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15
A general method for errorsinvariables problems in computer vision.
 Proc CVPR IEEE, 2:2018 —
, 2021
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Optimal Rigid Motion Estimation and Performance Evaluation with Bootstrap
 Proc. Conf. Computer Vision and Pattern Recognition , Fort Collins Co
, 1999
"... A new method for 3D rigid motion estimation is derived under the most general assumption that the measurements are corrupted by inhomogeneous and anisotropic, i.e., heteroscedastic noise. This is the case, for example, when the motion of a calibrated stereohead is to be determined from image pairs. ..."
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Cited by 26 (5 self)
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A new method for 3D rigid motion estimation is derived under the most general assumption that the measurements are corrupted by inhomogeneous and anisotropic, i.e., heteroscedastic noise. This is the case, for example, when the motion of a calibrated stereohead is to be determined from image pairs. Linearization in the quaternion space transforms the problem into a multivariate, heteroscedastic errorsin variables (HEIV) regression, from which the rotation and translation estimates are obtained simultaneously. The significant performance improvementisillustrated, for real data, by comparison with the results of quaternion, subspace and renormalization basedapproaches described in the literature. Extensive use is made of bootstrap, an advanced numerical tool from statistics, both to estimate the covariances of the 3D data points and to obtain confidence regions for the rotation and translation estimates. Bootstrap enables an accurate recovery of these information using only the two image pairs serving as input.
Performance Assessment by Resampling: Rigid Motion Estimators Bogdan Matei
 IN K.W. Bowyer and P.J. Phillips, Empirical Evaluation Techniques in Computer Vision, IEEE
, 1998
"... Quantitative assessment of performance in image understanding tasks with real data is difficult since the data is complex and the different computational modules most often interact. Employing modern statistical techniques we have developed a set of numerical tools which provide rigorous performance ..."
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Cited by 13 (3 self)
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Quantitative assessment of performance in image understanding tasks with real data is difficult since the data is complex and the different computational modules most often interact. Employing modern statistical techniques we have developed a set of numerical tools which provide rigorous performance measures derived solely from the given input. Covariance matrices and confidence intervals are computed for the estimated parameters and individually for the corrected data points. As an example, the proposed methodology is applied to compare rigid motion estimators. 1: Performance assessment in image understanding The lack of universally accepted, rigorous performance assessment methodology is considered by many as one of the major bottlenecks of progress in image understanding. In a recent paper Christensen and Forstner [7] discuss several objections against the widespread use of evaluation techniques. Most of these objections are well justified.
Trajectory Triangulation over Conic Sections
 In ICCV99
, 1999
"... We consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera, i.e., to reconstruct moving objects from lineofsight measurements only. The task is feasible only when some constraints are placed on the shape of the trajectory of the moving point. ..."
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Cited by 9 (5 self)
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We consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera, i.e., to reconstruct moving objects from lineofsight measurements only. The task is feasible only when some constraints are placed on the shape of the trajectory of the moving point. We coin the family of such tasks as "trajectory triangulation". In this paper we focus on trajectories whose shape is a conicsection and show that generally 9 views are sufficient for a unique reconstruction of the moving point and fewer views when the conic is a known type (like a circle in 3D Euclidean space for which 7 views are sufficient.) Experiments demonstrate that our solutions are practical. The paradigm of Trajectory Triangulation in general pushes the envelope of processing dynamic scenes forward. Thus static scenes become a particular case of a more general task of reconstructing scenes rich with moving objects (where an object could be a single point.)
An analysis of linear subspace approaches for computer vision and pattern recognition
 International Journal of Computer Vision (IJCV
, 2006
"... Abstract. Linear subspace analysis (LSA) has become rather ubiquitous in a wide range of problems arising in pattern recognition and computer vision. The essence of these approaches is that certain structures are intrinsically (or approximately) low dimensional: for example, the factorization approa ..."
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Cited by 8 (2 self)
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Abstract. Linear subspace analysis (LSA) has become rather ubiquitous in a wide range of problems arising in pattern recognition and computer vision. The essence of these approaches is that certain structures are intrinsically (or approximately) low dimensional: for example, the factorization approach to the problem of structure from motion (SFM) and principal component analysis (PCA) based approach to face recognition. In LSA, the singular value decomposition (SVD) is usually the basic mathematical tool. However, analysis of the performance, in the presence of noise, has been lacking. We present such an analysis here. First, the “denoising capacity ” of the SVD is analysed. Specifically, given a rankr matrix, corrupted by noise—how much noise remains in the rankr projected version of that corrupted matrix? Second, we study the “learning capacity ” of the LSAbased recognition system in a noisecorrupted environment. Specifically, LSA systems that attempt to capture a data class as belonging to a rankr column space will be affected by noise in both the training samples (measurement noise will mean the learning samples will not produce the “true subspace”) and the test sample (which will also have measurement noise on top of the ideal clean sample belonging to the “true subspace”). These results should help one to predict aspects of performance and to design more optimal systems in computer vision, particularly in tasks, such as SFM and face recognition. Our analysis agrees with certain observed phenomenon, and these observations, together with our simulations, verify the correctness of our theory.
Heteroscedastic Hough Transform (HtHT): An efficient method for robust line fitting in the `Errors in the Variables' problem
, 2000
"... this paper we present an efficient method for robust line fitting in the heteroscedastic `errors in the variables' problem, with correlated noise. It is assumed that the covariance matrix associated with each data point is known. The method suggested is easy to implement, fast to compute, and p ..."
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Cited by 4 (1 self)
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this paper we present an efficient method for robust line fitting in the heteroscedastic `errors in the variables' problem, with correlated noise. It is assumed that the covariance matrix associated with each data point is known. The method suggested is easy to implement, fast to compute, and provides a systematic solution to this important practical problem. The organization of the paper is as follows. In section 2 the problem is defined and formulated as a global optimization problem and the general approach to solving it is outlined. In section 3 it is shown that the objective function can be simplified and has a special structure. It is further shown that this special structure leads to an elegant, efficient computational solution. Experimental results are presented in section 4. In section 5 an alternative definition of the problem is considered and limitations of the method are discussed
Parameterized Image Varieties and Estimation with Bilinear Constraints
 In: Proc. IEEE Conf. Comp. Vision Patt. Recog. Fort Collins, CO
, 1999
"... This paper addresses the problem of reliably estimating the coefficients of the parameterized image variety (PIV) [3] associated with the set of weak perspective images of a rigid scene, with applications in imagebased rendering. Exploiting the fact that the constraints defining the PIV are linear ..."
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Cited by 2 (2 self)
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This paper addresses the problem of reliably estimating the coefficients of the parameterized image variety (PIV) [3] associated with the set of weak perspective images of a rigid scene, with applications in imagebased rendering. Exploiting the fact that the constraints defining the PIV are linear in its coefficients and bilinear in the image data, the estimation procedure is cast in the errorsinvariables framework and solved using the method proposed in [10] for this type of problems. The proposed approach has been implemented, and experiments with real data are shown to yield much better prediction power than the original method based on singular value decomposition. Extensions to the more difficult case of paraperspective projection are briefly discussed.
A Bilinear Approach to the Parameter Estimation of a general Heteroscedastic Linear System with Application to Conic Fitting
"... A bilinear approach to the parameter estimation of a general heteroscedastic linear system, with application to conic fitting ..."
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A bilinear approach to the parameter estimation of a general heteroscedastic linear system, with application to conic fitting
Weak Calibration and ImageBased Rendering Algorithms
, 1999
"... This thesis introduces twonovel techniques for the analysis and synthesis of image sequences: a linear algorithm for weak calibration of a stereo rig from point correspondences, and an algorithm for imagebased rendering without explicit threedimensional reconstruction based on point and line corre ..."
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This thesis introduces twonovel techniques for the analysis and synthesis of image sequences: a linear algorithm for weak calibration of a stereo rig from point correspondences, and an algorithm for imagebased rendering without explicit threedimensional reconstruction based on point and line correspondences. By recasting the epipolar constraint in a projective setting with an appropriate basis choice, we first show that Jepson's and Heeger's linear subspace algorithm for infinitesimal motion estimation can be generalized to the finite motion case. This yields a linear method for weak calibration. The algorithm has been implemented and tested on both real and synthetic images, and it is compared to other linear and nonlinear approaches to weak calibration. We then show that the set of all images of a rigid scene taken by a Euclidean camera is a sixdimensional variety, and we introduce a parameterization (called parameterized image variety, or PIV in short) of this variety for weak perspective and paraperspective cameras in terms of the image positions of three reference points. This parameterization can be estimated via linear leastsquares and nonlinear leastsquares with lowdegree equations. We use parameterized image varieties of both point and line features to synthesize new images from a set of prerecorded pictures without actual threedimensional reconstruction (imagebased rendering) in an integrated framework. The method has been implemented and extensively tested on real data sets. Finally, we show how to adapt recent advances in statisticallyunbiased leastsquares methods to our imagebased rendering approach. The pointbased PIV involves equations with bilinear or higherorder data dependencies and we showhow to efficiently estimate its parameters by adapting L...
Camera Calibration and Euclidean Reconstruction from Known Observer Translations
, 1998
"... We present a technique for camera calibration and Euclidean reconstruction from multiple images of the same scene. Unlike standard Tsai's camera calibration from a known scene, we exploited controlled known motions of the camera to obtain its calibration and Euclidean reconstruction without any ..."
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Cited by 1 (0 self)
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We present a technique for camera calibration and Euclidean reconstruction from multiple images of the same scene. Unlike standard Tsai's camera calibration from a known scene, we exploited controlled known motions of the camera to obtain its calibration and Euclidean reconstruction without any knowledge about the scene. We consider three linearly independent translations of an uncalibrated camera mounted on a robot arm that provides us with four views of the scene. The translations of the robot arm are measured in a robot coordinate system. This special, but still realistic, arrangement allowed us to find a linear algorithm for recovering all intrinsic camera calibration parameters, the rotation of the camera with respect to the robot coordinate system, and proper scaling factors for all points allowing their Euclidean reconstruction. The experiments showed that an efficient and robust algorithm was obtained by exploiting Total Least Squares in combination with careful normalization o...