uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

spherical prediction variance properties of central composite and box behnken designs for quadratic regression on the hypercube a single number criterion such as a g efficiency that is based on the prediction variance is often included as one of the criteria when selecting a response surface design as an alternative to the single number criterion approach the variance dispersion graph presented by giovannitti jensen and myers is a graphical technique for evaluating prediction variance properties throughout the experimental region three properties of interest are the maximum minimum and average spherical prediction variances given the spherical radius as an alternative to the computer based approach requiring an optimization algorithm to evaluate these properties the maximum minimum and spherical prediction variances for central composite and box behnken designs can be determined analytically and are functions only of the radius and the design parameters these functions yield the exact values of the spherical prediction variance properties of central composite and box behnke

what tasks can be performed with an uncalibrated stereo vision system this article studies the following question when is it possible to decide on the basis of images of point features observed by an imprecisely modeled two camera stereo vision system whether or not a prescribed robot positioning task has been precisely accomplished results are shown for three camera model classes injective cameras weakly calibrated projective cameras and uncalibrated projective cameras in particular given a weakly calibrated stereo pair it is shown that a positioning task can be precisely accomplished if and only if the task specification is invariant to projective transformations it is shown that injective and uncalibrated projective cameras can accomplish fewer tasks but are still able to accomplish tasks involving point coincidences the same formal framework is applied to the problem of determining the set of tasks which can be precisely accomplished with the well known position based control architecture it is shown that for any class of camera models the set of tasks which can be precisely accomplished using a position based control architecture is a subset of the complete set of tasks which can be decided on the set but includes all positioning tasks based on point coincidences two ways of extending the idea of position based control to accomplish more tasks are also presented

a large sample confidence region useful in characterizing the stationary point of a quadratic response surface a second order response surface model is often used to approximate the relationship between a response variable and a set of explanatory variables the nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model s second order terms since the elements of this matrix are estimated from the data however it follows that the eigenvalues are random variables hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point in this article it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface the delta method is used to construct an approximate confidence region for these eigenvalues box and draper 1987 gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs a simulation study was performed to

self calibration from one circular motion sequence and two images this paper describes a new method for self calibration of camera with constant internal parameters under circular motion using one sequence and two images captured with different camera orientations unlike the previous method in which three circular motion sequences are needed with known motion the new method computes the rotation angles and the projective reconstructions of the sequence and the images with circular constraint enforced which is called a circular projective reconstruction using a factorization based method it is then shown that the images of the circular points of each circular projective reconstruction can be readily obtained subsequently the image of the absolute conic and the calibration matrix of the camera can be determined experiments on both synthetic and real image sequence are given showing the accuracy and robustness of the new algorithm

a measure of rotatability for response surface designs a measure that quantifies the amount of rotatability in a given response surface design is introduced in this article the measure which is expressible as a percentage takes the value 100 if and only if the design is rotatable one of the main advantages of this measure is that it can be used to repair a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest four numerical examples are given to illustrate the applications of this measure

two dimensional lattice vesicles and polygons we consider a lattice model of two dimensional vesicles in which the boundary of the vesicle is the perimeter of a self avoiding polygon embeddable in the square lattice with fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon we derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram by deriving exact values ofthe numbers of polygons with m edges which enclose area n and analysing the resulting series we obtain the free energy the phase boundary and various scaling exponents and amplitudes numerically

self calibration under the cayley framework the cayley framework here is meant to tackle the vision problems under the infinite cayley transformation ict its main advantage lies in its numerical stability in this work the stratified self calibration under the cayley framework is investigated it is well known that the main difficulty of the stratified self calibration in multiple view geometry is to upgrade a projective reconstruction to an affine one in other words to estimate the unknown 3 vector of the plane at infinity called the normal vector to our knowledge without any prior knowledge about the scene or the camera motion the only available constraint on a moving camera with constant intrinsic parameters is the well known modulus constraint in the literature do other kinds of constraints exist if yes what they are how could they be used in this work such questions will be systematically investigated under the cayley framework our key contributions include 1 the original projective expression of the ict is simplified and a new projective expression is derived to make the upgrade easier from a projective reconstruction to a metric reconstruction 2 the constraints on the normal vector are systematically investigated for two views two constraints on the normal vector are derived one of them is the well known modulus constraint while the other is a new inequality constraint there are only these two constraints for two views for three views besides the constraints for two views two groups of new constraints are derived and each of them contains three constraints in other words there are 12 constraints in total for three views 3 based on our projective expression and these constraints a stratified cayley algorithm and a total cayley algorithm are proposed for the metric reconstruction from images it is experimentally shown that they both improve significantly the numerical stability of the classical algorithms compared with the global optimal algorithm under the infinite homography framework the cayley algorithms have comparable calibration accuracy but substantially reduce the computational load

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

motion estimation in image sequences using the deformation of apparent contours the problem of determining the camera motion from apparent contours or silhouettes of a priori unknown curved 3d surfaces is considered in a sequence of images it is shown how to use the generalized epipolar constraint on apparent contours one such constraint is obtained for each epipolar tangency point in each image pair an accurate algorithm for computing the motion is presented based on a maximum likelihood estimate it is shown how to generate initial estimates on the camera motion using only the tracked contours it is also shown that in theory the motion can be calculated from the deformation of a single contour the algorithm has been tested on several real image sequences for both euclidean and projective reconstruction the resulting motion estimate is compared to motion estimates calculated independently using standard feature based methods the motion estimate is also used to classify the silhouettes as curves or apparent contours the statistical evaluation shows that the technique gives accurate and stable results

ray saliency bottom up visual saliency for a rotating and zooming camera we extend the classical notion of computational visual saliency to multi image data collected using a stationary pan tilt zoom ptz camera by introducing the concept of consistency the requirement that the set of generated saliency maps should each assign the same saliency value to unique regions of the environment that appear in more than one image we show that processing each image independently will often fail to provide a consistent measure of saliency and that using an image mosaic to quantify saliency suffers from several drawbacks we then propose ray saliency and an immediate extension approximate ray saliency a mosaic free method for calculating a consistent measure of bottom up saliency experimental results demonstrating the effectiveness of the proposed approach are presented

statistical bias in 3 d reconstruction from a monocular video the present state of the art in computing the error statistics in three dimensional 3 d reconstruction from video concentrates on estimating the error covariance a different source of error which has not received much attention is the fact that the reconstruction estimates are often significantly statistically biased in this paper we derive a precise expression for the bias in the depth estimate based on the continuous differentiable version of structure from motion sfm many sfm algorithms or certain portions of them can be posed in a linear least squares ls framework ax b examples include initialization procedures for bundle adjustment or algorithms that alternately estimate depth and camera motion it is a well known fact that the ls estimate is biased if the system matrix a is noisy in sfm the matrix a contains point correspondences which are always difficult to obtain precisely thus it is expected that the structure and motion estimates in such a formulation of the problem would be biased existing results on the minimum achievable variance of the sfm estimator are extended by deriving a generalized cramer rao lower bound a detailed analysis of the effect of various camera motion parameters on the bias is presented we conclude by presenting the effect of bias compensation on reconstructing 3 d face models from rendered images

i optimal design of mixture experiments abstractin mixture experiments the factors under study are proportions of the ingredients of a mixture the special nature of the factors necessitates specific types of regression models and specific types of experimental designs although mixture experiments usually are intended to predict the response s for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients little research has been done concerning their i optimal design this is surprising given that i optimal designs minimize the average variance of prediction and therefore seem more appropriate for mixture experiments than the commonly used d optimal designs which focus on a precise model estimation rather than precise predictions in this article we provide the first detailed overview of the literature on the i optimal design of mixture experiments and identify several contradictions for the second order and the special cubic model we present continuous i optimal designs and contrast the

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

the 3d line motion matrix and alignment of line reconstructions we study the problem of aligning two 3d line reconstructions in projective affine metric or euclidean space we introduce the 6 6 3d line motion matrix that acts on plucker coordinates we characterize its algebraic properties and its relation to the usual 4 4 point motion matrix and propose various methods for estimating 3d motion from line correspondences based on cost functions defined in images or 3d space we assess the quality of the different estimation methods using simulated data and real images

projective reconstruction and invariants from multiple images this correspondence investigates projective reconstruction of geometric configurations seen in two or more perspective views and the computation of projective invariants of these configurations from their images a basic tool in this investigation is the fundamental matrix that describes the epipolar correspondence between image pairs it is proven that once the epipolar geometry is known the configurations of many geometric structures for instance sets of points or lines are determined up to a collineation of projective 3 space spl pscrsup 3 by their projection in two independent images this theorem is the key to a method for the computation of invariants of the geometry invariants of six points in spl pscrsup 3 and of four lines in spl pscrsup 3 are defined and discussed an example with real images shows that they are effective in distinguishing different geometrical configurations since the fundamental matrix is a basic tool in the computation of these invariants new methods of computing the fundamental matrix from seven point correspondences in two images or six point correspondences in three images are given

sequential updating of projective and affine structure from motion a structure from motion algorithm is described which recovers structure and camera position modulo a projective ambiguity camera calibration is not required and camera parameters such as focal length can be altered freely during motion the structure is updated sequentially over an image sequence in contrast to schemes which employ a batch process a specialisation of the algorithm to recover structure and camera position modulo an affine transformation is described together with a method to periodically update the affine coordinate frame to prevent drift over time we describe the constraint used to obtain this specialisation structure is recovered from image corners detected and matched automatically and reliably in real image sequences results are shown for reference objects and indoor environments and accuracy of recovered structure is fully evaluated and compared for a number of reconstruction schemes a specific application of the work is demonstrated affine structure is used to compute free space maps enabling navigation through unstructured environments and avoidance of obstacles the path planning involves only affine constructions

motion and structure of four points from one motion of a stereo rig with unknown extrinsic parameters we describe an analytical method for recovering 3d motion and structure of four or more points from one motion of a stereo rig the extrinsic parameters are unknown the motion of the stereo rig is also unknown because of the exploitation of information redundancy the approach gains over the traditional motion and structure from motion approach in that less features and less motions are required and thus more robust estimation of motion and structure can be obtained since the constraint on the rotation matrix is not fully exploited in the analytical method nonlinear minimization can be used to improve the result we propose to estimate directly the motion and structure by minimizing the difference between the measured positions and the predicted ones in the image plane both computer simulated data and real data are used to validate the proposed algorithm and very promising results are obtained

what can two images tell us about a third one this paper discusses the problem of predicting image features in an image from image features in two other images and the epipolar geometry between the three images we adopt the most general camera model of perspective projection and show that a point can be predicted in the third image as a bilinear function of its images in the first two cameras that the tangents to three corresponding curves are related by a trilinear function and that the curvature of a curve in the third image is a linear function of the curvatures at the corresponding points in the other two images our analysis relies heavily on the use of the fundamental matrix which has been recently introduced faugeras et al 1992 and on the properties of a special plane which we call the trifocal plane though the trinocular geometry of points and lines has been very recently addressed our use of the differential properties of curves for prediction is unique we thus completely solve the following problem given two views of an object predict what a third view would look like the problem and its solution bear upon several areas of computer vision stereo motion analysis and model based object recognition our answer is quite general since it assumes the general perspective projection model for image formation and requires only the knowledge of the epipolar geometry for the triple of views we show that in the special case of orthographic projection our results for points reduce to those of ullman and basri ullman and basri 1991 we demonstrate on synthetic as well as on real data the applicability of our theory

graphical tools for assessing the sensitivity of response surface designs to model misspecification when comparing different designs for an experiment optimality criteria and other measures often depend on the correctness of the assumed model in this article we develop and illustrate an approach for comparing designs given the potential effect of bias due to an underspecified model we illustrate this approach using graphical summaries of the expected mean squared error emse that allow assessment of the robustness of designs to model misspecification for response surface designs in cuboidal regions the excellent performance of the central composite designs when the quadratic model is correct is tempered when bias is present the box behnken designs demonstrate superiority in the presence of missing cubic terms for a screening design scenario the amount of bias present leads to different conclusions as to which design is best

self calibration of stationary cameras a new practical method is given for the self calibration of a camera in this method at least three images are taken from the same point in space with different orientations of the camera and calibration is computed from an analysis of point matches between the images the method requires no knowledge of the orientations of the camera calibration is based on the image correspondences only this method differs fundamentally from previous results by maybank and faugeras on self calibration using the epipolar structure of image pairs in the method of this paper there is no epipolar structure since all images are taken from the same point in space and so maybank and faugeras s method does not apply since the images are all taken from the same point in space determination of point matches is considerably easier than for images taken with a moving camera since problems of occlusion or change of aspect or illumination do not occur a non iterative calibration algorithm is given that works with any number of images an iterative refinement method that may be used with noisy data is also described the algorithm is implemented and validated on several sets of synthetic and real image data

matching two perspective views a computational approach to image matching is described it uses multiple attributes associated with each image point to yield a generally overdetermined system of constraints taking into account possible structural discontinuities and occlusions in the algorithm implemented intensity edgeness and cornerness attributes are used in conjunction with the constraints arising from intraregional smoothness field continuity and discontinuity and occlusions to compute dense displacement fields and occlusion maps along the pixel grids the intensity edgeness and cornerness are invariant under rigid motion in the image plane in order to cope with large disparities a multiresolution multigrid structure is employed coarser level edgeness and cornerness measures are obtained by blurring the finer level measures the algorithm has been tested on real world scenes with depth discontinuities and occlusions a special case of two view matching is stereo matching where the motion between two images is known the algorithm can be easily specialized to perform stereo matching using the epipolar constraint

invariants of six points and projective reconstruction from three uncalibrated images there are three projective invariants of a set of six points in general position in space it is well known that these invariants cannot be recovered from one image however an invariant relationship does exist between space invariants and image invariants this invariant relationship is first derived for a single image then this invariant relationship is used to derive the space invariants when multiple images are available this paper establishes that the minimum number of images for computing these invariants is three and the computation of invariants of six points from three images can have as many as three solutions algorithms are presented for computing these invariants in closed form the accuracy and stability with respect to image noise selection of the triplets of images and distance between viewing positions are studied both through real and simulated images applications of these invariants are also presented both the results of faugeras 1992 and hartley et al 1992 for projective reconstruction and sturm s method 1869 for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points

a computational approach for corner and vertex detection corners and vertexes are strong and useful features in computer vision for scene analysis stereo matching and motion analysis here we deal with the development of a computational approach to these important features we consider first a corner model and study analytically its behavior once it has been smoothed using the well known gaussian filter this allows us to clarify the behavior of some well knowncornerness measure based approaches used to detect these points of interest most of these classical approaches appear to detect points that do not correspond to the exact position of the corner a new scale space based approach that combines useful properties from the laplacian and beaudet s measure beaudet 1978 is then proposed in order to correct and detect exactly the corner position an extension of this approach is then developed to solve the problem of trihedral vertex characterization and detection in particular it is shown that a trihedral vertex has two elliptic maxima on extremal contrast surfaces if the contrast is sufficient and this allows us to classify trihedral vertexes in 2 classes vertex and vertex as corner the corner detection approach developed is applied to accurately detect trihedral vertexes using an additional test in order to make a distinction between trihedral vertexes and corners many experiments have been carried out using noisy synthetic data and real images containing corners and vertexes most of the promising results obtained are used to illustrate the experimental section of this paper

learning the dynamics and time recursive boundary detection of deformable objects we propose a principled framework for recursively segmenting deformable objects across a sequence of frames we demonstrate the usefulness of this method on left ventricular segmentation across a cardiac cycle the approach involves a technique for learning the system dynamics together with methods of particle based smoothing as well as nonparametric belief propagation on a loopy graphical model capturing the temporal periodicity of the heart the dynamic system state is a low dimensional representation of the boundary and the boundary estimation involves incorporating curve evolution into recursive state estimation by formulating the problem as one of state estimation the segmentation at each particular time is based not only on the data observed at that instant but also on predictions based on past and future boundary estimates although this paper focuses on left ventricle segmentation the method generalizes to temporally segmenting any deformable object

a optimality for regression designs abstract use is made of a result of horn amer j math 76 1954 620 630 on the existence of a symmetric matrix with prescribed diagonal elements and eigenvalues a necessary and sufficient condition is then given for the existence of an a optimal design for a regression experiment in the dorogovcev selected transl math statist probab 10 1971 35 41 setting

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

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 highly robust estimator through partially likelihood function modeling and its application in computer vision the authors present a highly robust estimator known as the model fitting mf estimator for general regression they explain that high robustness becomes possible through partially but completely modeling the unknown log likelihood function the partial modeling takes place by taking the bayesian statistical decision rule and a number of important heuristics into consideration while maximizing the log likelihood function applications include the automatic selection of multiple thresholds single rigid motion estimation or multiple rigid motion segmentation and estimation from two perspective views it is believed that the proposed mf estimator will aid in solving many robust estimation problems that demand an estimator that is either highly robust or capable of handling contaminated gaussian mixture models

stereo correspondence through feature grouping and maximal cliques the authors propose a method for solving the stereo correspondence problem the method consists of extracting local image structures and matching similar such structures between two images linear edge segments are extracted from both the left and right images each segment is characterized by its position and orientation in the image as well as its relationships with the nearby segments a relational graph is thus built from each image for each segment in one image as set of potential assignments is represented as a set of nodes in a correspondence graph arcs in the graph represent compatible assignments established on the basis of segment relationships stereo matching becomes equivalent to searching for sets of mutually compatible nodes in this graph sets are found by looking for maximal cliques the maximal clique best suited to represent a stereo correspondence is selected using a benefit function numerous results obtained with this method are shown

robust parameter estimation in computer vision estimation techniques in computer vision applications must estimate accurate model parameters despite small scale noise in the data occasional large scale measurement errors outliers and measurements from multiple populations in the same data set increasingly robust estimation techniques some borrowed from the statistics literature and others described in the computer vision literature have been used in solving these parameter estimation problems ideally these techniques should effectively ignore the outliers and measurements from other populations treating them as outliers when estimating the parameters of a single population two frequently used techniques are least median of squares lms p j rousseeuw j amer statist assoc 79 1984 pp 871 880 and m estimators robust statistics the approach based on influence functions f r hampel et al john wiley 1986 robust statistics p j huber john wiley 1981 lms handles large fractions of outliers up to the theoretical limit of 50 for estimators invariant to affine changes to the data but has low statistical efficiency m estimators have higher statistical efficiency but tolerate much lower percentages of outliers unless properly initialized while robust estimators have been used in a variety of computer vision applications three are considered here in analysis of range images images containing depth or x y z measurements at each pixel instead of intensity measurements robust estimators have been used successfully to estimate surface model parameters in small image regions in stereo and motion analysis they have been used to estimate parameters of what is called the fundamental matrix which characterizes the relative imaging geometry of two cameras imaging the same scene recently robust estimators have been applied to estimating a quadratic image to image transformation model necessary to create a composite mosaic image from a series of images of the human retina in each case a straightforward application of standard robust estimators is insufficient and carefully developed extensions are used to solve the problem

a voting based computational framework for visual motion analysis and interpretation most approaches for motion analysis and interpretation rely on restrictive parametric models and involve iterative methods which depend heavily on initial conditions and are subject to instability further difficulties are encountered in image regions where motion is not smooth typically around motion boundaries this work addresses the problem of visual motion analysis and interpretation by formulating it as an inference of motion layers from a noisy and possibly sparse point set in a 4d space the core of the method is based on a layered 4d representation of data and a voting scheme for affinity propagation the inherent problem caused by the ambiguity of 2d to 3d interpretation is usually handled by adding additional constraints such as rigidity however enforcing such a global constraint has been problematic in the combined presence of noise and multiple independent motions by decoupling the processes of matching outlier rejection segmentation and interpretation we extract accurate motion layers based on the smoothness of image motion and then locally enforce rigidity for each layer in order to infer its 3d structure and motion the proposed framework is noniterative and consistently handles both smooth moving regions and motion discontinuities without using any prior knowledge of the motion model

graphical assessment of the prediction capability of response surface designs measures of the quality of prediction at locations on the surface of a hyperspherc are presented these measures are used to form a graphical method of assessing the overall prediction capability of an experimental design throughout the region of interest a plot of the spherical variance and the maximum and minimum prediction variances for locations on a sphere against the radius of the sphere a vuriance dispersion graph is used to give a comprehensive picture of the behavior of the prediction variances throughout a region and hence of the quality of the predicted responses obtained with a particular design such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher

comparison of design optimality criteria of reduced models for response surface designs in the hypercube results from the comparison of d g and a efficiencies and the scaled average prediction variance iv criterion are presented for the central composite small composite notz hoke box draper and computer generated designs these design optimality criteria are evaluated over the cuboidal design region for three four and five design variables

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

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 probabilistic criterion to detect rigid point matches between two images and estimate the fundamental matrix the perspective projections of n physical points on two views stereovision are constrained as soon as n 8 however to prove in practice the existence of a rigid motion between two images more than 8 point matches are desirable in order to compensate for the limited accuracy of the matches in this paper we propose a computational definition of rigidity and a probabilistic criterion to rate the meaningfulness of a rigid set as a function of both the number of pairs of points n and the accuracy of the matches this criterion yields an objective way to compare say precise matches of a few points and approximate matches of a lot of points it gives a yes no answer to the question could this rigid points correspondence have occurred by chance since it guarantees that the expected number of meaningful rigid sets found by chance in a random distribution of points is as small as desired it also yields absolute accuracy requirements for rigidity detection in the case of non matched points and optimal values of n depending on the expected accuracy of the matches and on the proportion of outliers we use it to build an optimized random sampling algorithm that is able to detect a rigid motion and estimate the fundamental matrix when the set of point matches contains up to 90 of outliers which outperforms the best currently known methods like m estimators lmeds classical ransac and tensor voting

uniform asymptotics of area weighted dyck paths using the generalized method of steepest descents for the case of two coalescing saddle points we derive an asymptotic expression for the bivariate generating function of dyck paths weighted according to their length and their area in the limit of the area generating variable tending towards 1 the result is valid uniformly for a range of the length generating variable including the tricritical point of the model

area width scaling in generalised motzkin paths abstract we consider a generalised version of motzkin paths where horizontal steps have length l with l being a fixed positive integer we first give the general functional equation for the area width generating function of this model using a heuristic ansatz we then derive the area width scaling behaviour in terms of a scaling function in one variable for the special cases of dyck standard motzkin and schroder paths before generalising our approach to arbitrary l we then rigorously derive the tricritical scaling of schroder paths by applying the generalised method of steepest descents to the known exact solution for their area width generating function our results show that for dyck and schroder paths the heuristic scaling ansatz reproduces the rigorous results

a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction one of the fundamental purposes of response surface methodology is to develop a relatively simple linear model to explain a more complex phenomenon since the proposed model is only an approximation analysts almost always must address the potential of bias due to model misspecification this article proposes a graphical approach for evaluating response surface designs in terms of the mean squared error of prediction in particular this article develops measures of the maximum and minimum mean squared error of prediction over the surface of spheres through a region of interest these measures are then plotted against radius the resulting plots reveal at a glance how a response surface design performs in the presence of model misspecification these plots are used to investigate the performances of several common response surface designs

spherical mosaics with quaternions and dense correlation we describe an algorithm for generating spherical mosaics from a collection of images acquired from a common optical center the algorithm takes as input an arbitrary number of partially overlapping images an adjacency map relating the images initial estimates of the rotations relating each image to a specified base image and approximate internal calibration information for the camera the algorithm s output is a rotation relating each image to the base image and revised estimates of the camera s internal parameters our algorithm is novel in the following respects first it requires no user input our image capture instrumentation provides both an adjacency map for the mosaic and an initial rotation estimate for each image second it optimizes an objective function based on a global correlation of overlapping image regions third our representation of rotations significantly increases the accuracy of the optimization finally our representation and use of adjacency information guarantees globally consistent rotation estimates the algorithm has proved effective on a collection of nearly four thousand images acquired from more than eighty distinct optical centers the experimental results demonstrate that the described global optimization strategy is superior to non global aggregation of pair wise correlation terms and that it successfully generates high quality mosaics despite significant error in initial rotation estimates

using geometric constraints through parallelepipeds for calibration and 3d modeling this paper concerns the incorporation of geometric information in camera calibration and 3d modeling using geometric constraints enables more stable results and allows us to perform tasks with fewer images our approach is motivated and developed within a framework of semi automatic 3d modeling where the user defines geometric primitives and constraints between them in this paper first a duality that exists between the shape parameters of a parallelepiped and the intrinsic parameters of a camera is described then a factorization based algorithm exploiting this relation is developed using images of parallelepipeds it allows us to simultaneously calibrate cameras recover shapes of parallelepipeds and estimate the relative pose of all entities besides geometric constraints expressed via parallelepipeds our approach simultaneously takes into account the usual self calibration constraints on cameras the proposed algorithm is completed by a study of the singular cases of the calibration method a complete method for the reconstruction of scene primitives that are not modeled by parallelepipeds is also briefly described the proposed methods are validated by various experiments with real and simulated data for single view as well as multiview cases

cayley transformation and numerical stability of calibration equation the application of cayley transformation to enhance the numerical stability of camera calibration is investigated first a new calibration equation called the standard calibration equation is introduced using the cayley transformation and its analytical solution is obtained the standard calibration equation is equivalent to the classical calibration equation but it exhibits remarkable better numerical stability second a one parameter calibration family called the cayley calibration family which is equivalent to the standard calibration equation is obtained using also the cayley transformation and it is found that this family is composed of those infinite homographies whose rotation has the same axis with the rotation between the two given views the condition number of equations in the cayley calibration family varies with the parameter value and an algorithm to determine the best parameter value is provided third the generalized cayley calibration families equivalent to the standard calibration equation are also introduced via generalized cayley transformations an example of the generalized cayley transformations is illustrated called the s cayley calibration family as in the cayley calibration family the numerical stability of equations in a generalized cayley calibration family also depends on the parameter value in addition a more generic calibration family is also proposed and it is proved that the standard calibration equation the cayley calibration family and the s cayley calibration family are all some special cases of this generic calibration family

continuous stereo self calibration by camera parameter tracking this paper presents a consistent framework for continuous stereo self calibration based on a practical analysis of the sensitivity of stereo reconstruction to camera calibration uncertainties we identify important parameters for self calibration we evaluate different geometric constraints for estimation and tracking of these parameters bundle adjustment with reduced structure representation relating corresponding points in image sequences the epipolar constraint between stereo image pairs and trilinear constraints between image triplets continuous recursive calibration refinement is obtained with a robust adapted iterated extended kalman filter to achieve high accuracy physically relevant geometric optimization criteria are formulated in a gauss helmert type model the self calibration framework is tested on an active stereo system experiments with synthetic data as well as on natural indoor and outdoor imagery indicate that the different constraints are complementing each other and thus a method combining two of the above constraints is proposed while reduced order bundle adjustment gives by far the most accurate results and might suffice on its own in some environments the epipolar constraint yields instantaneous calibration that is not affected by independently moving objects in the scene hence it expedites and stabilizes the calibration process

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

higher order airy scaling in deformed dyck paths we introduce a deformed version of dyck paths ddp where additional to the steps allowed for dyck paths jumps orthogonal to the preferred direction of the path are permitted we consider the generating function of ddp weighted with respect to their half length area and number of jumps this represents the first example of an exactly solvable two dimensional lattice vesicle model showing a higher order multicritical point applying the generalized method of steepest descents we see that the associated two variable scaling function is given by the logarithmic derivative of a generalized higher order airy integral

an index of rotatability an index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second order designs values of the index are tabulated for the central composite designs for two to eight factors three designs of roquemore 1976 are assessed via the index comparisons are made with an entirely different index suggested by khuri 1988 it is concluded that both indexes are useful and sensibly consistent

algebraic functions for recognition in the general case a trilinear relationship between three perspective views is shown to exist the trilinearity result is shown to be of much practical use in visual recognition by alignment yielding a direct reprojection method that cuts through the computations of camera transformation scene structure and epipolar geometry moreover the direct method is linear and sets a new lower theoretical bound on the minimal number of points that are required for a linear solution for the task of reprojection the proof of the central result may be of further interest as it demonstrates certain regularities across homographics of the plane and introduces new view invariants experiments on simulated and real image data were conducted including a comparative analysis with epipolar intersection and the linear combination methods with results indicating a greater degree of robustness in practice and a higher level of performance in reprojection tasks

motion of points and lines in the uncalibrated case in the present paper we address the problem of computing structure and motion given a set point and or line correspondences in a monocular image sequence when the camera is not calibrated

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

response surface designs in flexible regions abstract when both variance and bias errors are present the choice of an experimental design to fit a response surface depends on the choice of the region of interest box and draper 1959 1963 and draper and lawrence 1965a 1966 investigated spherical and cuboidal regions respectively both of these regions can however be regarded as special cases of a more general flexible region family depending on a single parameter in this manner the sensitivity of the design criterion can be studied and more general results can be obtained the problem is set out here for k predictor variables and specific designs are provided for k 2 predictor variables when variance and bias errors are of roughly the same order a broad rule of thumb that emerges from this work is that for practical situations involving both variance and bias errors and for designs of the composite type used for fitting a second order surface the design points should be near the boundary of the region of interest over which a uni

determining the epipolar geometry and its uncertainty a review two images of a single scene object are related by the epipolar geometry which can be described by a 3 3 singular matrix called the essential matrix if images internal parameters are known or the fundamental matrix otherwise it captures all geometric information contained in two images and its determination is very important in many applications such as scene modeling and vehicle navigation this paper gives an introduction to the epipolar geometry and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty a well founded measure is proposed to compare these techniques projective reconstruction is also reviewed the software which we have developed for this review is available on the internet

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

heteroscedastic regression in computer vision problems with bilinear constraint we present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise i e each data point having a different covariance matrix the algorithm is motivated by the recovery of bilinear forms one of the fundamental problems in computer vision which appears whenever the epipolar constraint is imposed or a conic is fit to noisy data points we employ the errors in variables eiv model and show why already at moderate noise levels most available methods fail to provide a satisfactory solution the improved behavior of the new algorithm is due to two factors taking into account the heteroscedastic nature of the errors arising from the linearization of the bilinear form and the use of generalized singular value decomposition gsvd in the computations the performance of the algorithm is compared with several methods proposed in the literature for ellipse fitting and estimation of the fundamental matrix it is shown that the algorithm achieves the accuracy of nonlinear optimization techniques at much less computational cost

mixed response surface models with heterogeneous within block error variances the experimental error variance in a response surface model with a block effect has traditionally been assumed to be constant in some experimental situations however this variance may be different for the different blocks that compose the associated design this article discusses the analysis of a response surface model with a random block effect and heterogeneous error variances among the blocks the model includes interactions between the fixed polynomial effects and the block effect procedures for estimating and testing the fixed effects and the random effects in the model are described in particular anova based procedures for testing equality of the error variances and testing the random effects are introduced

new measurements and corner guidance for curve matching with probabilistic relaxation reliable curve matching is a difficult yet important problem in many vision based applications including image based modeling we describe in this paper two aspects of our research in this area a new algorithm for curve matching including lines within a probabilistic relaxation framework and an approach of incorporating previously matched points corners to guide curve matching we propose similarity invariant unary and binary measurements suitable for curves and introduce an additional measurement to model the uncertainty of the binary measurements the uncertainty measure is proven to be very important in computing the matching support from neighboring matches we also show how to use a set of previously matched points corners to guide the curve matching the role of the corner guidance is explicitly modeled by a set of unary measurements and a similarity function under the same relaxation framework preprocessing techniques contributing to the success of our curve matching techniques are also developed and discussed experiments with complex real scenes show that the rate of correct matching is higher than 98

projective structure from uncalibrated images structure from motion and recognition address the problem of reconstructing 3 d space in a projective framework from two or more views and the problem of artificially generating novel views of the scene from two given views reprojection the author describes an invariance relation that provides a new description of structure which the author calls projective depth that is captured by a single equation relating image point correspondences across two or more views and the homographics of two arbitrary virtual planes the framework is based on knowledge of correspondence of features across views is linear and extremely simple and the computations of structure readily extend to overdetermination using multiple views experimental results demonstrate a high degree of accuracy in both tasks reconstruction and reprojection

epipolar geometry and linear subspace methods a new approach to weak calibration this paper addresses the problem of estimating the epipolar geometry from point correspondences between two images taken by uncalibrated perspective cameras it is shown that jepson s and heeger s linear subspace technique for infinitesimal motion estimation can be generalized to the finite motion case by choosing an appropriate basis for projective space this yields a linear method for weak calibration the proposed algorithm has been implemented and tested on both real and synthetic images and it is compared to other linear and non linear approaches to weak calibration

stereo calibration from rigid motions we describe a method for calibrating a stereo pair of cameras using general or planar motions the method consists of upgrading a 3d projective representation to affine and to euclidean without any knowledge neither about the motion parameters nor about the 3d layout we investigate the algebraic properties relating projective representation to the plane at infinity and to the intrinsic camera parameters when the camera pair is considered as a moving rigid body we show that all the computations can be carried out using standard linear resolutions techniques an error analysis reveals the relative importance of the various steps of the calibration process projective to affine and affine to metric upgrades extensive experiments performed with calibrated and natural data confirm the error analysis as well as the sensitivity study performed with simulated data

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series