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31
The Fundamental matrix: theory, algorithms, and stability analysis
 International Journal of Computer Vision
, 1995
"... 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 th ..."
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Cited by 233 (14 self)
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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 commonly used formalism of the Essential matrix introduced by LonguetHiggins [40]. This paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 \Theta 3 ma...
Understanding noise sensitivity in structure from motion

, 1996
"... Solutions to the structure from motion problem have been shown to be very sensitive to measurement noise and the respective motion and geometry configuration. Statistical error analysis has become an invaluable tool in analyzing the sensitivity phenomenon. This paper presents a unifying approach to ..."
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Cited by 54 (5 self)
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Solutions to the structure from motion problem have been shown to be very sensitive to measurement noise and the respective motion and geometry configuration. Statistical error analysis has become an invaluable tool in analyzing the sensitivity phenomenon. This paper presents a unifying approach to the problems of statistical bias, correlated noise, choice of error metrics, geometric instabilities and information fusion exploring several assumptions commonly used in motion estimation and reviews several promising techniques for motion estimation. The techniques are based on a small number of principles of statistics and perturbation theory. The analyticity of the approach enables the design of alternatives overcoming the observed instabilities.
Linear differential algorithm for motion recovery: A geometric approach
 International Journal of Computer Vision
, 2000
"... The aim of this paper is to explore a linear geometric algorithm for recovering the three dimensional motion of a moving camera from image velocities. Generic similarities and differences between the discrete approach and the differential approach are clearly revealed through a parallel development ..."
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Cited by 35 (7 self)
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The aim of this paper is to explore a linear geometric algorithm for recovering the three dimensional motion of a moving camera from image velocities. Generic similarities and differences between the discrete approach and the differential approach are clearly revealed through a parallel development of an analogous motion estimation theory previously explored in [24, 26]. We present a precise characterization of the space of differential essential matrices, which gives rise to a novel eigenvaluedecompositionbased 3D velocity estimation algorithm from the optical flow measurements. This algorithm gives a unique solution to the motion estimation problem and serves as a differential counterpart of the wellknown SVDbased 3D displacement estimation algorithm for the discrete case. Since the proposed algorithm only involves linear algebra techniques, it may be used to provide a fast initial guess for more sophisticated nonlinear algorithms [13]. Extensive simulation results are presented for evaluating the performance of our algorithm in terms of bias and sensitivity of the estimates with respect to di erent noise levels in image velocity measurements.
Structure from Controlled Motion
 IEEE Trans. on Pattern Analysis and Machine Intelligence
, 1996
"... This paper deals with the recovery of 3D information using a single mobile camera in the context of active vision. First, we propose a general revisited formulation of the structurefromknownmotion issue. Within the same formalism, we handle various kinds of 3D geometrical primitives such as point ..."
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Cited by 33 (15 self)
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This paper deals with the recovery of 3D information using a single mobile camera in the context of active vision. First, we propose a general revisited formulation of the structurefromknownmotion issue. Within the same formalism, we handle various kinds of 3D geometrical primitives such as points, lines, cylinders, spheres, etc. We also aim at minimizing effects of the different measurement errors which are involved in such a process. More precisely, we mathematically determine optimal camera configurations and motions which lead to a robust and accurate estimation of the 3D structure parameters. We apply the visual servoing approach to perform these camera motions using a control law in closedloop with respect to visual data. Realtime experiments dealing with 3D structure estimation of points and cylinders are reported. They demonstrate that this active vision strategy can very significantly improve the estimation accuracy. Index TermsComputer vision, robotics, active vision,...
The Coupling of Rotation and Translation in Motion Estimation of Planar Surfaces
"... This paper studies the error sensitivity in the estimation of the 3Dmotion and the normal of a planar surface from an instantaneous motion field. We use the statistical theory of the CramerRao lower bound for the error covariance in the estimated motion and structure parameters which enables the d ..."
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Cited by 32 (0 self)
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This paper studies the error sensitivity in the estimation of the 3Dmotion and the normal of a planar surface from an instantaneous motion field. We use the statistical theory of the CramerRao lower bound for the error covariance in the estimated motion and structure parameters which enables the derivation of results valid for any unbiased estimator under the assumption of Gaussian noise in the motion eld. The obtained lowerboundmatrix is studied analytically with respect to the measurement noise, size of the field of view and the motiongeometry configuration. The main result of this analysis is the coupling between translation and rotation which is exacerbated if the field of view and the slant of the plane become smaller and the deviation of the translation from the viewing direction becomes larger. Byproducts of this study are the relationships of the uncertainty bounds for every unknown motion parameter to the angle between translation and the planenormal, the size of the field of view, the distance from the perceived plane and the translation magnitude.
Optimization criteria and geometric algorithms for motion and structure estimation
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2001
"... 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 setting ..."
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Cited by 29 (6 self)
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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 socalled “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
A multiframe structurefrommotion algorithm under perspective projection
 International Journal of Computer Vision
, 1999
"... Abstract. We present a fast, robust algorithm for multiframe 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 rea ..."
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Cited by 25 (2 self)
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Abstract. We present a fast, robust algorithm for multiframe 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 twoimage 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 basrelief ambiguity and exemplifies how such an analysis can be exploited to improve reconstruction. Also, we demonstrate a structurefrommotion 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 focallength variations strengthen the effects of the basrelief ambiguity. This paper includes extensive experimental studies of twoframe reconstruction and the Tomasi/Kanade approach in comparison to our algorithm. We find that twoframe algorithms are surprisingly robust and accurate, despite some problems with local minima. We demonstrate experimentally that a nearly optimal
Stochastic Approximation and RateDistortion Analysis for Robust Structure and Motion Estimation
 INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2003
"... Recent research on structure and motion recovery has focused on issues related to sensitivity and robustness of existing techniques. One possible reason is that in practical applications, the underlying assumptions made by existing algorithms are often violated. In this paper, we propose a framework ..."
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Cited by 22 (12 self)
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Recent research on structure and motion recovery has focused on issues related to sensitivity and robustness of existing techniques. One possible reason is that in practical applications, the underlying assumptions made by existing algorithms are often violated. In this paper, we propose a framework for 3D reconstruction from short monocular video sequences taking into account the statistical errors in reconstruction algorithms. Detailed error analysis is especially important for this problem because the motion between pairs of frames is small and slight perturbations in its estimates can lead to large errors in 3D reconstruction. We focus on the following issues: physical sources of errors, their experimental and theoretical analysis, robust estimation techniques and measures for characterizing the quality of the final reconstruction. We derive a precise relationship between the error in the reconstruction and the error in the image correspondences. The error analysis is used to design a robust, recursive multiframe fusion algorithm using "stochastic approximation" as the framework since it is capable of dealing with incomplete information about errors in observations. Ratedistortion analysis is proposed for evaluating the quality of the final reconstruction as a function of the number of frames and the error in the image correspondences. Finally, to demonstrate the e#ectiveness of the algorithm, examples of depth reconstruction are shown for different video sequences.
On The Geometry Of Visual Correspondence
 International Journal of Computer Vision
, 1994
"... Image displacement fieldsoptical flow fields, stereo disparity fields, normal flow fieldsdue to rigid motion possess a global geometric structure which is independent of the scene in view. Motion vectors of certain lengths and directions are constrained to lie on the imaging surface at particu ..."
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Cited by 19 (12 self)
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Image displacement fieldsoptical flow fields, stereo disparity fields, normal flow fieldsdue to rigid motion possess a global geometric structure which is independent of the scene in view. Motion vectors of certain lengths and directions are constrained to lie on the imaging surface at particular loci whose location and form depends solely on the 3D motion parameters. If optical flow fields or stereo disparity fields are considered, then equal vectors are shown to lie on conic sections. Similarly, for normal motion fields, equal vectors lie within regions whose boundaries also constitute conics. By studying various properties of these curves and regions and their relationships, a characterization of the structure of rigid motion fields is given. The goal of this paper is to introduce a concept underlying the global structure of image displacement fields. This concept gives rise to various constraints that could form the basis of algorithms for the recovery of visual information f...
An Algebraic/analytic Method for Reconstruction From Image Correspondences
 In Theory & Applications of Image Analysis, Machine Perception Articial Intelligence, World Scientic Publishing Co (Selected papers from SCIA'91
, 1991
"... A method for the reconstruction and motion problems is presented, under the assumption that a number of point correspondences in a pair of images are known.. A geometric and algebraic theory is presented, based on invariancy properties of point conøguration under aOEne and projective transformations ..."
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Cited by 13 (5 self)
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A method for the reconstruction and motion problems is presented, under the assumption that a number of point correspondences in a pair of images are known.. A geometric and algebraic theory is presented, based on invariancy properties of point conøguration under aOEne and projective transformations. In particular, a characterization of those image pairs which correspond to the same point conøguration in an unknown scene is given. An algorithm along these lines is presented, yielding complete sets of solutions to the reconstruction problem. A main idea is to exploit the aOEne structure of the problem, and work with relative locations, before turning the the metrical structure, and absolute locations. It turns out that relative depth information can be achieved at a low computational cost. 1 Introduction The reconstruction of a scene and/or the determination of the movement of a camera from a pair (or sequence) of images are central topics in computer vision, with applications e.g. in ...