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46
Determining the Epipolar Geometry and its Uncertainty: A Review
 International Journal of Computer Vision
, 1998
"... 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, an ..."
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Cited by 322 (7 self)
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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 wellfounded measure is proposed to compare these techniques. Projective reconstruction is also reviewed. The software which we have developed for this review is available on the Internet.
A Framework for Uncertainty and Validation of 3D Registration Methods based on Points and Frames
 Int. Journal of Computer Vision
, 1997
"... In this paper, we propose and analyze several methods to estimate a rigid transformation from a set of 3D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributi ..."
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Cited by 75 (23 self)
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In this paper, we propose and analyze several methods to estimate a rigid transformation from a set of 3D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributions are: an intrinsic model of noise for transformations based on composition rather than addition; a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences, and a statistical validation method to verify the error estimation, which applies even when no "ground truth" is available. We analyze and demonstrate on synthetic data that our scheme is well behaved. The practical contribution of the paper is the validation of our transformation estimation method in the case of 3D medical images, which shows that an accuracy of the registration far below the size of a voxel can be achieved, and in the case of protein substructure matching, where frame features drastically improve both selectivity and complexity. 1.
A Plane Measuring Device
, 1997
"... A requirement of a visual measurement device is that both measurements and their uncertainties can be determined. This paper develops an uncertainty analysis which includes both the errors in image localization and the uncertainty in the imaging transformation. The matrix representing the imaging tr ..."
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Cited by 54 (4 self)
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A requirement of a visual measurement device is that both measurements and their uncertainties can be determined. This paper develops an uncertainty analysis which includes both the errors in image localization and the uncertainty in the imaging transformation. The matrix representing the imaging transformation is estimated from imagetoworld point correspondences. A general expression is derived for the covariance of this matrix. This expression is valid if the matrix is over determined and also if the minimum number of correspondences are used. A bound on the errors of the first order approximations involved is also derived. Armed with this covariance result the uncertainty of any measurement can be predicted, and furthermore the distribution of correspondences can be chosen to achieve a particular bound on the uncertainty. Examples are given of measurements such as distance and parallelism for several applications. These include indoor scenes and architectural measurements. Key word...
A General Method for ErrorsinVariables Problems in Computer Vision
 In Proceedings, CVPR 2000, IEEE Computer Society Conference on Computer Vision and Pattern Recognition
, 2000
"... The ErrorsinVariables (EIV) model from statistics is often employed in computer vision though only rarely under this name. In an EIV model all the measurements are corrupted by noise while the a priori information is captured with a nonlinear constraint among the true (unknown) values of these mea ..."
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Cited by 47 (9 self)
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The ErrorsinVariables (EIV) model from statistics is often employed in computer vision though only rarely under this name. In an EIV model all the measurements are corrupted by noise while the a priori information is captured with a nonlinear constraint among the true (unknown) values of these measurements. To estimate the model parameters and the uncorrupted data, the constraint can be linearized, i.e., embedded in a higher dimensional space. We show that linearization introduces datadependent (heteroscedastic) noise and propose an iterative procedure, the heteroscedastic EIV (HEIV) estimator to obtain consistent estimates in the most general, multivariate case. Analytical expressions for the covariances of the parameter estimates and corrected data points, a generic method for the enforcement of ancillary constraints arising from the underlying geometry are also given. The HEIV estimator minimizes the first order approximation of the geometric distances between the measurements and the true data points, and thus can be a substitute for the widely used LevenbergMarquardt based direct solution of the original, nonlinear problem. The HEIV estimator has however the advantage of a weaker dependence on the initial solution and a faster convergence. In comparison to Kanatani's renormalization paradigm (an earlier solution of the same problem) the HEIV estimator has more solid theoretical foundations which translate into better numerical behavior. We show that the HEIV estimator can provide an accurate solution to most 3D vision estimation tasks, and illustrate its performance through two case studies: calibration and the estimation of the fundamental matrix.
3D Reconstruction of Urban Scenes from Sequences of Images
 Automatic Extraction of ManMade Objects from Aerial and Space Images. Birkhauser
, 1995
"... In this paper, we address the problem of the recovery of the Euclidean geometry of a scene from a sequence of images without any prior knowledge either about the parameters of the cameras, or about the motion of the camera(s). We do not require any knowledge of the absolute coordinates of some contr ..."
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Cited by 42 (1 self)
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In this paper, we address the problem of the recovery of the Euclidean geometry of a scene from a sequence of images without any prior knowledge either about the parameters of the cameras, or about the motion of the camera(s). We do not require any knowledge of the absolute coordinates of some control points in the scene to achieve this goal. Using various computer vision tools, we establish correspondences between images and recover the epipolar geometry of the set of images, from which we show how to compute the complete set of perspective projection matrices for each camera position. These being known, we proceed to reconstruct the scene. This reconstruction is defined up to an unknown projective transformation (i.e. is parameterized with 15 arbitrary parameters). Next we show how to go from this reconstruction to a more constrained class of reconstructions, defined up to an unknown affine transformation (i.e. parameterized with 12 arbitrary parameters) by exploiting known geometr...
Scalable Extrinsic Calibration of OmniDirectional Image Networks
 International Journal of Computer Vision
, 2002
"... We describe a lineartime algorithm that recovers absolute camera orientations and positions, along with uncertainty estimates, for networks of terrestrial image nodes spanning hundreds of meters in outdoor urban scenes. The algorithm produces pose estimates globally consistent to roughly 0.1 # (2 ..."
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Cited by 36 (6 self)
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We describe a lineartime algorithm that recovers absolute camera orientations and positions, along with uncertainty estimates, for networks of terrestrial image nodes spanning hundreds of meters in outdoor urban scenes. The algorithm produces pose estimates globally consistent to roughly 0.1 # (2 milliradians) and 5 centimeters on average, or about four pixels of epipolar alignment.
Robust Detection of Degenerate Configurations whilst Estimating the Fundamental Matrix
, 1998
"... We present a new method for the detection of multiple solutions or degeneracy when estimating the Fundamental Matrix, with specific emphasis on robustness to data contamination (mismatches). The Fundamental Matrix encapsulates all the information on camera motion and internal parameters available f ..."
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Cited by 31 (3 self)
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We present a new method for the detection of multiple solutions or degeneracy when estimating the Fundamental Matrix, with specific emphasis on robustness to data contamination (mismatches). The Fundamental Matrix encapsulates all the information on camera motion and internal parameters available from image feature correspondences between two views. It is often used as a first step in structure from motion algorithms. If the set of correspondences is degenerate, then this structure cannot be accurately recovered and many solutions explain the data equally well. It is essential that we are alerted to such eventualities. As current feature matchers are very prone to mismatching the degeneracy detection method must also be robust to outliers. In this paper a definition of degeneracy is given and all two view nondegenerate and degenerate cases are catalogued in a logical way by introducing the language of varieties from algebraic geometry. It is then shown how each of the cases can be ro...
Overall view regarding fundamental matrix estimation
 Image and Vision Computing
, 2003
"... Epipolar geometry is a key point in computer vision and the fundamental matrix estimation is the only way to compute it. This article is a fresh look in the subject that overview classic and latest presented methods of fundamental matrix estimation which have been classified into linear methods, ite ..."
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Cited by 29 (3 self)
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Epipolar geometry is a key point in computer vision and the fundamental matrix estimation is the only way to compute it. This article is a fresh look in the subject that overview classic and latest presented methods of fundamental matrix estimation which have been classified into linear methods, iterative methods and robust methods. All of these methods have been programmed and their accuracy analyzed in synthetic and real images. A summary including experimental results and algorithmic details is given and the whole code is available in Internet.
Estimating the Fundamental Matrix by Transforming Image Points in Projective Space
 COMPUTER VISION AND IMAGE UNDERSTANDING
, 2001
"... This paper proposes a novel technique for estimating the fundamental matrix by transforming the image points in projective space. We therefore only need to perform nonlinear optimization with one parameterization of the fundamental matrix, rather than considering 36 distinct parameterizations as in ..."
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Cited by 19 (0 self)
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This paper proposes a novel technique for estimating the fundamental matrix by transforming the image points in projective space. We therefore only need to perform nonlinear optimization with one parameterization of the fundamental matrix, rather than considering 36 distinct parameterizations as in previous work. We also show how to preserve the characteristics of the data noise model from the original image space.
Estimating the jacobian of the singular value decomposition: Theory and applications
 In Proc. European Conf. on Computer Vision, ECCV’00
, 2000
"... Abstract. The Singular Value Decomposition (SVD) of a matrix is a linear algebra tool that has been successfully applied to a wide variety of domains. The present paper is concerned with the problem of estimating the Jacobian of the SVD components of a matrix with respect to the matrix itself. An ex ..."
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Cited by 18 (1 self)
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Abstract. The Singular Value Decomposition (SVD) of a matrix is a linear algebra tool that has been successfully applied to a wide variety of domains. The present paper is concerned with the problem of estimating the Jacobian of the SVD components of a matrix with respect to the matrix itself. An exact analytic technique is developed that facilitates the estimation of the Jacobian using calculations based on simple linear algebra. Knowledge of the Jacobian of the SVD is very useful in certain applications involving multivariate regression or the computation of the uncertainty related to estimates obtained through the SVD. The usefulness and generality of the proposed technique is demonstrated by applying it to the estimation of the uncertainty for three different vision problems, namely selfcalibration, epipole computation and rigid motion estimation. 1