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## Estimation of nonlinear errors-in-variables models for computer vision applications (2006)

Venue: | IEEE Trans. Patt. Anal. Mach. Intell |

Citations: | 33 - 6 self |

### Citations

4680 | Multiple View Geometry in Computer Vision - Hartley - 2000 |

965 | Closed-form solution of absolute orientation using orthonormal matrices
- Horn, Hilden, et al.
- 1988
(Show Context)
Citation Context ... popular closed-form estimators are based on the SVD method proposed originally by Arun et al. [1] and refined later by Umeyama [40], respectively, using the quaternion method proposed by Horn et al. =-=[17]-=-. Both estimators were shown to numerically yield the same results [10], [30] since they utilize different representation for the same constraint solved by the SVD and the eigenvalue decomposition. Th... |

710 | Introductory techniques for 3-D computer vision - Trucco, Verri - 1998 |

631 | The geometry of algorithms with orthogonality constraints
- Edelman, Arias, et al.
- 1998
(Show Context)
Citation Context ...choices made for the objective functions applied to solve 3D computer vision problems (structure from motion, triangulation) was published by Ma et al. [24] using optimization on Riemannian manifolds =-=[9]-=-. The task of direct calibration provides a simple example illustrating the importance of a correct treatment of the statistical model selected. Since, in this paper, we are not concerned with the iss... |

628 | Measurement Error Models - Fuller - 1987 |

571 |
Least-squares fitting of two 3-D point sets
- Arun, Huang, et al.
- 1987
(Show Context)
Citation Context ...oblem in 3D scene reconstruction from unregistered 3D data acquired by range sensors or stereo heads. Two popular closed-form estimators are based on the SVD method proposed originally by Arun et al. =-=[1]-=- and refined later by Umeyama [40], respectively, using the quaternion method proposed by Horn et al. [17]. Both estimators were shown to numerically yield the same results [10], [30] since they utili... |

353 |
Kronecker Products and Matrix Calculus with Applications
- Graham
- 1981
(Show Context)
Citation Context ...etero.html. APPENDIX The Kronecker product and the vec operator provide an elegant mathematical framework for vector calculus and are extensively used throughout the paper. The survey [2] or the book =-=[15]-=- contain all the necessary background and also cover vector calculus. Given a matrix A 2 IR p q , the vectorization of A is defined as the pq-dimensional vector 2 3 A a1a2aqŠ ai 2 IR p vecðAÞ 6 4... |

332 |
Least-squares estimation of transformation parameters between two point patterns
- Umeyama
- 1991
(Show Context)
Citation Context ...from unregistered 3D data acquired by range sensors or stereo heads. Two popular closed-form estimators are based on the SVD method proposed originally by Arun et al. [1] and refined later by Umeyama =-=[40]-=-, respectively, using the quaternion method proposed by Horn et al. [17]. Both estimators were shown to numerically yield the same results [10], [30] since they utilize different representation for th... |

283 |
Statistical Optimization for Geometric Computation: Theory and Practice
- Kanatani
- 1996
(Show Context)
Citation Context ...ndled. Linear(ized) algorithms have the appeal of yielding close form solutions; however, they are notoriously sensitive in practice to errors in the measurements and result in biased solutions [19], =-=[20]-=-. Nonlinear objective functions have a better performance, however, they result in iterative algorithms which may not converge if initialized far away from the correct solution. A rigorous study on va... |

272 |
Kronecker products and matrix calculus in system theory
- Brewer
- 1978
(Show Context)
Citation Context .../riul/research/hetero.html. APPENDIX The Kronecker product and the vec operator provide an elegant mathematical framework for vector calculus and are extensively used throughout the paper. The survey =-=[2]-=- or the book [15] contain all the necessary background and also cover vector calculus. Given a matrix A 2 IR p q , the vectorization of A is defined as the pq-dimensional vector 2 3 A a1a2aqŠ ai 2 ... |

194 |
Geometric computation for machine vision
- Kanatani
- 1993
(Show Context)
Citation Context ...are handled. Linear(ized) algorithms have the appeal of yielding close form solutions; however, they are notoriously sensitive in practice to errors in the measurements and result in biased solutions =-=[19]-=-, [20]. Nonlinear objective functions have a better performance, however, they result in iterative algorithms which may not converge if initialized far away from the correct solution. A rigorous study... |

154 |
Bundle adjustment-a modern synthesis. Vision algorithms: theory and practice
- Triggs, McLauchlan, et al.
- 2000
(Show Context)
Citation Context ...uardt Method A widely employed optimization technique for least squares problems is the Levenberg-Marquardt (LM) method [34, pp. 683-689]. An excellent review of the topic is done by Triggs et al. in =-=[38]-=-. Assume that the ideal values of the measurements yio 2 IR q , xio 2 IR m , i 1; ...;n obey the constraint y io gðxio; oÞ i 1; ...;n ð96Þs1546 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE ... |

150 |
Estimating 3D rigid body transformations: a comparison of four major algorithms
- Eggert, Lorusso, et al.
- 1997
(Show Context)
Citation Context ...iginally by Arun et al. [1] and refined later by Umeyama [40], respectively, using the quaternion method proposed by Horn et al. [17]. Both estimators were shown to numerically yield the same results =-=[10]-=-, [30] since they utilize different representation for the same constraint solved by the SVD and the eigenvalue decomposition. The SVD and quaternion-based representation are optimal only when the 3D ... |

132 |
Fitting conic sections to “very scattered” data: An iterative refinement of the Bookstein algorithm
- Sampson
- 1982
(Show Context)
Citation Context ...VISION APPLICATIONS 1541 we will show that different particular cases of (40) appear in several techniques used in computer vision: the generalized total least squares (GTLS) [18], the Sampson method =-=[35]-=-, [16, p. 97], and the renormalization method of Kanatani [20]. 3 SOLVING THE HEIV EQUATION The HEIV equation (40) is nonlinear since the unknown ^ also appears in the matrices Sð^Þ and Cð^Þ. An itera... |

93 | P.: Heteroscedastic regression in computer vision: Problems with bilinear constraint
- Leedan, Meer
- 2000
(Show Context)
Citation Context ...two matrices, since, for any a 6 0, Sða ^ Þa 2 Sð ^ Þ and Cða ^ Þa 2 Cð ^ Þ. The HEIV equation (40) generalizes for multivariate constraints with a separable parameter the scalar case described in =-=[23]-=-, which is also equivalent to the fundamental numerical scheme (FNS) of Chojnacki et al. [7]. Being the most general expression for solving separable EIV models linear in the parameter , the HEIV equa... |

88 | The Total Least Squares Problem - Huffel, Vandewalle - 1991 |

75 | On the fitting of surfaces to data with covariances
- Chojnacki, Brooks, et al.
- 2000
(Show Context)
Citation Context ...d values of the coefficients used in the HEIV equation, the proportionality factor being given by the equivalent noise variance 2 . 4.4 The Fundamental Numerical Scheme Chojnacki et al. proposed [5], =-=[6]-=-, [7] a different approach called Fundamental Numerical Scheme (FNS) as an alternative for solving (11). The FNS was derived only for scalar constraints, m 1. Assuming that, in the objective functio... |

58 | A general method for errors-invariables problems in computer vision
- Matei, Meer
(Show Context)
Citation Context ...s violated since, in each pair of rows of B, the noisy image coordinates z1i or z2i are multiplied by a different vector Mio, as seen in (7). For such a noise process, the TLS estimate becomes biased =-=[28]-=- and should only be used as the initial solution in a nonlinear estimation procedure minimizing geometric distances, i.e., the distance between a measured image point and the unknown projection of the... |

48 | Optimization criteria and geometric algorithms for motion and structure estimation
- Ma, Kosecka, et al.
- 2001
(Show Context)
Citation Context ...e correct solution. A rigorous study on various choices made for the objective functions applied to solve 3D computer vision problems (structure from motion, triangulation) was published by Ma et al. =-=[24]-=- using optimization on Riemannian manifolds [9]. The task of direct calibration provides a simple example illustrating the importance of a correct treatment of the statistical model selected. Since, i... |

39 | Robust regression with projection based m-estimators, in
- Chen, Meer
(Show Context)
Citation Context ...not concerned with the issue of outlier tolerance, we will assume that all the data points obey the model. Robust techniques, especially developed for the employed class of models, exist, e.g., [43], =-=[4]-=-, and can be easily integrated into the proposed estimator. In direct calibration, the relation between a 3D point mio on a calibration grid and its 2D projection in the image plane, zio z1io z2ioŠ ... |

33 |
The generic bilinear calibrationestimation problem
- Koenderink, Doorn
- 1997
(Show Context)
Citation Context ...ny computer vision problems can already be solved without using an explicit reparameterization in . It can be demonstrated that (4) can be generalized to account for the gauge matrix H 2 IR p p [21], =-=[22]-=-, [31] ðzioÞHH 1 o 0 ð5Þ when the gauge matrix is orthonormal, H > H Ip. The constraint (4) also has an ambiguity to multiplication with a nonzero constant. To remove this ambiguity, an additional... |

25 | Optimal rigid motion estimation and performance evaluation with bootstrap
- Matei, Meer
- 1999
(Show Context)
Citation Context ...t of Qo is equal to q0o. Because q0o 0 represents a discontinuity in the quaternion representation, the application of this approach is not recommended for cases when the rotation angle is close to =-=[27]-=-. Note that (106) is of the form (4) with z > i u> i v> i q > o Š and > o > o Š and that ðzioÞ is linear in the measurements zi. The quaternion method is a TLS estimator in which the quaternion ^... |

20 | From FNS to HEIV: A link between two vision parameter estimation methods
- Chojnacki, Brooks, et al.
- 2004
(Show Context)
Citation Context ...tion (40) generalizes for multivariate constraints with a separable parameter the scalar case described in [23], which is also equivalent to the fundamental numerical scheme (FNS) of Chojnacki et al. =-=[7]-=-. Being the most general expression for solving separable EIV models linear in the parameter , the HEIV equation plays a central role in many computer vision problems. In Section 4,sMATEI AND MEER: ES... |

20 |
Heteroscedastic Errors-In-Variables Models in Computer Vision
- Matei
- 2001
(Show Context)
Citation Context ... better numerical behavior for the iterative procedure. 3.2 Statistical Properties To assess the accuracy of the HEIV estimates, the first two moments can be computed in first order approximation. In =-=[25]-=-, it is shown that the HEIV estimate is consistent for linear functions ðziÞ and heteroscedastic measurements and that the parameter ^ is unbiased in the first order approximation. The theoretical cov... |

18 | A new constrained parameter estimator for computer vision applications
- Chojnacki, Brooks, et al.
- 2004
(Show Context)
Citation Context ... finding the solution ^ which satisfies (10) and (57) simultaneously. A special case of scalar additional constraints, homogeneous of degree , ðt ^ Þt ð ^ Þ, was explored in the CFNS algorithm [41], =-=[8]-=-, which showed that the constraint can be elegantly included in the optimization scheme. The approach proposed in the paper is generally applicable to multivariate constraints which are not necessaril... |

18 | Optimal Estimation of Three-Dimensional Rotation and Reliability Evaluation,” in Computer Vision — ECCV’98: 5th European Conference on Computer Vision
- Ohta, Kanatani
(Show Context)
Citation Context ...e ^R of Ro must satisfy the constraint ^R^R > I3. An elegant rotation parametrization is obtained by using quaternions, which are four-dimensional unit vectors q q0 q1 q2 q3 Š > . It can be shown =-=[33]-=- that the constraint (105) is equivalent to Zioq o þ o 0; ð106Þ where Zio 2 IR 3 4 is the matrix of the carriers 2 v1io u1io Zio 4 v2io u2io 0 v3io þ u3io v3io 0 3 u3io v2io þ u2io v1io u1io5 v3io ... |

17 |
den Hengel, A.: Rationalising the Renormalisation Method of Kanatani
- Chojnacki, Brooks, et al.
- 2001
(Show Context)
Citation Context ...pected values of the coefficients used in the HEIV equation, the proportionality factor being given by the equivalent noise variance 2 . 4.4 The Fundamental Numerical Scheme Chojnacki et al. proposed =-=[5]-=-, [6], [7] a different approach called Fundamental Numerical Scheme (FNS) as an alternative for solving (11). The FNS was derived only for scalar constraints, m 1. Assuming that, in the objective fu... |

16 | Gauge and gauge transformations for uncertainty description of geometric structure with indeterminacy
- Kanatani, Morris
(Show Context)
Citation Context ...re, many computer vision problems can already be solved without using an explicit reparameterization in . It can be demonstrated that (4) can be generalized to account for the gauge matrix H 2 IR p p =-=[21]-=-, [22], [31] ðzioÞHH 1 o 0 ð5Þ when the gauge matrix is orthonormal, H > H Ip. The constraint (4) also has an ambiguity to multiplication with a nonzero constant. To remove this ambiguity, an addi... |

15 | Gauge Fixing for Accurate 3D Estimation
- Morris, Kanatani, et al.
- 2001
(Show Context)
Citation Context ...puter vision problems can already be solved without using an explicit reparameterization in . It can be demonstrated that (4) can be generalized to account for the gauge matrix H 2 IR p p [21], [22], =-=[31]-=- ðzioÞHH 1 o 0 ð5Þ when the gauge matrix is orthonormal, H > H Ip. The constraint (4) also has an ambiguity to multiplication with a nonzero constant. To remove this ambiguity, an additional restr... |

14 | Reduction of bias in maximum likelihood ellipse fitting
- Matei, Meer
- 2000
(Show Context)
Citation Context ...ut of the box optimization packages, such as fmincon without addressing the sparseness of the Hessian matrix induced by the nuisance parameters would result in inefficient and slow execution time. In =-=[29]-=-, the HEIV algorithm was applied to conic fitting. Simulations with synthetic and real data showed significantly improved performance compared with the direct least squares (DLS) method and was more n... |

13 | Performance assessment by resampling: rigid motion estimators
- Matei, Meer, et al.
- 1998
(Show Context)
Citation Context ...ly by Arun et al. [1] and refined later by Umeyama [40], respectively, using the quaternion method proposed by Horn et al. [17]. Both estimators were shown to numerically yield the same results [10], =-=[30]-=- since they utilize different representation for the same constraint solved by the SVD and the eigenvalue decomposition. The SVD and quaternion-based representation are optimal only when the 3D measur... |

8 | Balanced recovery of 3d structure and camera motion from uncalibrated image sequences
- Georgescu, Meer
- 2002
(Show Context)
Citation Context ...ance matrices. The HEIV algorithm was used by Georgescu and Meer to initialize the bundle adjustment of parameters for the recovery of 3D structure and camera motion from uncalibrated video sequences =-=[13]-=-. Experiments using simulated TABLE 2 Performance Evaluation: PUMA Sequence and real sequences showed the improved behavior of the final estimates when the HEIV was employed at various stages, such as... |

7 |
Registration Via Direct Methods: A Statistical Approach
- Bride, Meer
- 2001
(Show Context)
Citation Context ... algorithm. For stable configurations of the scene and camera, the HEIV solution and the Gold Standard method were shown to be numerically very close, with the HEIV providing a faster convergence. In =-=[3]-=-, the HEIV estimator was applied to 2D rigid motion between two images. The approach was further extended by Nestares and Fleet for affine motion estimation [32]. The authors reported a significant im... |

7 |
A versatile method for trifocal tensor estimation
- Matei, Georgescu, et al.
- 2001
(Show Context)
Citation Context ...es ð^Þ ^ . However, in some cases, minimizing the unconstrained objective function can lead to convergence to the incorrect solution as shown by Matei et al. in the case of trifocal tensor estimation =-=[26]-=-. Instead, the HEIV algorithm derived in Section 2 can be extended to allow enforcing the constraint ð^Þ ^ at each iteration of the HEIV algorithm. Expressing (39) in the parameter ^ , the cost functi... |

6 |
Error-in-variables likelihood functions for motion estimation
- Nestares, Fleet
- 2003
(Show Context)
Citation Context ...HEIV providing a faster convergence. In [3], the HEIV estimator was applied to 2D rigid motion between two images. The approach was further extended by Nestares and Fleet for affine motion estimation =-=[32]-=-. The authors reported a significant improvement of the accuracy of the optical flow estimated with the HEIV scheme due to the non i.i.d. nature of the noise affecting the measurements. Förstner publi... |

6 | A balanced approach to 3D tracking from image streams
- Subbarao, Meer, et al.
- 2005
(Show Context)
Citation Context ... camera resectioning, in terms of faster convergence of the bundle adjustment when initialized with the HEIV estimates, and no instances of failed convergence to the correct solution. Subbarao et al. =-=[37]-=- employed the HEIV algorithm for estimating the camera pose for 3D tracking within an augmented reality application. The authors reported superior accuracy compared to using the Levenberg-Marquardt op... |

3 |
Robust estimation in the errors-in-variables model
- Zamar
- 1989
(Show Context)
Citation Context ...e are not concerned with the issue of outlier tolerance, we will assume that all the data points obey the model. Robust techniques, especially developed for the employed class of models, exist, e.g., =-=[43]-=-, [4], and can be easily integrated into the proposed estimator. In direct calibration, the relation between a 3D point mio on a calibration grid and its 2D projection in the image plane, zio z1io z... |

2 |
Errors in Variables for Numerical Analysis
- Stewart
- 1997
(Show Context)
Citation Context ...tatistical models. The difference between structural and functional affects models more their respective statistical analysis, rather than the manner in which the estimates of parameters are obtained =-=[36]-=-. Numerous estimation techniques are available in the literature depending on the objective function selected, the manner in which the objective function is minimized, or on how the intrinsic constrai... |

2 |
A New Constrained Parameter Estimator
- Hengel, Chojnacki, et al.
- 2002
(Show Context)
Citation Context ...rectly finding the solution ^ which satisfies (10) and (57) simultaneously. A special case of scalar additional constraints, homogeneous of degree , ðt ^ Þt ð ^ Þ, was explored in the CFNS algorithm =-=[41]-=-, [8], which showed that the constraint can be elegantly included in the optimization scheme. The approach proposed in the paper is generally applicable to multivariate constraints which are not neces... |

1 |
On Estimating 3D Points and Lines from 2D Points and Lines,” Festschrift Anläßlich des 60. Geburtstages von
- Förstner
- 2001
(Show Context)
Citation Context ...imated with the HEIV scheme due to the non i.i.d. nature of the noise affecting the measurements. Förstner published an optimal algorithm for 3D points or 3D lines recovery from 2D points or 2D lines =-=[11]-=-. The recovery of lines was achieved with Plücker coordinates. The author followed the HEIV algorithm presented by Matei and Meer in [28], however, he obtained an ordinary eigenvalue problem instead o... |

1 |
Analysis and Properties of GTLS in Problem AX B When Some or All Columns of A Are Subject to Errors
- Huffel, Vandewalle
- 1989
(Show Context)
Citation Context ...BLES MODELS FOR COMPUTER VISION APPLICATIONS 1541 we will show that different particular cases of (40) appear in several techniques used in computer vision: the generalized total least squares (GTLS) =-=[18]-=-, the Sampson method [35], [16, p. 97], and the renormalization method of Kanatani [20]. 3 SOLVING THE HEIV EQUATION The HEIV equation (40) is nonlinear since the unknown ^ also appears in the matrice... |

1 |
Image Matching Software,” 1995, http://www. inria.fr/robotvis/personnel/zzhang. Bogdan C. Matei received the DiplEngn and MSc degrees in electrical and computer engineering from the Polytechnic University of Bucharest
- Zhang
- 1994
(Show Context)
Citation Context ...s labeled. (a) and (b) Castle frames from the CIL-CMU database. (c) and (d) Frames from the PUMA sequence. sequence are presented in Table 1. There were 93 points matched across the four frames using =-=[44]-=-. The estimation errors for the HEIV are much smaller than for the quaternion method which is heavily biased under heteroscedastic noise. The HEIV errors are also smaller than the renormalization meth... |