feature matching in stereo images encouraging uniform spatial distribution finding feature correspondences between a pair of stereo images is a key step in computer vision for 3d reconstruction and object recognition in practice a larger number of correct correspondences and a higher percentage of correct matches are beneficial previous researches show that the spatial distribution of correspondences are also very important especially for fundamental matrix estimation so far no existing feature matching method considers the spatial distribution of correspondences in our research we develop a new algorithm to find good correspondences in all the three aspects mentioned i e larger number of correspondences higher percentage of correct correspondences and better spatial distribution of correspondences our method consists of two processes an adaptive disparity smoothing filter to remove false correspondences based on the disparities of neighboring correspondences and a matching exploration algorithm to find more correspondences according to the spatial distribution of correspondences so that the correspondences are as uniformly distributed as possible in the images to find correspondences correctly and efficiently we incorporate the cheirality constraint under an epipole polar transformation together with the epipolar constraint to predict the potential location of matching point experiments demonstrate that our method performs very well on both the number of correct correspondences and the percentage of correct correspondences and the obtained correspondences are also well distributed over the image space highlightswe propose a feature matching method incorporating epipolar geometry estimation uniformly distributed correspondences improve epipolar geometry estimation we propose an adaptive weighted median filter for removing false correspondences good results are obtained on both epipolar geometry estimation and correspondences

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

computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

estimating the fundamental matrix via constrained least squares a convex approach in this paper a new method for the estimation of the fundamental matrix from point correspondences in stereo vision is presented the minimization of the algebraic error is performed while taking explicitly into account the rank two constraint on the fundamental matrix it is shown how this nonconvex optimization problem can be solved avoiding local minima by using recently developed convexification techniques the obtained estimate of the fundamental matrix turns out to be more accurate than the one provided by the linear criterion where the rank constraint of the matrix is imposed after its computation by setting the smallest singular value to zero this suggests that the proposed estimate can be used to initialize nonlinear criteria such as the distance to epipolar lines and the gradient criterion in order to obtain a more accurate estimate of the fundamental matrix

estimating three dimensional motion parameters of a rigid planar patch ii singular value decomposition we show that the three dimensional 3 d motion parameters of a rigid planar patch can be determined by computing the singular value decomposition svd of a 3 3 matrix containing the eight so called pure parameters furthermore aside from a scale factor for the translation parameters the number of solutions is either one or two depending on the multiplicity of the singular values of the matrix

registration based reconstruction of four dimensional cone beam computed tomography we present a new method for reconstruction of 4 d cone beam computed tomography from an undersampled set of x ray projections the novelty of the proposed method lies in utilizing optical flow based registration to facilitate that each temporal phase is reconstructed from the full set of acquired projections the reconstruction of each phase thus exhibits limited aliasing despite significant intra phase undersampling the method is fully self contained initially an approximate 4 d volume is reconstructed and an inter phase registration based hereon a subsequent reconstruction pass integrates the optical flow estimation in a cost function formulation in which the x ray projections from all temporal phases are considered for the reconstruction of each individual phase quantitative and qualitative evaluations were performed through reconstruction of both a numerical phantom and a clinical dataset the obtained reconstructions are compared to the state of the art alternatives of total variation regularization and prior image constrained compressed sensing our studies show that the proposed method is the better overall compromise in the depiction of both moving and stationary anatomical structures

globally convergent autocalibration using interval analysis we address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed in general to address this problem we have developed a method where an interval branch and bound method is employed for numerical minimization thanks to the properties of interval analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy and the only input information it requires from the user are a set of point correspondences and a search interval the cost function is based on the huang faugeras constraint of the essential matrix a recently proposed interval extension based on bernstein polynomial forms has been investigated to speed up the search for the solution finally experimental results are presented

epipolar geometry from profiles under circular motion addresses the problem of motion estimation from profiles apparent contours of an object rotating on a turntable in front of a single camera a practical and accurate technique for solving this problem from profiles alone is developed it is precise enough to reconstruct the shape of the object no correspondences between points or lines are necessary symmetry of the surface of revolution swept out by the rotating object is exploited to obtain the image of the rotation axis and the homography relating epipolar lines in two views robustly and elegantly these together with geometric constraints for images of rotating objects are used to obtain first the image of the horizon which is the projection of the plane that contains the camera centers and then the epipoles thus fully determining the epipolar geometry of the image sequence the estimation of this geometry by this sequential approach avoids many of the problems found in other algorithms the search for the epipoles by far the most critical step is carried out as a simple 1d optimization parameter initialization is trivial and completely automatic at all stages after the estimation of the epipolar geometry the euclidean motion is recovered using the fixed intrinsic parameters of the camera obtained either from a calibration grid or from self calibration techniques finally the spinning object is reconstructed from its profiles using the motion estimated in the previous stage results from real data are presented demonstrating the efficiency and usefulness of the proposed methods

compensation of some time dependent deformations in tomography this work concerns 2d t dynamic tomography we show that a much larger class of deformations than the affine transforms can be compensated analytically within filtered back projection algorithms in 2d parallel beam and fan beam dynamic tomography we present numerical experiments on the shepp and logan phantom showing that nonaffine deformations can be compensated a generalization to 3d cone beam tomography is proposed

on the fitting of surfaces to data with covariances we consider the problem of estimating parameters of a model described by an equation of special form specific models arise in the analysis of a wide class of computer vision problems including conic fitting and estimation of the fundamental matrix we assume that noisy data are accompanied by known covariance matrices characterizing the uncertainty of the measurements a cost function is first obtained by considering a maximum likelihood formulation and applying certain necessary approximations that render the problem tractable a newton like iterative scheme is then generated for determining a minimizer of the cost function unlike alternative approaches such as sampson s method or the renormalization technique the new scheme has as its theoretical limit the minimizer of the cost function furthermore the scheme is simply expressed efficient and unsurpassed as a general technique in our testing an important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches

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

unbiased estimation of ellipses by bootstrapping a general method for eliminating the bias of nonlinear estimators using bootstrap is presented instead of the traditional mean bias we consider the definition of bias based on the median the method is applied to the problem of fitting ellipse segments to noisy data no assumption beyond being independent identically distributed is made about the error distribution and experiments with both synthetic and real data prove the effectiveness of the technique

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

the ramification of centres lie algebras in positive characteristic and quantised enveloping algebras let h be a hopf algebra over the field k which is a finite module over a central affine sub hopf algebra r examples include enveloping algebras u mathfrak g of finite dimensional k lie algebras mathfrak g in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity the ramification behaviour of the maximal ideals of z h with respect to the subalgebra r is studied and the conclusions are then applied to the cases of classical and quantised enveloping algebras in the case of u mathfrak g for mathfrak g semisimple a conjecture of humphreys 28 on the block structure of u mathfrak g is confirmed in the case of u epsilon mathfrak g for mathfrak g semisimple and epsilon an odd root of unity we obtain a quantum analogue of a result of mirkovic and rumynin 35 and we fully describe the factor algebras lying over the regular sheet 9 the blocks of u epsilon mathfrak g are determined and a necessary condition which may also be sufficient for a baby verma u epsilon mathfrak g module to be simple is obtained

from lines to epipoles through planes in two views this paper addresses the computation of the fundamental matrix between two views when camera motion and 3d structure are unknown but planar surfaces can be assumed we use line features which are automatically matched in two steps firstly with image based parameters a set of matches are obtained to secondly compute homographies which allows to reject wrong ones and to grow good matches in a final stage the inclusion of projective transformations gives much better results to match features with short computing overload when two or more planes are observed different homographies can be computed segmenting simultaneously the corresponding planar surfaces these can be used to obtain the fundamental matrix which gives constraints for the whole scene the results show that the global process is robust enough turning out stable and useful to obtain matches and epipolar geometry from lines in man made environments

a cone beam reconstruction algorithm using shift variant filtering and cone beam backprojection an exact inversion formula written in the form of shift variant filtered backprojection fbp is given for reconstruction from cone beam data taken from any orbit satisfying h k tuy s 1983 sufficiency conditions the method is based on a result of p grangeat 1987 involving the derivative of the three dimensional 3d radon transform but unlike grangeat s algorithm no 3d rebinning step is required data redundancy which occurs when several cone beam projections supply the same values in the radon domain is handled using an elegant weighting function and without discarding data the algorithm is expressed in a convenient cone beam detector reference frame and a specific example for the case of a dual orthogonal circular orbit is presented when the method is applied to a single circular orbit even though tuy s condition is not satisfied it is shown to be equivalent to the well known algorithm of l a feldkamp et al 1984

on quasi class a contractions abstract let qa denote the class of bounded linear hilbert space operators t which satisfy the operator inequality t t 2 t t t 2 t it is proved that if t qa is a contraction then either t has a nontrivial invariant subspace or t is a proper contraction and the nonnegative operator d t t 2 t 2 t is strongly stable it is shown that if t qa is a contraction with hilbert schmidt defect operator such that t 1 0 t 1 0 then t is completely non normal if and only if t c 10 and a commutativity theorem is proved for contractions t qa let t u and t c denote the unitary part and the cnu part of a contraction t respectively we prove that if a a u a c and b b u b c are qa contractions such that a c then a and b are quasi similar if and only a u and b u are unitarily equivalent and a c and b c are quasi similar

estimation of f matrix and image rectification by double quaternion fundamental matrix or f matrix is one of the most important and elemental tools in the field of computer vision in conventional methods for estimating the f matrix an eight point algorithm is adopted first an approximate f matrix is calculated by a linear solver using at least eight corresponding pairs since this linear optimization method excludes an essential property the rank 2 constraint a method based on a singular value decomposition svd is applied to impose the constraint this last step with svd however provides additional noise in the f matrix several methods introduce parameterizations taking into account the rank 2 constraint and optimized nonlinearly without svd in this paper we propose a novel parameterization for the nonlinear optimization which includes this constraint we adopt double quaternion dq and a scalar as the parameter set experimental results show that the nonlinear optimization with our parameterization is competitive with other parameterization methods moreover through the proposed parameterization we can obtain two transformations for the two input images these transformations lead to a novel method to estimate epipolar lines and to rectify the image pairs this rectification method can deal with any image pairs in the same manner whether the epipoles are inside or outside the images

data driven affine deformation estimation and correction in cone beam computed tomography in computed tomography ct motion and deformation during the acquisition lead to streak artefacts and blurring in the reconstructed images to remedy these artefacts we introduce an efficient algorithm to estimate and correct for global affine deformations directly on the cone beam projections the proposed technique is data driven and thus removes the need for markers and or a tracking system a relationship between affine transformations and the cone beam transform is proved and used to correct the projections the deformation parameters that describe deformation perpendicular to the projection direction are estimated for each projection by minimizing a plane based inconsistency criterion the criterion compares each projection of the main scan with all projections of a fast reference scan which is acquired prior or posterior to the main scan experiments with simulated and experimental data show that the proposed affine deformation estimation method is able to substantially reduce motion artefacts in cone beam ct images

detecting motion regions in the presence of a strong parallax from a moving camera by multiview geometric constraints we present a method for detecting motion regions in video sequences observed by a moving camera in the presence of a strong parallax due to static 3d structures the proposed method classifies each image pixel into planar background parallax or motion regions by sequentially applying 2d planar homographies the epipolar constraint and a novel geometric constraint called the structure consistency constraint the structure consistency constraint being the main contribution of this paper is derived from the relative camera poses in three consecutive frames and is implemented within the plane parallax framework unlike previous planar parallax constraints proposed in the literature the structure consistency constraint does not require the reference plane to be constant across multiple views it directly measures the inconsistency between the projective structures from the same point under camera motion and reference plane change the structure consistency constraint is capable of detecting moving objects followed by a moving camera in the same direction a so called degenerate configuration where the epipolar constraint fails we demonstrate the effectiveness and robustness of our method with experimental results of real world video sequences

some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive image frames this paper describes the results of some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive tv images and discusses several factors which affect the accuracy of the results these factors include the sizes of the motion parameters themselves the accuracy of the raw data and the number of point correspondences in addition we address two related topics determining corner positions to subpixel accuracy and matching point patterns with different scales

calibrating a video camera pair with a rigid bar abstract in this paper a new procedure to determine all the geometrical parameters of a stereo system is presented it is based on surveying a rigid bar carrying two markers on its extremities moved inside the working volume and it does not require grids or complex calibration structures the external parameters are estimated through the epipolar geometry up to a scale factor which is determined from the true length of the bar the focal lengths are determined using the properties of the absolute conic in the projective space the principal points are computed through a non linear minimisation carried out through an evolutionary optimisation the accuracy of the method is assessed on real data and it compares favourably with that obtained through classical approaches based on control points of known 3d coordinates

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

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

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

quantum flag varieties equivariant quantum d modules and localization of quantum groups abstract let o q g be the algebra of quantized functions on an algebraic group g and o q b its quotient algebra corresponding to a borel subgroup b of g we define the category of sheaves on the quantum flag variety of g to be the o q b equivariant o q g modules and prove that this is a proj category we construct a category of equivariant quantum d modules on this quantized flag variety and prove the beilinson bernstein s localization theorem for this category in the case when q is transcendental

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

self calibration of a moving camera from pointcorrespondences and fundamental matrices we address the problem of estimating three dimensional motion and structure from motion with an uncalibrated moving camera we show that point correspondences between three images and the fundamental matrices computed from these point correspondences are sufficient to recover the internal orientation of the camera its calibration the motion parameters and to compute coherent perspective projection matrices which enable us to reconstruct 3 d structure up to a similarity in contrast with other methods no calibration object with a known 3 d shape is needed and no limitations are put upon the unknown motions to be performed or the parameters to be recovered as long as they define a projective camera the theory of the method which is based on the constraint that the observed points are part of a static scene thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic is first detailed several algorithms are then presented and their performances compared by means of extensive simulations and illustrated by several experiments with real images

optimization criteria and geometric algorithms for motion and structure estimation prevailing efforts to study the standard formulation of motion and structure recovery have recently been focused on issues of sensitivity and robustness of existing techniques while many cogent observations have been made and verified experimentally many statements do not hold in general settings and make a comparison of existing techniques difficult with an ultimate goal of clarifying these issues we study the main aspects of motion and structure recovery the choice of objective function optimization techniques and sensitivity and robustness issues in the presence of noise we clearly reveal the relationship among different objective functions such as normalized epipolar constraints reprojection error or triangulation all of which can be unified in a new optimal triangulation procedure regardless of various choices of the objective function the optimization problems all inherit the same unknown parameter space the so called essential manifold based on recent developments of optimization techniques on riemannian manifolds in particular on stiefel or grassmann manifolds we propose a riemannian newton algorithm to solve the motion and structure recovery problem making use of the natural differential geometric structure of the essential manifold we provide a clear account of sensitivity and robustness of the proposed linear and nonlinear optimization techniques and study the analytical and practical equivalence of different objective functions the geometric characterization of critical points and the simulation results clarify the difference between the effect of bas relief ambiguity rotation and translation confounding and other types of local minima this leads to consistent interpretations of simulation results over a large range of signal to noise ratio and variety of configurations

trinocular geometry revisited when do the visual rays associated with triplets of point correspondences converge that is intersect in a common point classical models of trinocular geometry based on the fundamental matrices and trifocal tensor associated with the corresponding cameras only provide partial answers to this fundamental question in large part because of underlying but seldom explicit general configuration assumptions this paper uses elementary tools from projective line geometry to provide necessary and sufficient geometric and analytical conditions for convergence in terms of transversals to triplets of visual rays without any such assumptions in turn this yields a novel and simple minimal parameterization of trinocular geometry for cameras with non collinear or collinear pinholes which can be used to construct a practical and efficient method for trinocular geometry parameter estimation we present numerical experiments using synthetic and real data

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

circular motion geometry using minimal data circular motion or single axis motion is widely used in computer vision and graphics for 3d model acquisition this paper describes a new and simple method for recovering the geometry of uncalibrated circular motion from a minimal set of only two points in four images this problem has been previously solved using nonminimal data either by computing the fundamental matrix and trifocal tensor in three images or by fitting conics to tracked points in five or more images it is first established that two sets of tracked points in different images under circular motion for two distinct space points are related by a homography then we compute a plane homography from a minimal two points in four images after that we show that the unique pair of complex conjugate eigenvectors of this homography are the image of the circular points of the parallel planes of the circular motion subsequently all other motion and structure parameters are computed from this homography in a straightforward manner the experiments on real image sequences demonstrate the simplicity accuracy and robustness of the new method

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

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

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

the local projective shape of smooth surfaces and their outlines this article examines projectively invariant local geometric properties of smooth curves and surfaces oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines it is applied to two problems 1 the projective generalization of koenderink s famous characterization of convexities concavities and inflections of the apparent contours of solids bounded by smooth surfaces and 2 the image based construction of rim meshes which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it

estimation three dimensional motion of rigid objects from noisy observations an estimate is made of the motion of a rigid body from two noisy 2 d perspective projections using the least squares method and the algebra of r y tsai and t s huang 1984 the accuracy of the estimated motion parameters is influenced by the position of the features of the object used in the calculation four test variables are derived that indicate how the accuracy is affected and they are used for discarding inaccurate estimates monte carlo tests demonstrate the obtained accuracy

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

a new approach to 3d reconstruction without camera calibration in this paper we present a new approach for 3d scene reconstruction based on projective geometry without camera calibration previous works use at least six points to build two projective reference planes our contribution is to reduce the number of reference points to four by exploiting some geometrical shapes contained in the scene the first implemented algorithm allows the reconstruction of a fourth point on each reference plane the second algorithm is devoted to the 3d reconstruction we obtained the expected good results and the proposed method is to equip a mobile robot moving in a structured environment

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

the jordan form of a bitriangular operator abstract a jordan operator is a infinite direct sum of a norm bounded sequence of finite dimensional jordan blocks an operator t is quasisimilar to a jordan operator if and only if both t and t are triangularizable whence bitriangular the canonical jordan model of such an operator is determined just by the numerical data dim ker t i n 1 ker t i n an analysis of hyperinvariant subspaces similarity orbits and other aspects of these operators are analyzed using this model

simultaneous two view epipolar geometry estimation and motion segmentation by 4d tensor voting we address the problem of simultaneous two view epipolar geometry estimation and motion segmentation from nonstatic scenes given a set of noisy image pairs containing matches of n objects we propose an unconventional efficient and robust method 4d tensor voting for estimating the unknown n epipolar geometries and segmenting the static and motion matching pairs into n independent motions by considering the 4d isotropic and orthogonal joint image space only two tensor voting passes are needed and a very high noise to signal ratio up to five can be tolerated epipolar geometries corresponding to multiple rigid motions are extracted in succession only two uncalibrated frames are needed and no simplifying assumption such as affine camera model or homographic model between images other than the pin hole camera model is made our novel approach consists of propagating a local geometric smoothness constraint in the 4d joint image space followed by global consistency enforcement for extracting the fundamental matrices corresponding to independent motions we have performed extensive experiments to compare our method with some representative algorithms to show that better performance on nonstatic scenes are achieved results on challenging data sets are presented

linearly estimating all parameters of affine motion using radon transform fast and accurate motion estimation takes an important place in many fields of computer vision and image processing using radon transform to compute projections of the images along specified directions is an effective way to show the relationship between the 2d image object and its projections and estimate motions between the images all existing projection based motion estimation methods without the use of iteration have a severe defect that only five of the six affine parameters can be estimated there are some other methods that can estimate the six parameters but most of them are usually based on a certain iterative framework which is computationally intensive and sensitively dependent on the initial values in this paper a novel method based on radon transform is proposed to estimate all the six affine parameters directly the relationship in the projection domain between a pair of images connected by an affine motion is studied and a linear model is established by which all the six affine parameters can be directively found the employment of a hierarchical framework can produce more accurate results the experimental results reveal that the proposed method has a much better performance than the state of the art methods in this field

recovering multiple view geometry from mutual projections of multiple cameras in this paper we analyze the computation of epipolar geometry in some special cases where multiple cameras are projected each other in their images in such cases epipoles can be obtained directly from images as the projection of cameras as the result the epipolar geometry can be computed from less image correspondences with higher stability in this paper we analyze the number of corresponding points required for computing bifocal trifocal and quadrifocal tensors linearly in the case where cameras are projected mutually we next show a practical linear method for computing multifocal tensors by using the mutual projection of cameras the degenerate configurations of points and cameras is also studied and it is shown that some degenerate configurations in general cases are no longer degenerate under the mutual projection of cameras

localization of modules for a semisimple lie algebra in prime characteristic with an appendix by r bezrukavnikov and s riche computation for sl 3 we show that on the level of derived categories representations of the lie algebra of a semisimple algebraic group over a field of finite characteristic with a given generalized regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding generalized springer fiber the first step is to observe that the derived functor of global sections provides an equivalence between the derived category of d modules with no divided powers on the flag variety and the appropriate derived category of modules over the corresponding lie algebra thus the derived version of the beilinson bernstein localization theorem holds in sufficiently large positive characteristic next one finds that for any smooth variety this algebra of differential operators is an azumaya algebra on the cotangent bundle in the case of the flag variety it splits on springer fibers and this allows us to pass from d modules to coherent sheaves the argument also generalizes to twisted d modules as an application we prove lusztig s conjecture on the number of irreducible modules with a fixed central character we also give a formula for behavior of dimension of a module under translation functors and reprove the kac weisfeiler conjecture the sequel to this paper bmr2 treats singular infinitesimal characters

epipolar consistency in transmission imaging this paper presents the derivation of the epipolar consistency conditions ecc between two x ray images from the beer lambert law of x ray attenuation and the epipolar geometry of two pinhole cameras using grangeat s theorem we motivate the use of oriented projective geometry to express redundant line integrals in projection images and define a consistency metric which can be used for instance to estimate patient motion directly from a set of x ray images we describe in detail the mathematical tools to implement an algorithm to compute the epipolar consistency metric and investigate its properties with detailed random studies on both artificial and real fd ct data a set of six reference projections of the ct scan of a fish were used to evaluate accuracy and precision of compensating for random disturbances of the ground truth projection matrix using an optimization of the consistency metric in addition we use three x ray images of a pumpkin to prove applicability to real data we conclude that the metric might have potential in applications related to the estimation of projection geometry by expression of redundancy between two arbitrary projection views we in fact support any device or acquisition trajectory which uses a cone beam geometry we discuss certain geometric situations where the ecc provide the ability to correct 3d motion without the need for 3d reconstruction

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

dense versus sparse approaches for estimating the fundamental matrix there are two main strategies for solving correspondence problems in computer vision sparse local feature based approaches and dense global energy based methods while sparse feature based methods are often used for estimating the fundamental matrix by matching a small set of sophistically optimised interest points dense energy based methods mark the state of the art in optical flow computation the goal of our paper is to show that this separation into different application domains is unnecessary and can be bridged in a natural way as a first contribution we present a new application of dense optical flow for estimating the fundamental matrix comparing our results with those obtained by feature based techniques we identify cases in which dense methods have advantages over sparse approaches motivated by these promising results we propose as a second contribution a new variational model that recovers the fundamental matrix and the optical flow simultaneously as the minimisers of a single energy functional in experiments we show that our coupled approach is able to further improve the estimates of both the fundamental matrix and the optical flow our results prove that dense variational methods can be a serious alternative even in classical application domains of sparse feature based approaches

adaptive outlier elimination in image registration using genetic programming abstract in feature based methods outlier removal plays an important role in attaining a reasonable accuracy for image registration in this paper we propose a genetic programming gp based adaptive method for outlier removal first features are extracted through the scale invariant feature transform sift from the reference and sensed images which were initially matched using euclidean distance the classification of feature points into inliers and outliers is done in two stages in the first stage feature vectors are computed using various distance and angle information feature points are categorized into three groups inliers outliers and non classified feature ncf points in the second stage a gp based classifier is developed to classify ncf points into inliers and outliers the gp based function takes features as an input feature vector and provides a scalar output by combining features with arithmetic operations finally registration is done by eliminating the outliers the effectiveness of the proposed outlier removal method is analyzed through the classification and positional accuracy the experimental results show a considerable improvement in the registration accuracy

multiple interpretations of a pair of images of a surface it is known that if two optical images of a visually textured surface projected from finitely separated viewpoints allow more than one 3d interpretation then the surface must be part of a quadric passing through the two viewpoints it is here shown that this quadric is either a plane or a ruled surface of a type first considered by maybank in a study of ambiguous optic flow fields and that in the latter case three is the maximum number of distinct interpretations that the two images can sustain

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

hyponormal operators quasisimilar to an isometry an expression for the multiplicity of an arbitrary contraction is pre sented it is in terms of the isometries which can be densely intertwined to the given contraction this is then used to obtain a generalization of a result of sz nagy and foias concerning the existence of a c 0 contraction which is a quasiaffine transform of a contraction we then consider the problem when a hyponormal operator is quasisimilar to an isometry or more generally when two hyponormal contractions are quasisimilar to each other our main results in this respect generalize previous ones obtained by hastings and the author for quasinormal and certain subnormal operators quasisimilarity or similarity to an isometry may even imply unitary equivalence 1 introduction let f and f2 be bounded linear operators on the complex separable hilbert spaces 77 and h2 respectively we say that f is densely d intertwined to t2 tx 0 resp t x 0 for all x t is of class c resp cx if t x 0 resp t x 0 for all x 0 caj3 ca n c p for a b 0 1 a contraction f can be decomposed as us ffi ua ffi t where us and ua axe singular and absolutely continuous unitary operators and t is completely nonunitary c n u us ua and t are called the unitary part and c ai w part of f respectively following alexander 1 we say that the contraction f is of analytic type if it has no singular unitary direct summand for such a

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

image based rendering using parameterized image varieties this paper addresses the problem of characterizing the set of all images of a rigid set of m points and n lines observed by a weak perspective or paraperspective camera by taking explicitly into account the euclidean constraints associated with calibrated cameras we show that the corresponding image space can be represented by a six dimensional variety embedded in cal r 2 m n and parameterized by the image positions of three reference points the coefficients defining this parameterized image variety or piv for short can be estimated from a sample of images of a scene via linear and non linear least squares the piv provides an integrated framework for using both point and line features to synthesize new images from a set of pre recorded pictures image based rendering the proposed technique does not perform any explicit three dimensional scene reconstruction but it supports hidden surface elimination texture mapping and interactive image synthesis at frame rate on ordinary pcs it has been implemented and extensively tested on real data sets

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

outward looking circular motion analysis of large image sequences this work presents a novel and simple method of analyzing the motion of a large image sequence captured by a calibrated outward looking video camera moving on a circular trajectory for large scale environment applications previous circular motion algorithms mainly focus on inward looking turntable like setups they are not suitable for outward looking motion where the conic trajectory of corresponding points degenerates to straight lines the circular motion of a calibrated camera essentially has only one unknown rotation angle for each frame the motion recovery for the entire sequence computes only one fundamental matrix of a pair of frames to extract the angular motion of the pair using laguerre s formula and then propagates the computation of the unknown rotation angles to the other frames by tracking one point over at least three frames finally a maximum likelihood estimation is developed for the optimization of the whole sequence extensive experiments demonstrate the validity of the method and the feasibility of the application in image based rendering

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

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

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