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## Revisiting Hartley's Normalized Eight-Point Algorithm (2003)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 16 - 4 self |

### Citations

4680 |
Multiple View Geometry in Computer Vision
- Hartley
- 2000
(Show Context)
Citation Context ...omputation and, as such, is of limited use. Many alternative methods have been advanced since LonguetHiggins’ proposal, including more sophisticated and computationally intensive iterative algorithms =-=[4]-=-, [5]. Hartley [6] discovered that the accuracy of the eight-point algorithm can be greatly improved if, prior to applying the method, a simple normalization of image data is performed. This fundament... |

782 |
A computer algorithm for reconstructing a scene from two projections
- Longuet-Higgins
- 1981
(Show Context)
Citation Context ... estimation techniques within a coherent framework. Index Terms—Epipolar equation, fundamental matrix, eight-point algorithm, data normalization. 1 INTRODUCTION æ IN a landmark paper, Longuet-Higgins =-=[1]-=- proposed the eight-point algorithm—a simple direct method for computation of the essential matrix. The algorithm extends straightforwardly to the computation of the fundamental matrix, the uncalibrat... |

543 |
What can be seen in three dimensions with an uncalibrated stereo rig
- Faugeras
- 1992
(Show Context)
Citation Context ...a simple direct method for computation of the essential matrix. The algorithm extends straightforwardly to the computation of the fundamental matrix, the uncalibrated analogue of the essential matrix =-=[2]-=-, [3]. While simple and fast, the algorithm is very sensitive to noise in the specification of the image coordinates serving as input for computation and, as such, is of limited use. Many alternative ... |

396 | The Geometry of Multiple Images
- Faugeras, Luong
- 2001
(Show Context)
Citation Context ...ation and, as such, is of limited use. Many alternative methods have been advanced since LonguetHiggins’ proposal, including more sophisticated and computationally intensive iterative algorithms [4], =-=[5]-=-. Hartley [6] discovered that the accuracy of the eight-point algorithm can be greatly improved if, prior to applying the method, a simple normalization of image data is performed. This fundamental mo... |

321 | Estimation of relative camera position for uncalibrated cameras
- Hartley
- 1992
(Show Context)
Citation Context ...ple direct method for computation of the essential matrix. The algorithm extends straightforwardly to the computation of the fundamental matrix, the uncalibrated analogue of the essential matrix [2], =-=[3]-=-. While simple and fast, the algorithm is very sensitive to noise in the specification of the image coordinates serving as input for computation and, as such, is of limited use. Many alternative metho... |

283 |
Statistical Optimization for Geometric Computation: Theory and Practice
- Kanatani
- 1996
(Show Context)
Citation Context ...ht-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of estimation techniques within a coherent framework (e.g., see [10], [11], =-=[12]-=-, [13]). 2 ESTIMATION PROBLEM A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point represented by a pair ðm1;m2Þ of coordinates, or equivalently,... |

203 | defense of the eight-point algorithm
- Hartley, In
- 1997
(Show Context)
Citation Context ... such, is of limited use. Many alternative methods have been advanced since LonguetHiggins’ proposal, including more sophisticated and computationally intensive iterative algorithms [4], [5]. Hartley =-=[6]-=- discovered that the accuracy of the eight-point algorithm can be greatly improved if, prior to applying the method, a simple normalization of image data is performed. This fundamental modification dr... |

116 | Outlier detection and motion segmentation
- Torr, Murray
- 1993
(Show Context)
Citation Context ...terms leads to a statistically more appropriate expression for minimization, and this in turn translates into a more accurate estimator. The proposed approach continues a line of research due to Torr =-=[7]-=-, Mühlich and Mester [8], and Torr and Fitzgibbon [9], in which variants of the normalized eight-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort t... |

93 | P.: Heteroscedastic regression in computer vision: Problems with bilinear constraint
- Leedan, Meer
- 2000
(Show Context)
Citation Context ...nt algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of estimation techniques within a coherent framework (e.g., see [10], [11], [12], =-=[13]-=-). 2 ESTIMATION PROBLEM A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point represented by a pair ðm1;m2Þ of coordinates, or equivalently, by th... |

75 | On the fitting of surfaces to data with covariances
- Chojnacki, Brooks, et al.
- 2000
(Show Context)
Citation Context ...rmalized eight-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of estimation techniques within a coherent framework (e.g., see =-=[10]-=-, [11], [12], [13]). 2 ESTIMATION PROBLEM A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point represented by a pair ðm1;m2Þ of coordinates, or e... |

55 |
The role of total least squares in motion analysis
- Muhlich, Mester
- 1998
(Show Context)
Citation Context ...ically more appropriate expression for minimization, and this in turn translates into a more accurate estimator. The proposed approach continues a line of research due to Torr [7], Mühlich and Mester =-=[8]-=-, and Torr and Fitzgibbon [9], in which variants of the normalized eight-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of est... |

23 |
On the optimization criteria used in two-view motion analysis
- Zhang
- 1998
(Show Context)
Citation Context ...st function Pn i1 r2i =varriŠ, where varrŠ denotes the variance of r. The latter function is closer in form to a natural cost function derivable from the principle of maximum likelihood (cf. [10], =-=[17]-=-). We show that under a certain statistical model of data distribution, Pn i1 r2 i =varriŠ can be identified with JNALS. Assume that, for each i 1; ...;n, the observed location mi is a realization... |

17 |
den Hengel, A.: Rationalising the Renormalisation Method of Kanatani
- Chojnacki, Brooks, et al.
- 2001
(Show Context)
Citation Context ...ed eight-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of estimation techniques within a coherent framework (e.g., see [10], =-=[11]-=-, [12], [13]). 2 ESTIMATION PROBLEM A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point represented by a pair ðm1;m2Þ of coordinates, or equival... |

8 | A new approach to constrained parameter estimation applicable to some computer vision problems
- Chojnacki, Brooks, et al.
- 2002
(Show Context)
Citation Context ...clusively the unconstrained minimizer, as this will prove critical to rationalizing the Hartley method. For alternative integrated approaches to computing a constrained minimizer, see the CFNS method =-=[14]-=-, [15] or the Gold Standard Method [4]. 3 ALGEBRAIC LEAST SQUARES A straightforward estimation method employs the cost function JALSðF; x1; ...;xnÞ jjFjj 2 X F n m i1 0T i Fmi 2 ð3Þ with kF kF ð P ... |

6 | Invariant fitting of two view geometry or ”in defiance of the 8 point algorithm
- Torr, Fitzgibbon
- 2002
(Show Context)
Citation Context ...ssion for minimization, and this in turn translates into a more accurate estimator. The proposed approach continues a line of research due to Torr [7], Mühlich and Mester [8], and Torr and Fitzgibbon =-=[9]-=-, in which variants of the normalized eight-point algorithm are analyzed statistically. The work presented here also forms part of a wider effort to place a variety of estimation techniques within a c... |

2 |
A New Constrained Parameter Estimator
- Hengel, Chojnacki, et al.
- 2002
(Show Context)
Citation Context ...ely the unconstrained minimizer, as this will prove critical to rationalizing the Hartley method. For alternative integrated approaches to computing a constrained minimizer, see the CFNS method [14], =-=[15]-=- or the Gold Standard Method [4]. 3 ALGEBRAIC LEAST SQUARES A straightforward estimation method employs the cost function JALSðF; x1; ...;xnÞ jjFjj 2 X F n m i1 0T i Fmi 2 ð3Þ with kF kF ð P i;j f2... |

1 |
Handbook of Matrices. Chicester
- Lütkepol
- 1996
(Show Context)
Citation Context ... ŠT . Let vecðF T Þ andsIEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 9, SEPTEMBER 2003 1173 uðxÞ vecðmm 0T Þ. Then, by applying standard rules for matrix operations =-=[16]-=-, we find that m 0T Fm tr m 0T Fm tr mm 0T F vec F T T vec mm 0T T ð4Þ uðxÞ; where tr denotes trace. With this formula, JALS can be written as where JALSð ; x1; ...;xnÞ k k 2 T A ; ð5Þ A Xn... |