## Autocalibration from planar scenes (1998)

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Venue: | European Conference on Computer Vision |

Citations: | 126 - 2 self |

### BibTeX

@INPROCEEDINGS{Triggs98autocalibrationfrom,

author = {Bill Triggs},

title = {Autocalibration from planar scenes},

booktitle = {European Conference on Computer Vision},

year = {1998}

}

### Years of Citing Articles

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### Abstract

This paper describes a theory and a practical algorithm for the autocalibration of a moving projective camera, from views of a planar scene. The unknown camera calibration, and (up to scale) the unknown scene geometry and camera motion are recovered from the hypothesis that the camera’s internal parameters remain constant during the motion. This work extends the various existing methods for non-planar autocalibration to a practically common situation in which it is not possible to bootstrap the calibration from an intermediate projective reconstruction. It also extends Hartley’s method for the internal calibration of a rotating camera, to allow camera translation and to provide 3D as well as calibration information. The basic constraint is that the projections of orthogonal direction vectors (points at infinity) in the plane must be orthogonal in the calibrated camera frame of each image. Abstractly, since the two circular points of the 3D plane (representing its Euclidean structure) lie on the 3D absolute conic, their projections into each image must lie on the absolute conic’s image (representing the camera calibration). The resulting numerical algorithm optimizes this constraint over all circular points and projective calibration parameters, using the inter-image homographies as a projective scene representation.

### Citations

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Citation Context ...arametrization of the problem guarantees that all variables are of O(1) and fairly well decoupled, so preconditioning is not necessary. The Choleski routine uses diagonal pivoting and Gill & Murray's =-=[4]-=- minimum-diagonal-value regularization to provide local stability. The regularizer is also manipulated in much the same way as a Levenberg-Marquardt parameter to ensure that each step actually reduces... |

366 | Camera self-calibration: Theory and experiments
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(Show Context)
Citation Context ...example the knowledge that the camera's internal parameters are constant during the motion, and the inter-image consistency constraints that this entails. Since the seminal work of Maybank & Faugeras =-=[14, 3]-=-, a number of different approaches to autocalibration have been developed [5, 6, 1, 27, 26, 2, 13, 9, 16, 15, 22, 10]. For the `classical' problem of 4/3/98. This is an extension of a preliminary vers... |

295 |
Faugeras. A theory of selfcalibration of a moving camera
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(Show Context)
Citation Context ...example the knowledge that the camera's internal parameters are constant during the motion, and the inter-image consistency constraints that this entails. Since the seminal work of Maybank & Faugeras =-=[14, 3]-=-, a number of different approaches to autocalibration have been developed [5, 6, 1, 27, 26, 2, 13, 9, 16, 15, 22, 10]. For the `classical' problem of 4/3/98. This is an extension of a preliminary vers... |

245 |
Statistical Optimization for Geometric Computation: Theory and Practice
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Citation Context ...t such a validation step. One very useful simplification is to ignore the dependence of V \Gamma1 e onsin cost function derivatives. This gives self-consistent or iterative re-weighting schemes (e.g. =-=[12]-=-), where V e is treated as a constant within each optimization step, but updated at the end of it. One can show that the missing terms effectively displace the cost derivative evaluation point from th... |

218 | Autocalibration and the absolute quadric
- Triggs
- 1997
(Show Context)
Citation Context ...he motion, and the inter-image consistency constraints that this entails. Since the seminal work of Maybank & Faugeras [14, 3], a number of different approaches to autocalibration have been developed =-=[5, 6, 1, 27, 26, 2, 13, 9, 16, 15, 22, 10]-=-. For the `classical' problem of 4/3/98. This is an extension of a preliminary version of my ECCV'98 paper [23]. It goes into a little more detail on various topics, notably Kruppa instability, and co... |

216 | A factorization based algorithm for multi-image projective structure and motion
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Citation Context ...ng appropriate scale factors for e and F, the Kruppa constraint can be written as F! F ? = [ e ] \Theta ! [ e ] ? \Theta . Since x is fixed, x ? F x = 0 and by the projective depth recovery relations =-=[20]-=- F x =s[ e ] \Theta x wheresis the relative projective depth (projective scale factor) of x in the two images. Hence F(! +sx x ? )F ? = [ e ] \Theta (! +sx x ? )[ e ] ? \Theta . With these normalizati... |

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Algebraic projective geometry
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Citation Context ...nfinity"), or dually the dot product between (normals to) hyperplanes. The former leads to the stratified "hyperplane at infinity + absolute (direction) conic" formulation (affine + met=-=ric structure) [17], the latt-=-er to the "absolute (hyperplane) quadric" one [22]. These are just dual ways of saying the same thing. The hyperplane formalism is preferable for `pure' autocalibration where there is no a p... |

177 | Canonic representations for the geometries of multiple projective views
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Citation Context ...he motion, and the inter-image consistency constraints that this entails. Since the seminal work of Maybank & Faugeras [14, 3], a number of different approaches to autocalibration have been developed =-=[5, 6, 1, 27, 26, 2, 13, 9, 16, 15, 22, 10]-=-. For the `classical' problem of 4/3/98. This is an extension of a preliminary version of my ECCV'98 paper [23]. It goes into a little more detail on various topics, notably Kruppa instability, and co... |

153 | Self-calibration from multiple views with a rotating camera
- Hartley
- 1994
(Show Context)
Citation Context ...he motion, and the inter-image consistency constraints that this entails. Since the seminal work of Maybank & Faugeras [14, 3], a number of different approaches to autocalibration have been developed =-=[5, 6, 1, 27, 26, 2, 13, 9, 16, 15, 22, 10]-=-. For the `classical' problem of 4/3/98. This is an extension of a preliminary version of my ECCV'98 paper [23]. It goes into a little more detail on various topics, notably Kruppa instability, and co... |

136 | In Defence of the 8-point Algorithm
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- 1995
(Show Context)
Citation Context ...nd (ii) severe constriction of the domain of convergence of the optimization method. Appropriate balancing or preconditioning (numerical scaling of the variables and constraints, e.g. as advocated in =-=[7, 8] or any nu-=-merical optimization text) is the first step towards eliminating such problems, but it is not the whole story. In any case it begs the question of what is "balanced". It is not always approp... |

108 | Euclidean reconstruction from images sequences with varying and unknown focal length and principal point - Heyden, Åström |

104 | Self-calibration from Image Triplets
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- 1996
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Citation Context |

104 | Critical motion sequences for monocular self-calibration and uncalibrated Euclidean reconstruction
- Sturm
- 1997
(Show Context)
Citation Context ...ntly [9, 15, 22]. The main problem is that classical autocalibration has some restrictive intrinsic degeneracies --- classes of motion for which no algorithm can recover a full unique solution. Sturm =-=[18, 19]-=- has given a catalogue of these. In particular, at least 3 views, some translation and some rotation about at least two non-aligned axes are required. Planar Autocalibration: All of the existing appro... |

72 | Euclidean reconstruction from constant intrinsic parameters
- Heyden, Astrom
- 1996
(Show Context)
Citation Context |

48 | Stratification of 3-d vision: Projective, affine, and metric representations
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- 1995
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Citation Context |

48 | Camera self-calibration from video sequences: the Kruppa equations revisited. Research Report 2793
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Citation Context |

43 | A stratified approach to metric self-calibration
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- 1997
(Show Context)
Citation Context |

41 | Self-calibration and Euclidean reconstruction using motions of a stereo rig
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(Show Context)
Citation Context ...e projective reconstruction [5, 9, 15, 22, 10]. More specialized methods exist for particular types of motion and simplified calibration models [6, 24, 1, 16]. Stereo heads can also be autocalibrated =-=[27, 11]-=-. Solutions are still --- in theory --- possible if some of the intrinsic parameters are allowed to vary [9, 15]. Hartley [6] has given a particularly simple internal calibration method for the case o... |

37 | Motion of an uncalibrated stereo rig: self-calibration and metric reconstruction
- Zhang, Luong, et al.
- 1996
(Show Context)
Citation Context |

33 |
Motion analysis with a camera with unknown, and possibly varying intrinsic parameters
- Vieville, Faugeras
- 1995
(Show Context)
Citation Context ...pproaches based on the `rectification' of an intermediate projective reconstruction [5, 9, 15, 22, 10]. More specialized methods exist for particular types of motion and simplified calibration models =-=[6, 24, 1, 16]-=-. Stereo heads can also be autocalibrated [27, 11]. Solutions are still --- in theory --- possible if some of the intrinsic parameters are allowed to vary [9, 15]. Hartley [6] has given a particularly... |

32 | Euclidean 3D reconstruction from image sequences with variable focal lengths
- Pollefeys, Gool, et al.
- 1996
(Show Context)
Citation Context |

23 |
Minimizing algebraic error
- Hartley
- 1998
(Show Context)
Citation Context ...nd (ii) severe constriction of the domain of convergence of the optimization method. Appropriate balancing or preconditioning (numerical scaling of the variables and constraints, e.g. as advocated in =-=[7, 8] or any nu-=-merical optimization text) is the first step towards eliminating such problems, but it is not the whole story. In any case it begs the question of what is "balanced". It is not always approp... |

20 |
Vision 3D Non Calibrée: Contributions à la Reconstruction Projective et Étude des Mouvements Critiques pour l’Auto-Calibrage
- Sturm
- 1997
(Show Context)
Citation Context ...cantly [9, 15, 22]. The main problem is that classical autocalibration has some restrictive intrinsic degeneracies — classes of motion for which no algorithm can recover a full unique solution. Sturm =-=[18, 19]-=- has given a catalogue of these. In particular, at least 3 views, some translation and some rotation about at least two non-aligned axes are required. Planar Autocalibration: All of the existing appro... |

9 |
A new approach to geometric fitting, Available from http://www.inrialpes.fr/movi/people/- Triggs
- Triggs
(Show Context)
Citation Context ... updated at the end of it. One can show that the missing terms effectively displace the cost derivative evaluation point from the measured x to a first order estimate of the true underlying value x 0 =-=[21]-=-. For the most part this makes little difference unless the constraints are strongly curved on the scale of V x . For our autocalibration method, the statistical error splits into independent terms fo... |

9 |
Rechnerische Rekonstruktion eines ebenen Objekts aus zwei Photographien,” Mitteilungen der geodaetischen
- Wunderlich
(Show Context)
Citation Context ... have preferred to implement a coarse numerical search over f . The search uses a new SVD-based planar relative orientation method (see appendix 1) related to Wunderlich's eigendecomposition approach =-=[25]-=-. The camera pose and planar structure are recovered directly from the SVD of the inter-image homography. As always with planar relative orientation, there is a two-fold ambiguity in the solution, so ... |

7 |
Euclidean reconstruction from multiple views
- Hartley
- 1993
(Show Context)
Citation Context |

3 |
Vision 3D non calibree—contributions a la reconstruction projective et etude des mouvements critiques pour lÕauto-calibrage
- Sturm
- 1997
(Show Context)
Citation Context ...ntly [9, 15, 22]. The main problem is that classical autocalibration has some restrictive intrinsic degeneracies --- classes of motion for which no algorithm can recover a full unique solution. Sturm =-=[18, 19]-=- has given a catalogue of these. In particular, at least 3 views, some translation and some rotation about at least two non-aligned axes are required. Planar Autocalibration: All of the existing appro... |