 4 n we also consider these problems when a is restricted to be a toeplitz triangular circulant or 1 1 matrix besides the obvious application to finding the most ill conditioned 0 1 matrices there are connections with weighing designs number theory and geometry

multiple interpretations of the shape and motion of objects from two perspective images the following problem is investigated given the position coordinates in two images of all points on an object obtained from two different camera positions under what conditions can there be more than one interpretation for the shape of the object and the transformation between the coordinate systems at the two camera positions it is shown that only certain hyperboloids of one sheet and their degeneracies such as hyperbolic paraboloids circular cylinders and intersecting planes that are viewed from a point on their surface can give rise to an ambiguity in the case of hyperboloids of one sheet and hyperbolic paraboloids there can be three possible solutions in the case of circular cylinders and intersecting planes there are at most two solutions the author gives the relationship among the multiple interpretations and determines them all in closed form in terms of the true solution

differential algebraic systems as differential equations on manifolds based on the theory of differential equations on manifolds existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications both the autonomous and nonautonomous case are considered moreover a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower dimensional manifolds this class includes systems for which the application of ode solvers is known to lead to difficulties finally some solution approach based on continuation techniques is outlined

explicit bounds for multidimensional linear recurrences with restricted coefficients this note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions the results imply optimal zero free polydisks for multivariable power series with 0 1 coefficients among the applications is a result that states that the optimal zero free polydisk has radius approximately 1 v 1 for large dimensions v

numerical solution of differential algebraic equations for constrained mechanical motion the two most popular formulations of the equations of constrained mechanical motion thedescriptor andstate space forms each have severe practical limitations in this paper we discuss and relate some proposed reformulations of the equations which have improved numerical properties

determining motion parameters using intensity guided range sensing abstract this paper describes a method for the direct computation of rotation and translation parameters of an object using multisensory data gathered from intensity and range sensors intensity images which can be gathered rapidly are used as the basis for guiding the extraction of a small set of correspondent points whose three dimensional coordinates can be determined by a range sensing device library model centered motion transformations are computed using a least squares approach object centered motion parameters are derived by computing the transformation between successive positions of the object results are presented for range and intensity images of synthetic and laboratory scenes

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

epipolar parameterization for reconstructing 3d rigid curve abstract this paper describes a new method for reconstructing a 3d rigid curve from a sequence of uncalibrated images using 31 epipolar parameterization the approach can be divided into the following two steps first a nonlinear discrete method is presented for point by point reconstruction of the curve instead of whole curve reconstruction projective and euclidean geometric tools are used second the parametric representation of the curve is defined by 31 b spline curves a linear method is proposed to interpolate the reconstructed points to obtain a complete curve thus it is proved that the 31 curve interpolation is equivalent to determining a set of control points of 3d regularized b spline curves it is shown that the 3d epipolar parameterization is an efficient method for reconstructing a 31 curve from image sequences experimental results are presented for real data

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

scene segmentation from visual motion using global optimization this paper presents results from computer experiments with an algorithm to perform scene disposition and motion segmentation from visual motion or optic flow the maximum a posteriori map criterion is used to formulate what the best segmentation or interpretation of the scene should be where the scene is assumed to be made up of some fixed number of moving planar surface patches the bayesian approach requires first specification of prior expectations for the optic flow field which here is modeled as spatial and temporal markov random fields and secondly a way of measuring how well the segmentation predicts the measured flow field the markov random fields incorporate the physical constraints that objects and their images are probably spatially continuous and that their images are likely to move quite smoothly across the image plane to compute the flow predicted by the segmentation a recent method for reconstructing the motion and orientation of planar surface facets is used the search for the globally optimal segmentation is performed using simulated annealing

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

a bound for linear recurrence relations with unbounded order this paper considers linear recurrence relations with unbounded order where the coefficients are restricted to intervals of negative real numbers the optimal inequalities under the given constraints exhibit a second order structure

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

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

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

the numerical solution of delay differential algebraic equations of retarded and neutral type in this paper we consider the numerical solution of initial value delay differential algebraic equations ddaes of retarded and neutral types with a structure corresponding to that of hessenberg daes we give conditions under which the ddae is well conditioned and show how the ddae is related to an underlying retarded or neutral delay ordinary differential equation dode we present convergence results for linear multistep and runge kutta methods applied to ddaes of index 1 and 2 and show how higher index hessenberg ddaes can be formulated in as stable a way as index 2 hessenberg ddaes we also comment on some practical aspects of the numerical solution of these problems

the development and comparison of robust methodsfor estimating the fundamental matrix this paper has two goals the first is to develop a variety of robust methods for the computation of the fundamental matrix the calibration free representation of camera motion the methods are drawn from the principal categories of robust estimators viz case deletion diagnostics m estimators and random sampling and the paper develops the theory required to apply them to non linear orthogonal regression problems although a considerable amount of interest has focussed on the application of robust estimation in computer vision the relative merits of the many individual methods are unknown leaving the potential practitioner to guess at their value the second goal is therefore to compare and judge the methods comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery in contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth a variety of error measures are examined the experiments allow a statistically satisfying and quasi optimal method to be synthesized which is shown to be stable with up to 50 percent outlier contamination and may still be used if there are more than 50 percent outliers performance bounds are established for the method and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined the results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers

contour matching using epipolar geometry matching features computed in images is an important process in multiview image analysis when the motion between two images is large the matching problem becomes very difficult in this paper we propose a contour matching algorithm based on geometric constraints with the assumption that the contours are obtained from images taken from a moving camera with static scenes we apply the epipolar constraint between two sets of contours and compute the corresponding points on the contours from the initial epipolar constraints obtained from corner point matching candidate contours are selected according to the epipolar geometry contour end point constraints and contour distance measures in order to reduce the possibility of false matches the number of match points on a contour is also used as a selection measure the initial epipolar constraint is refined from the matched sets of contours the algorithm can be applied to a pair or two pairs of images all of the processes are fully automatic and successfully implemented and tested with various real images

correspondence problem in image sequence analysis abstract an algorithm to address the correspondence problem in image sequence analysis is proposed the underlying philosophy of this algorithm is reductionist and is based on the assertion that measurements pertaining to motion that are obtained from the whole image sequence must be very similar to the measurements that are obtained from corresponding portions of the image sequence the algorithm has been extensively tested on synthesized data as well as on real image sequences it appears that the assumption underlying the proposed algorithm is quite useful and applicable to a large class of images the proposed algorithm has a high degree of inherent parallelism and is thus suited for implementation on parallel machines

a review on egomotion by means of differential epipolar geometry applied to the movement of a mobile robot the estimation of camera egomotion is an old problem in computer vision since the 1980s many approaches based on both the discrete and the differential epipolar constraint have been proposed the discrete case is used mainly in self calibrated stereoscopic systems whereas the differential case deals with a single moving camera this article surveys several methods for 3d motion estimation unifying the mathematics convention which are then adapted to the common case of a mobile robot moving on a plane experimental results are given on synthetic data covering more than 0 5 million estimations these surveyed algorithms have been programmed and are available on the internet

optimal visual motion estimation a note the problem of estimating 3d motion in an optimal manner using correspondences of features in two views is analyzed the importance of having an optimal estimator is twofold first for the estimation itself and second for the bound it offers on how much sensitivity one can expect from a two frame point based motion algorithm the optimal estimator turns out to be nonlinear and for that reason techniques that provide very good initial guesses for the iterative computation of the optimal estimator are developed

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

passive navigation as a pattern recognition problem the most basic visual capabilities found in living organisms are based on motion machine vision of course does not have to copy animal vision but the existence of reliably functioning vision modules in nature gives us some reason to believe that it is possible for an artificial system to work in the same or a similar way in this article it is argued that many navigational capabilities can be formulated as pattern recognition problems an appropriate retinotopic representation of the image would make it possible to extract the information necessary to solve motion related tasks through the recognition of a set of locations on the retina this argument is illustrated by introducing a representation of image motion by which an observer s egomotion could be derived from information globally encoded in the image motion field in the past the problem of determining a system s own motion from dynamic imagery has been considered as one of the classical visual reconstruction problems wherein local constraints have been employed to compute from exact 2 d image measurements correspondence optical flow the relative 3 d motion and structure of the scene in view the approach introduced here is based on new global constraints defined on local normal flow measurements the spatio temporal derivatives of the image intensity function classifications are based on orientations of normal flow vectors which allows selection of vectors that form global patterns in the image plane the position of these patterns is related to the 3 d motion of the observer and their localization provides the axis of rotation and the direction of translation the constraints introduced are utilized in algorithmic procedures formulated as search techniques these procedures are very stable since they are not affected by small perturbations in the image measurements as a matter of fact the solution to the two directions of translation and rotation is not affected as long as the measurement of the sign of the normal flow is correct

bayesian algorithms for simultaneous structure from motion estimation of multiple independently moving objects the problem of simultaneous structure from motion estimation for multiple independently moving objects from a monocular image sequence is addressed two bayesian algorithms are presented for solving this problem using the sequential importance sampling sis technique the empirical posterior distribution of object motion and feature separation parameters is approximated by weighted samples the first algorithm addresses the problem when only two moving objects are present a singular value decomposition svd based sample clustering algorithm is shown to be capable of separating samples related to different objects a pair of sis procedures is used to track the posterior distribution of the motion parameters in the second algorithm a balancing step is added into the sis procedure to preserve samples of low weights so that all objects have enough samples to propagate empirical motion distributions by using the proposed algorithms the relative motions of all the moving objects with respect to the camera can be simultaneously estimated both algorithms have been tested on synthetic and real image sequences improved results have been achieved

rigidity checking of 3d point correspondences under perspective projection an algorithm is described which rapidly verifies the potential rigidity of three dimensional point correspondences from a pair of two dimensional views under perspective projection the output of the algorithm is a simple yes or no answer to the question could these corresponding points from two views be the projection of a rigid configuration potential applications include 3d object recognition from a single previous view and correspondence matching for stereo or motion over widely separated views the rigidity checking problem is different from the structure from motion problem because it is often the case that two views cannot provide an accurate structure from motion estimate due to ambiguity and ill conditioning whereas it is still possible to give an accurate yes no answer to the rigidity question rigidity checking verifies point correspondences using 3d recovery equations as a matching condition the proposed algorithm improves upon other methods that fall under this approach because it works with as few as six corresponding points under full perspective projection handles correspondences from widely separated views makes full use of the disparity of the correspondences and is integrated with a linear algorithm for 3d recovery due to kontsevich 1993 results are given for experiments with synthetic and real image data a complete implementation of this algorithm is being made publicly available

recent developments in the numerical solution of differential algebraic systems abstract in this paper we survey some recent developments in the numerical solution of nonlinear differential algebraic equation dae systems of the form 0 f t y y where f y may be singular initial value problems in daes arise in a wide variety of applications including circuit and control theory chemical kinetics modeling of constrained mechanical systems fluid dynamics and robotics dae systems include standard form odes as a special case but they also include problems which are in many ways quite different from odes we explore some classes of initial value problems which can be solved by backward differentiation formulas and discuss some results on the order of convergence of implicit runge kutta methods applied to dae systems finding a consistent set of initial conditions is often a problem for dae systems arising in applications we outline some recent work on a general algorithm for finding consistent initial conditions finally we discuss some new developments in the numerical solution of dae boundary value problems

numerical solution of differential algebraic equations and computation of consistent initial boundary conditions boundary value methods for the solution of differential algebraic equations are described we consider both initial and boundary value problems and derive an algorithm that does not require additional information from the user but only the initial or boundary conditions needed in theory to obtain a unique solution

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

approximation methods for the consistent initialization of differential algebraic equations the algebraic constraints in a system of differential algebraic equations daes impose a consistency requirement on the initial values that can be difficult to satisfy in this thesis the consistency requirement is characterized by a system of equations an approximation method is introduced for these equations and the numerical solution of the resulting system is analyzed for certain important classes of daes finally some numerical experiments are described

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

epipolar line estimation and rectification for stereo image pairs the assumption that epipolar lines are parallel to image scan lines is made in many algorithms for stereo analysis if valid it enables the search for corresponding image features to be confined to one dimension and hence simplified an algorithm that generates a vertically aligned stereo pair by warped resampling is described the method uses grey scale image matching between the components of the stereo pair but confined to feature points

estimating the heading direction using normal flow if an observer is moving rigidly with bounded rotation then normal flow measurements i e the spatiotemporal derivatives of the image intensity function give rise to a constraint on the oberver s translation this novel constraint gives rise to a robust qualitative solution to the problem of recovering the observer s heading direction by providing an area where the focus of expansion lies if the rotation of the observer is large then the solution area is large too while small rotation causes the solution area to be small thus giving rise to a robust solution in the paper the relationship between the solution area and the rotation and translation vectors is studied and experimental results using synthetic and real calibrated image sequences are presented this work demonstrates that the algorithm developed in horn and weldon 1987 for the case of pure translation if appropriately modified results in a robust algorithm that works in the case of general rigid motion with bounded rotation subsequently it has the potential to replace expensive accelerometers inertial systems and inaccurate odometers in practical navigational systems for the problem of kinetic stabilization which is a prerequisite for any other navigational ability

double roots of 1 1 power series and related matters for a given collection of distinct arguments vec theta theta 1 ldots theta t multiplicities vec k k 1 ldots k t and a real interval i u v containing zero we are interested in determining the smallest r for which there is a power series f x 1 sum n 1 infty a i x i with coefficients a i in i and roots alpha 1 re 2 pi i theta 1 ldots alpha t re 2 pi i theta t of order k 1 ldots k t respectively we denote this by r vec theta vec k i we describe the usual form of the extremal series we give a sufficient condition which is also necessary when the extremal series possesses at least left sum i 1 t delta theta i k i right 1 coefficients strictly inside i where delta theta i is 1 or 2 as alpha i is real or complex we focus particularly on r theta 2 1 1 the size of the smallest double root of a 1 1 power series lying on a given ray of interest in connection with the complex analogue of work of boris solomyak on the distribution of the random series sum pm lambda n we computed the value of r theta 2 1 1 for the rationals theta in 0 1 2 of denominator less than fifty the smallest value we encountered was r 4 29 2 1 1 0 7536065594 for the one sided intervals i 0 1 and 1 0 the corresponding smallest values were r 11 30 2 0 1 8237251991 and r 1 3 2 1 0 8656332072

on the geometry of visual correspondence image displacement fields optical flow fields stereo disparity fields normal flow fields due to rigid motion possess a global geometric structure which is independent of the scene in view motion vectors of certain lengths and directions are constrained to lie on the imaging surface at particular loci whose location and form depends solely on the 3d motion parameters if optical flow fields or stereo disparity fields are considered then equal vectors are shown to lie on conic sections similarly for normal motion fields equal vectors lie within regions whose boundaries also constitute conics by studying various properties of these curves and regions and their relationships a characterization of the structure of rigid motion fields is given the goal of this paper is to introduce a concept underlying the global structure of image displacement fields this concept gives rise to various constraints that could form the basis of algorithms for the recovery of visual information from multiple views