## Simultaneous Linear Estimation of Multiple View Geometry and Lens Distortion (2001)

Citations: | 91 - 1 self |

### BibTeX

@MISC{Fitzgibbon01simultaneouslinear,

author = {Andrew W. Fitzgibbon},

title = {Simultaneous Linear Estimation of Multiple View Geometry and Lens Distortion},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

A bugbear of uncalibrated stereo reconstruction is that cameras which deviate from the pinhole model have to be pre-calibrated in order to correct for nonlinear lens distortion. If they are not, and point correspondence is attempted using the uncorrected images, the matching constraints provided by the fundamental matrix must be set so loose that point matching is significantly hampered. This paper shows how linear estimation of the fundamental matrix from two-view point correspondences may be augmented to include one term of radial lens distortion. This is achieved by (1) changing from the standard radiallens model to another which (as we show) has equivalent power, but which takes a simpler form in homogeneous coordinates, and (2) expressing fundamental matrix estimation as a Quadratic Eigenvalue Problem (QEP), for which efficient algorithms are well known. I derive the new estimator, and compare its performance against bundle-adjusted calibration-grid data. The new estimator is fast enough to be included in a RANSAC-based matching loop, and we show cases of matching being rendered possible by its use. I show how the same lens can be calibrated in a natural scene where the lack of straight lines precludes most previous techniques. The modification when the multi-view relation is a planar homography or trifocal tensor is described. 1.

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(Show Context)
Citation Context ...action. Our goal in this paper is not the accurate estimation of the lens distortion coefficients themselves. If accurate camera information is required, there is no recourse but to bundle adjustment =-=[18]-=-, initialized with (1) reasonable estimates of camera geometry and (2) good correspondences. It is in the provision of these two requisites that this paper contributes. 2. Notation We shall denote 2D ... |

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Citation Context ...o o x',, .2 o o y,,.2x,.,2s,.,x?.+?.,x].f + + A 2 0 0 0 0 0 0 0 0 ?j27.2 ]sf - o Gathering the three row vectors into three design matrices, we obtain the following quadratic eigenvalue problem (QEP) =-=[17]-=-: (Ol d- .O2 d- .203)f -- 0 (4) Such problems are analogous to standard 2nd order ODEs (replace . with partial derivative operators), and efficient numerical algorithms are readily available, for exam... |

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Citation Context ...are exemplified by the work of Zhang [20] and Stein [15], where the rigidity constraint is extended to include the parameters of the distortion model. Until now, however, these and related techniques =-=[4, 11, 12]-=- have relied on iterative methods to find the distortion parameters. As is usual wi[h such iterative methods, [heir convergence is not guaranteed, initial estimates must be found, and--although last w... |

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Citation Context ...ve the new model and show that it performs as well as the traditional approximation. True lens distortion curves are typically very complex, and systems which deal carefully with nonlinear distortion =-=[2, 19]-=- use high-order models or lookup tables to calibrate their cameras. For computer vision, however, and particularly for matching, accuracies of the order of a pixel are all that are required. Thus, it ... |

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Citation Context ... methods, which demand nothing more of the images than the rigidity assumptions which allow the computation of the fundamental matrix. Such methods are exemplified by the work of Zhang [20] and Stein =-=[15]-=-, where the rigidity constraint is extended to include the parameters of the distortion model. Until now, however, these and related techniques [4, 11, 12] have relied on iterative methods to find the... |

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Citation Context ...hive footage, or when using variable lens geometries. Previous work on the online estimation of lens distortion divides neatly into two strategic approaches. The first, known as the plumb line method =-=[2, 3, 10, 14, 16]-=- uses Figure 2: A 3D reconstruction of an 800-frame video of an office scene computed by a commercial camera tracker [1 ], without distortion correction (left) and with (right). The lines of the scene... |

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Citation Context ...are exemplified by the work of Zhang [20] and Stein [15], where the rigidity constraint is extended to include the parameters of the distortion model. Until now, however, these and related techniques =-=[4, 11, 12]-=- have relied on iterative methods to find the distortion parameters. As is usual wi[h such iterative methods, [heir convergence is not guaranteed, initial estimates must be found, and--although last w... |

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Citation Context ...second class of methods, which demand nothing more of the images than the rigidity assumptions which allow the computation of the fundamental matrix. Such methods are exemplified by the work of Zhang =-=[20]-=- and Stein [15], where the rigidity constraint is extended to include the parameters of the distortion model. Until now, however, these and related techniques [4, 11, 12] have relied on iterative meth... |

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Citation Context ...are exemplified by the work of Zhang [20] and Stein [15], where the rigidity constraint is extended to include the parameters of the distortion model. Until now, however, these and related techniques =-=[4, 11, 12]-=- have relied on iterative methods to find the distortion parameters. As is usual wi[h such iterative methods, [heir convergence is not guaranteed, initial estimates must be found, and--although last w... |

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Citation Context ...hive footage, or when using variable lens geometries. Previous work on the online estimation of lens distortion divides neatly into two strategic approaches. The first, known as the plumb line method =-=[2, 3, 10, 14, 16]-=- uses Figure 2: A 3D reconstruction of an 800-frame video of an office scene computed by a commercial camera tracker [1 ], without distortion correction (left) and with (right). The lines of the scene... |

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Citation Context ...hive footage, or when using variable lens geometries. Previous work on the online estimation of lens distortion divides neatly into two strategic approaches. The first, known as the plumb line method =-=[2, 3, 10, 14, 16]-=- uses Figure 2: A 3D reconstruction of an 800-frame video of an office scene computed by a commercial camera tracker [1 ], without distortion correction (left) and with (right). The lines of the scene... |

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Citation Context ...mation of r and Let us now derive this paper's main contribution. it is known that in order to compute the fundamental matrix from perfect point correspondences psp', we may use the 8-point algorithm =-=[6]-=-. In this section, we show how the 8-point algoritNn can be modified to include ,, the distortion parameter, and thus compute p from distorted, measurable, points xsx'. 4.1. Review: The 8-point algori... |

1 |
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Citation Context ...y when lens distortion would otherwise hinder the process. The most successful current techniques for matching interest points are based on the geometric constraints offered by multiple-view geometry =-=[8]-=-. These are effective because fast linear algorithms exist for the computation of the relationships, allowing their computation to form the kernel of RANSAC-based matching algorithms. However, when im... |

1 |
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