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

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

robust estimation of egomotion from normal flow this paper investigates the robustness of an algorithm due to horn and weldon 1988 for recovery of egomotion from optic flow assuming only normal components of flow vectors are availableand that 3d angular velocity is known tight constraints can be constructed on the direction of translational motion or equivalently on the focus of expansion in practice however this is unacceptably restrictive some allowance must be made for uncertainty in angular velocity we show that the algorithm can indeed be extended to cope with such uncertainty with graceful degradation in accuracy of estimated position of the focus of expansion the shape of the error distribution depends on whether the focus of expansion is inside or outside the field of view if it is inside the error distribution is isotropic as it moves outside the distribution becomes increasingly anisotropic results from an implementation of the algorithm confirm the validity of the error bounds

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

egomotion analysis based on the frenet serret motion model in this paper we propose a new model frenet serret motion for the motion of an observer in a stationary environment this model relates the motion parameters of the observer to the curvature and torsion of the path along which the observer moves screw motion equations for frenet serret motion are derived and employed for geometrical analysis of the motion normal flow is used to derive constraints on the rotational and translational velocity of the observer and to compute egomotion by intersecting these constraints in the manner proposed in duric and aloimonos 1991 the accuracy of egomotion estimation is analyzed for different combinations of observer motion and feature distance we explain the advantages of controlling feature distance to analyze egomotion and derive the constraints on depth which make either rotation or translation dominant in the perceived normal flow field the results of experiments on real image sequences are presented

multiple solutions of periodic boundary value problems for first order differential equations by applying the well known leggett williams multiple fixed point theorem and fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions of periodic boundary value problems for first order differential equations meanwhile two examples are worked out to demonstrate the main results

comparametric equations with practical applications in quantigraphic image processing it is argued that hidden within the flow of signals from typical cameras through image processing to display media is a homomorphic filter while homomorphic filtering is often desirable there are some occasions where it is not thus cancellation of this implicit homomorphic filter is proposed through the introduction of an antihomomorphic filter this concept gives rise to the principle of quantigraphic image processing wherein it is argued that most cameras can be modeled as an array of idealized light meters each linearly responsive to a semi monotonic function of the quantity of light received integrated over a fixed spectral response profile this quantity depends only on the spectral response of the sensor elements in the camera a particular class of functional equations called comparametric equations is introduced as a basis for quantigraphic image processing these are fundamental to the analysis and processing of multiple images differing only in exposure the gamma correction of an image is presented as a simple example of a comparametric equation for which it is shown that the underlying quantigraphic function does not pass through the origin thus it is argued that exposure adjustment by gamma correction is inherently flawed and alternatives are provided these alternatives when applied to a plurality of images that differ only in exposure give rise to a new kind of processing in the amplitude domain the theoretical framework presented in this paper is applicable to the processing of images from nearly all types of modern cameras this paper is a much revised draft of a 1992 peer reviewed but unpublished report by the author entitled lightspace and the wyckoff principle

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

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

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

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

quantitative planar region detection this paper presents a means of segmenting planar regions from two views of a scene using point correspondences the initial selection of groups of coplanar points is performed on the basis of conservation of two five point projective invariants groups for which this invariant is conserved are assumed to be coplanar the five point correspondences are used to estimate a projectivity which is used to predict the change in position of other points assuming they lie on the same plane as the original four the variance in any points new position is then used to define a distance threshold between actual and predicted position which is used as a coplanarity test to find extended planar regions if two distinct planar regions can be found then a novel motion direction estimator suggests itself the projection of the line of intersection of two planes in an image may also be recovered an analytical error model is derived which relates image uncertainty in a corner s position to genuine perpendicular height of a point above a given plane in the world the model may be used for example to predict the performance of given stereo ground plane prediction system or a monocular drivable region detection system on and agv the model may also be used in reverse to determine the camera resolution required if a vehicle in motion is to resolve obstacles of a given height a given distance from it

q warping direct computation of quadratic reference surfaces we consider the problem of wrapping around an object of which two views are available a reference surface and recovering the resulting parametric flow using direct computations via spatio temporal derivatives the well known examples are affine flow models and eight parameter flow models both describing a flow field of a planar reference surface we extend those classic flow models to deal with a quadric reference surface and work out the explicit parametric form of the flow field as a result we derive a simple warping algorithm that maps between two views and leaves a residual flow proportional to the 3d deviation of the surface from a virtual quadric surface the applications include image morphing model building image stabilization and disparate view correspondence

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

singular boundary value problems on a half line singular boundary value problems of nonlinear second order differential equations on a half line are considered the main methods used are modified upper and lower solution techniques necessary and sufficient conditions for the existence of solutions to a class of sublinear singular boundary value problems are obtained

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

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

robust ego motion estimation and 3 d model refinement using surface parallax we present an iterative algorithm for robustly estimating the ego motion and refining and updating a coarse depth map using parametric surface parallax models and brightness derivatives extracted from an image pair given a coarse depth map acquired by a range finder or extracted from a digital elevation map dem ego motion is estimated by combining a global ego motion constraint and a local brightness constancy constraint using the estimated camera motion and the available depth estimate motion of the three dimensional 3 d points is compensated we utilize the fact that the resulting surface parallax field is an epipolar field and knowing its direction from the previous motion estimates estimate its magnitude and use it to refine the depth map estimate the parallax magnitude is estimated using a constant parallax model cpm which assumes a smooth parallax field and a depth based parallax model dbpm which models the parallax magnitude using the given depth map we obtain confidence measures for determining the accuracy of the estimated depth values which are used to remove regions with potentially incorrect depth estimates for robustly estimating ego motion in subsequent iterations experimental results using both synthetic and real data both indoor and outdoor sequences illustrate the effectiveness of the proposed algorithm

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

robust egomotion estimation from the normal flow using search subspaces we address the problem of egomotion estimation for a monocular observer moving under arbitrary translation and rotation in an unknown environment the method we propose is uniquely based on the spatio temporal image derivatives or the normal flow we introduce a search paradigm which is based on geometric properties of the normal flow field and consists in considering a family of search subspaces to estimate the egomotion parameters various algorithms are proposed within this framework in order to decrease the noise sensitivity of the estimation methods we use statistical tools based on robust regression theory finally we present and discuss a wide variety of experiments with synthetic and real images for various kinds of camera motion

structure of positive solution sets of semi positone singular boundary value problems abstract in this paper we first obtain some results on the structure of positive solution sets of some differential boundary value problems then by using these results we obtain some existence and multiplicity results for positive solutions of differential boundary value problems the method used to show the main results is that of global bifurcation theories

observability of 3d motion this paper examines the inherent difficulties in observing 3d rigid motion from image sequences it does so without considering a particular estimator instead it presents a statistical analysis of all the possible computational models which can be used for estimating 3d motion from an image sequence these computational models are classified according to the mathematical constraints that they employ and the characteristics of the imaging sensor restricted field of view and full field of view regarding the mathematical constraints there exist two principles relating a sequence of images taken by a moving camera one is the epipolar constraint applied to motion fields and the other the positive depth constraint applied to normal flow fields 3d motion estimation amounts to optimizing these constraints over the image a statistical modeling of these constraints leads to functions which are studied with regard to their topographic structure specifically as regards the errors in the 3d motion parameters at the places representing the minima of the functions for conventional video cameras possessing a restricted field of view the analysis shows that for algorithms in both classes which estimate all motion parameters simultaneously the obtained solution has an error such that the projections of the translational and rotational errors on the image plane are perpendicular to each other furthermore the estimated projection of the translation on the image lies on a line through the origin and the projection of the real translation the situation is different for a camera with a full 360 degree field of view achieved by a panoramic sensor or by a system of conventional cameras in this case at the locations of the minima of the above two functions either the translational or the rotational error becomes zero while in the case of a restricted field of view both errors are non zero although some ambiguities still remain in the full field of view case the implication is that visual navigation tasks such as visual servoing involving 3d motion estimation are easier to solve by employing panoramic vision also the analysis makes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rotation algorithms that do the opposite and algorithms that estimate all motion parameters simultaneously thus providing a sound framework for the observability of 3d motion finally the introduced framework points to new avenues for studying the stability of image based servoing schemes

structure from motion beyond the epipolar constraint the classic approach to structure from motion entails a clear separation between motion estimation and structure estimation and between two dimensional 2d and three dimensional 3d information for the recovery of the rigid transformation between different views only 2d image measurements are used to have available enough information most existing techniques are based on the intermediate computation of optical flow which however poses a problem at the locations of depth discontinuities if we knew where depth discontinuities were we could using a multitude of approaches based on smoothness constraints accurately estimate flow values for image patches corresponding to smooth scene patchess but to know the discontinuities requires solving the structure from motion problem first this paper introduces a novel approach to structure from motion which addresses the processes of smoothing 3d motion and structure estimation in a synergistic manner it provides an algorithm for estimating the transformation between two views obtained by either a calibrated or uncalibrated camera the results of the estimation are then utilized to perform a reconstruction of the scene from a short sequence of images the technique is based on constraints on image derivatives which involve the 3d motion and shape of the scene leading to a geometric and statistical estimation problem the interaction between 3d motion and shape allows us to estimate the 3d motion while at the same time segmenting the scene if we use a wrong 3d motion estimate to compute depth we obtain a distorted version of the depth function the distortion however is such that the worse the motion estimate the more likely we are to obtain depth estimates that vary locally more than the correct ones since local variability of depth is due either to the existence of a discontinuity or to a wrong 3d motion estimate being able to differentiate between these two cases provides the correct motion which yields the least varying estimated depth as well as the image locations of scene discontinuities we analyze the new constraints show their relationship to the minimization of the epipolar constraint and present experimental results using real image sequences that indicate the robustness of the method

when discrete meets differential we provide a theoretical proof showing that under a proportional noise model the discrete eight point algorithm behaves similarly to the differential eight point algorithm when the motion is small this implies that the discrete algorithm can handle arbitrarily small motion for a general scene as long as the noise decreases proportionally with the amount of image motion and the proportionality constant is small enough this stability result extends to all normalized variants of the eight point algorithm using simulations we show that given arbitrarily small motions and proportional noise regime the normalized eight point algorithms outperform their differential counterparts by a large margin using real data we show that in practical small motion problems involving optical flow these discrete structure from motion sfm algorithms also provide better estimates than their differential counterparts even when the motion magnitudes reach sub pixel level the better performance of these normalized discrete variants means that there is much to recommend them as differential sfm algorithms that are linear and normalized

on the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction gradient based approaches to the computation of optical flow often use a minimization technique incorporating a smoothness constraint on the optical flow field the author derives the most general form of such a smoothness constraint that is quadratic in first derivatives of the grey level image intensity function based on three simple assumptions about the smoothness constraint 1 it must be expressed in a form that is independent of the choice of cartesian coordinate system in the image 2 it must be positive definite and 3 it must not couple different component of the optical flow it is shown that there are essentially only four such constraints any smoothness constraint satisfying 1 2 or 3 must be a linear combination of these four possibly multiplied by certain quantities invariant under a change in the cartesian coordinate system beginning with the three assumptions mentioned above the author mathematically demonstrates that all best known smoothness constraints appearing in the literature are special cases of this general form and in particular that the weight matrix introduced by h h nagel is essentially modulo invariant quantities the only physically plausible such constraint

bifurcation techniques for lidstone boundary value problems abstract by employing bifurcation techniques this paper investigates the existence of nontrivial solutions single and multiple for lidstone boundary value problems depending on higher order derivatives our results improve on those in the literature

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

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

existence and uniqueness of positive solution for singular boundary value problem this paper presents two sufficient conditions which guarantee the existence and uniqueness of a positive solution for a class of singular boundary value problems

positive solutions of singular sub linear boundary value problems for fourth order and second order differential equation systems abstract this paper investigates the existence of positive solutions of singular sub linear boundary value problems for fourth order and second order differential equation systems first of all we establish some important lemmas then we define a partial ordering in c 2 a b c 4 a b c a b c 2 a b and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 0 1 c 0 1 positive solutions as well as c 3 0 1 c 1 0 1 positive solutions our nonlinearity f t x 1 x 2 x 3 g t x 1 x 2 may be singular at x 1 0 x 2 0 x 3 0 t 0 and or t 1

parametric image alignment using enhanced correlation coefficient maximization in this work we propose the use of a modified version of the correlation coefficient as a performance criterion for the image alignment problem the proposed modification has the desirable characteristic of being invariant with respect to photometric distortions since the resulting similarity measure is a nonlinear function of the warp parameters we develop two iterative schemes for its maximization one based on the forward additive approach and the second on the inverse compositional method as it is customary in iterative optimization in each iteration the nonlinear objective function is approximated by an alternative expression for which the corresponding optimization is simple in our case we propose an efficient approximation that leads to a closed form solution per iteration which is of low computational complexity the latter property being particularly strong in our inverse version the proposed schemes are tested against the forward additive lucas kanade and the simultaneous inverse compositional algorithm through simulations under noisy conditions and photometric distortions our forward version achieves more accurate alignments and exhibits faster convergence whereas our inverse version has similar performance as the simultaneous inverse compositional algorithm but at a lower computational complexity

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

structure of a class of singular boundary value problem with superlinear effect abstract this paper investigates the structure of solutions of singular boundary value problem with superlinear effect it is proved that the closure of positive solution set possesses a maximal subcontinuum c i e a maximal closed connected subset of solutions which comes from 0 and tends to 0 finally as a corollary the existence of multiple positive solutions and the behavior of solutions according to parameter are obtained

positive solutions for a class of 2nth order singular boundary value problems abstract we establish sufficient and necessary conditions for guaranteeing the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions for 2 n th order singular boundary value problems the arguments are based upon the fixed point theorem an the maximal principle as well as constructing lower and upper solutions our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1 meanwhile an example is worked out to demonstrate the main results

iterative extensions of the sturm triggs algorithm convergence and nonconvergence we give the first complete theoretical convergence analysis for the iterative extensions of the sturm triggs algorithm we show that the simplest extension siesta converges to nonsense results another proposed extension has similar problems and experiments with balanced iterations show that they can fail to converge or become unstable we present ciesta an algorithm that avoids these problems it is identical to siesta except for one simple extra computation under weak assumptions we prove that ciesta iteratively decreases an error and approaches fixed points with one more assumption we prove it converges uniquely our results imply that ciesta gives a reliable way of initializing other algorithms such as bundle adjustment a descent method such as gauss newton can be used to minimize the ciesta error combining quadratic convergence with the advantage of minimizing in the projective depths experiments show that ciesta performs better than other iterations

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

direct estimation of range flow on deformable shape from a video rate range camera a method of estimating range flow space displacement vector field on nonrigid as well as rigid objects from a sequence of range images is described this method can directly estimate the deformable motion parameters by solving a system of linear equations that are obtained from substituting a linear transformation of nonrigid objects expressed by the jacobian matrix into motion constraints based on an extension of the conventional scheme used in intensity image sequences the range flow is directly computed by substituting these estimated motion parameters into the linear transformation the algorithm is supported by experimental estimations of range flow on a sheet of paper a piece of cloth human skin and a rubber balloon being inflated using real range image sequences acquired by a video rate range camera

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

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

positive solutions for singular boundary value problems of a coupled system of differential equations by using fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions for singular boundary value problems of a coupled system of nonlinear ordinary differential equations

photometric motion the author compares the photometric effects of motion which is defined as the variation of a point s imaged intensity as a consequence of motion and the geometric effects of motion which is defined as the variation in projected surface geometry as a consequence of motion it is shown that photometric motion provides a cue to surface shape that is potentially as useful as that provided by geometric motion a simple technique for using this photometric motion information to extract both surface shape and reflectance is developed and a biological implementation is proposed how this photometric motion mechanism can be integrated with and used to enhance existing structure motion algorithms is discussed intensity information is sometimes even more important than geometric distortion when estimating the shape of a single continuous surface that is rotating relative to the observer s frame of reference

some global generalizations of the birkhoff kellogg theorem and applications on etudie certaines generalisations globales du theoreme de birkhoff kellogg et on donne des applications

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

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

spline based image registration the problem of image registration subsumes a number of problems and techniques in multiframe image analysis including the computation of optic flow general pixel based motion stereo correspondence structure from motion and feature tracking we present a new registration algorithm based on spline representations of the displacement field which can be specialized to solve all of the above mentioned problems in particular we show how to compute local flow global parametric flow rigid flow resulting from camera egomotion and multiframe versions of the above problems using a spline based description of the flow removes the need for overlapping correlation windows and produces an explicit measure of the correlation between adjacent flow estimates we demonstrate our algorithm on multiframe image registration and the recovery of 3d projective scene geometry we also provide results on a number of standard motion sequences

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

global structure of solutions for a class of two point boundary value problems involving singular and convex or concave nonlinearities abstract this paper investigates the following two point singular boundary value problems bvp x t x q x p t 0 1 x t 0 t 0 1 x 0 x 1 0 where q 0 and p 0 are fixed given numbers r 0 is a parameter the results obtained are the global structure of solutions and exact number of solutions when p 1 or p 1 and q is sufficiently small

nonlinear superlinear singular and nonsingular second order boundary value problems abstract many new existence theorems are presented for singular and nonsingular second order boundary value problems singularities at y 0 t 0 and t 1 will be discussed

generalized epidemic process and tricritical dynamic percolation the renowned general epidemic process describes the stochastic evolution of a population of individuals which are either susceptible infected or dead a second order phase transition belonging to the universality class of dynamic isotropic percolation lies between the endemic and pandemic behavior of the process we generalize the general epidemic process by introducing a fourth kind of individuals viz individuals which are weakened by the process but not yet infected this weakening gives rise to a mechanism that introduces a global instability in the spreading of the process and therefore opens the possibility of a discontinuous transition in addition to the usual continuous percolation transition the tricritical point separating the lines of first and second order transitions constitutes an independent universality class namely the universality class of tricritical dynamic isotropic percolation using renormalized field theory we work out a detailed scaling description of this universality class we calculate the scaling exponents in an epsilon expansion below the upper critical dimension d c 5 for various observables describing tricritical percolation clusters and their spreading properties in a remarkable contrast to the usual percolation transition the exponents beta and beta governing the two order parameters viz the mean density and the percolation probability turn out to be different at the tricritical point in addition to the scaling exponents we calculate for all our static and dynamic observables logarithmic corrections to the mean field scaling behavior at d c 5