a new structure from motion ambiguity demonstrates the existence of an approximate intrinsic ambiguity in euclidean structure from motion sfm which occurs as generically as the bas relief ambiguity but unlike it strengthens for scenes with more depth variation the ambiguity does not occur in projective sfm but the reasons for this make projective reconstructions more likely to have large errors our analysis gives a semiquantitative characterization of the least squares error surface over a domain complementary to that analyzed by jepson heeger and maybank as part of our analysis we show that the least squares error for infinitesimal motion the optical flow error gives a good approximation to the least squares error for moderate finite motions we propose that many high error local minima occur for epipoles in or near the image we also establish the existence of a new local minimum in minimizing over the rotation given the translation direction

estimating relative camera motion from the antipodal epipolar constraint this paper introduces a novel antipodal epipolar constraint on relative camera motion by using antipodal points which are available in large field of view cameras the translational and rotational motions of a camera are geometrically decoupled allowing them to be separately estimated as two problems in smaller dimensions we present a new formulation based on discrete camera motions which works over a larger range of motions compared to previous differential techniques using antipodal points the use of our constraints is demonstrated with two robust and practical algorithms one based on ransac and the other based on hough like voting as an application of the motion decoupling property we also present a new structure from motion algorithm that does not require explicitly estimating rotation it uses only the translation found with our methods finally experiments involving simulations and real image sequences will demonstrate that our algorithms perform accurately and robustly with some advantages over the state of the art

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

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

generalised absolute stability and sum of squares this paper introduces a general framework for analysing nonlinear systems using absolute stability theory and sum of squares programming the technique decomposes a vector field into a system with a polynomial vector field in feedback with a nonlinear memoryless term which is contained in a generalised sector inequality with polynomial bounds this decomposition can be used to model uncertainty in the nonlinearity or to bound difficult to analyse terms by simpler polynomial functions such as time varying non polynomial or higher order nonlinearities conditions for stability and regions of attraction are found using polynomial and lur e type lyapunov functions which generalise those used for the derivation of the multivariable circle and popov criteria in classical absolute stability the technique extends both absolute stability theory and the applicability of sum of squares programming the usefulness of the technique is demonstrated with illustrative examples

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

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

globally optimal estimates for geometric reconstruction problems we introduce a framework for computing statistically optimal estimates of geometric reconstruction problems while traditional algorithms often suffer from either local minima or non optimality or a combination of both we pursue the goal of achieving global solutions of the statistically optimal cost function our approach is based on a hierarchy of convex relaxations to solve non convex optimization problems with polynomials these convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation the technique is applied to a number of classical vision problems triangulation camera pose homography estimation and last but not least epipolar geometry estimation experimental validation on both synthetic and real data is provided in practice only a few relaxations are needed for attaining the global optimum

ground state entropy in potts antiferromagnets abstract the q state potts antiferromagnet exhibits nonzero ground state entropy s0 g q 0 for sufficiently large q on a given n vertex lattice or graph g and its n limit g we present exact calculations of the zero temperature partition function z g q t 0 and w g q where s 0 k b ln w for this model on a number of families g these calculations have interesting connections with graph theory since z g q t 0 p g q where the chromatic polynomial p g q gives the number of ways of coloring the vertices of the graph g such that no adjacent vertices have the same color generalizing q from z to c we determine the accumulation set b of the zeros of p g q which constitute the continuous loci of points on which w is nonanalytic the potts antiferromagnet has a zero temperature critical point at the maximal value qc where b crosses the real q axis in particular exact solutions for w and b are given for infinitely long finite width strips of various lattices in addition to their intrinsic interest these yield insight into the approach to the 2d thermodynamic limit some corresponding results are presented for the exact finite temperature potts free energy on families of graphs finally we present rigorous upper and lower bounds on w for 2d lattices

adaptive tracking control for a class of uncertain switched nonlinear systems this paper is concerned with the problem of tracking control for a class of switched nonlinear systems in lower triangular form with unknown functions and arbitrary switchings two classes of state feedback controllers are constructed by adopting the adaptive backstepping technique and both of them are designed by using the common lyapunov function clf method the first controller is designed under multiple adaptive laws then the second one is designed based on constructing a maximum common adaptive parameter which can overcome the problem of over parameterization of the first controllers it is shown that the designed state feedback controllers can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin finally simulation results are presented to show the effectiveness of the proposed approaches

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

exponential stabilisation of nonlinear parameter varying systems with applications to conversion flight control of a tilt rotor aircraft abstractthe exponential stabilisation problem of nonlinear parameter varying npv systems with input constraints is investigated unlike the existing npv related works this paper directly address

a multi frame structure from motion algorithm under perspective projection we present a fast robust algorithm for multi frame structure from motion from point features which works for general motion and large perspective effects the algorithm is for point features but easily extends to a direct method based on image intensities experiments on synthetic and real sequences show that the algorithm gives results nearly as accurate as the maximum likelihood estimate in a couple of seconds on an iris 10000 the results are significantly better than those of an optimal two image estimate when the camera projection is close to scaled orthographic the accuracy is comparable to that of the tomasi kanade algorithm and the algorithms are comparably fast the algorithm incorporates a quantitative theoretical analysis of the bas relief ambiguity and exemplifies how such an analysis can be exploited to improve reconstruction also we demonstrate a structure from motion algorithm for partially calibrated cameras with unknown focal length varying from image to image unlike the projective approach this algorithm fully exploits the partial knowledge of the calibration it is given by a simple modification of our algorithm for calibrated sequences and is insensitive to errors in calibrating the camera center theoretically we show that unknown focal length variations strengthen the effects of the bas relief ambiguity this paper includes extensive experimental studies of two frame reconstruction and the tomasi kanade approach in comparison to our algorithm we find that two frame algorithms are surprisingly robust and accurate despite some problems with local minima we demonstrate experimentally that a nearly optimal two frame reconstruction can be computed quickly by a minimization in the motion parameters alone lastly we show that a well known problem with the tomasi kanade algorithm is often not a significant one

structure from motion with wide circular field of view cameras this paper presents a method for fully automatic and robust estimation of two view geometry autocalibration and 3d metric reconstruction from point correspondences in images taken by cameras with wide circular field of view we focus on cameras which have more than 180deg field of view and for which the standard perspective camera model is not sufficient e g the cameras equipped with circular fish eye lenses nikon fc e8 183deg sigma 8 mm f4 ex 180deg or with curved conical mirrors we assume a circular field of view and axially symmetric image projection to autocalibrate the cameras many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping examples are the above mentioned fish eye lenses and properly assembled catadioptric cameras with conical mirrors we show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem this allows the use of efficient ransac robust estimation to find the image projection model the epipolar geometry and the selection of true point correspondences from tentative correspondences contaminated by mismatches real catadioptric cameras are often slightly noncentral we show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3d scene reconstruction noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors

ground state entropy of potts antiferromagnets bounds series and monte carlo measurements we report several results concerning w ensuremath lambda q mathrm exp s 0 k b the exponent of the ground state entropy of the potts antiferromagnet on a lattice ensuremath lambda first we improve our previous rigorous lower bound on w mathrm hc q for the honeycomb hc lattice and find that it is extremely accurate it agrees to the first 11 terms with the large q series for w mathrm hc q second we investigate the heteropolygonal archimedean 4 ensuremath cdot 8 2 lattice derive a rigorous lower bound on w 4 ensuremath cdot 8 2 q and calculate the large q series for this function to o y 12 where y 1 q ensuremath 1 remarkably these agree exactly to all 13 terms calculated we also report monte carlo measurements and find that these are very close to our lower bound and series third we study the effect of non nearest neighbor couplings focusing on the square lattice with next nearest neighbor bonds

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

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

exact two image structure from motion for two image structure from motion we present a simple exact expression for a least squares image reprojection error for finite motion that depends only on the motion optimal estimates of the structure and motion can be computed by minimizing this expression just over the motion parameters also we present a solution to the triangulation problem an exact explicit expression for the optimal structure estimate given the motion we identify a new ambiguity in recovering the structure for known motion we study the exact error s properties experimentally and demonstrate that it often has several local minima for forward or backward motion estimates our experiments also show that the reflected local minimum of oliensis 2001 and soatto et al 1998 occurs for large translational motions our exact results assume that the camera is calibrated and use a least squares image reprojection error that applies most naturally to a spherical imaging surface we approximately extend our approach to the standard least squares error in the image plane and uncalibrated cameras we present an improved version of the sampson error which gives better results experimentally

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

quasiconvex optimization for robust geometric reconstruction geometric reconstruction problems in computer vision are often solved by minimizing a cost function that combines the reprojection errors in the 2d images in this paper we show that for various geometric reconstruction problems their reprojection error functions share a common and quasiconvex formulation based on the quasiconvexity we present a novel quasiconvex optimization framework in which the geometric reconstruction problems are formulated as a small number of small scale convex programs that are readily solvable our final reconstruction algorithm is simple and has intuitive geometric interpretation in contrast to existing local minimization approaches our algorithm is deterministic and guarantees a predefined accuracy of the minimization result the quasiconvexity also provides an intuitive method to handle directional uncertainties and outliers in measurements for a large scale problem or in a situation where computational resources are constrained we provide an efficient approximation that gives a good upper bound but not global minimum on the reconstruction error we demonstrate the effectiveness of our algorithm by experiments on both synthetic and real data

convex design control for practical nonlinear systems this paper describes a new control scheme for approximately optimal control aoc of nonlinear systems convex control design convcd the key idea of convcd is to transform the approximate optimal control problem into a convex semi definite programming sdp problem contrary to the majority of existing aoc designs where the problem that is addressed is to provide a control design which approximates the performance of the optimal controller by increasing the controller complexity the proposed approach addresses a different problem given a controller of fixed complexity it provides a control design that renders the controller as close to the optimal as possible and moreover the resulted closed loop system stable two numerical examples are used to show the effectiveness of the method

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

nonlinear gain scheduling output feedback control for polynomial nonlinear systems subject to actuator saturation this paper investigates nonlinear gain scheduling control approaches for a class of polynomial nonlinear systems containing an output dependent vector field with input saturation using the polytopic differential inclusion and norm bounded differential inclusion ndi of saturation and dead zone functions the nonlinear plants are transformed into systems with measurable parameters for the polytopic differential inclusion description a quasi linear parameter varying quasi lpv output feedback controller will be sought for saturation control on the other hand the ndi model leads to a nonlinear fractional transformation nft output feedback controller for saturated nonlinear systems the quasi lpv and nft output feedback control synthesis conditions are derived in the forms of output dependent matrix inequalities they can be reformulated as sum of squares sos optimisations and solved efficiently using sos programming the proposed nonlinear gain scheduling saturation control approaches will be dem

on the optimization criteria used in two view motion analysis the three best known criteria in two view motion analysis are based respectively on the distances between points and their corresponding epipolar lines on the gradient weighted epipolar errors and on the distances between points and the re projections of their reconstructed points the last one has a better statistical interpretation but is significantly slower than the first two the author shows that given a reasonable initial guess of the epipolar geometry the last two criteria are equivalent when the epipoles are at infinity and differ from each other only a little even when the epipoles are in the image as shown experimentally the first two criteria are equivalent only when the epipoles are at infinity and when the observed object scene has the same scale in the two images this suggests that the second criterion is sufficient in practice because of its computational efficiency experiments with several thousand computer simulations and four sets of real data confirm the analysis

fast and accurate algorithms for projective multi image structure from motion we describe algorithms for computing projective structure and motion from a multi image sequence of tracked points the algorithms are essentially linear work for any motion of moderate size and give accuracies similar to those of a maximum likelihood estimate they give better results than the factorization approach of sturm and triggs 1996 and are equally fast and they are much faster than bundle adjustment our experiments show that the iterated sturm triggs approach often fails for linear camera motions in addition we study experimentally the common situation where the calibration is fixed and approximately known comparing the projective versions of our algorithms to mixed projective euclidean strategies we clarify the nature of dominant plane compensation showing that it can be considered a small translation approximation rather than an approximation that the scene is planar we show that projective algorithms accurately recover the projected inverse depths and homographies despite the possibility of transforming the structure and motion by a projective transformation

sos based solution approach to polynomial lpv system analysis and synthesis problems based on sum of squares sos decomposition we propose a new solution approach for polynomial linear parameter varying lpv system analysis and control synthesis problems instead of solving matrix variables over positive cone the sos approach tries to find a suitable decomposition to verify the positiveness of given polynomials the complexity of the sos based numerical method is polynomial of the problem size and is computationally attractive this approach also leads to more accurate solutions to lpv systems than most existing relaxation methods several examples have been used to demonstrate benefits of the sos based solution approach

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly