## Autocalibration and the absolute quadric (1997)

### Cached

### Download Links

- [perception.inrialpes.fr]
- [www-ljk.imag.fr]
- [hal.inria.fr]
- [www.inrialpes.fr]
- DBLP

### Other Repositories/Bibliography

Venue: | in Proc. IEEE Conf. Computer Vision, Pattern Recognition |

Citations: | 222 - 7 self |

### BibTeX

@INPROCEEDINGS{Triggs97autocalibrationand,

author = {Bill Triggs},

title = {Autocalibration and the absolute quadric},

booktitle = {in Proc. IEEE Conf. Computer Vision, Pattern Recognition},

year = {1997},

pages = {609--614}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe a new method for camera autocalibration and scaled Euclidean structure and motion, from three or more views taken by a moving camera with fixed but unknown intrinsic parameters. The motion constancy of these is used to rectify an initial projective reconstruction. Euclidean scene structure is formulated in terms of the absolute quadric — the singular dual 3D quadric ( rank 3 matrix) giving the Euclidean dot-product between plane normals. This is equivalent to the traditional absolute conic but simpler to use. It encodes both affine and Euclidean structure, and projects very simply to the dual absolute image conic which encodes camera calibration. Requiring the projection to be constant gives a bilinear constraint between the absolute quadric and image conic, from which both can be recovered nonlinearly from images, or quasi-linearly from. Calibration and Euclidean structure follow easily. The nonlinear method is stabler, faster, more accurate and more general than the quasi-linear one. It is based on a general constrained optimization technique — sequential quadratic programming — that may well be useful in other vision problems.

### Citations

1490 |
Practical optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ...ches, but progress in tensorial and GrassmannCayley -like formalisms [19, 5] and computational nonlinear algebra (e.g. [2]) may soon make this much easier. Many constrained optimization schemes exist =-=[6]-=-. I will give a brief outline of the simple one used here, as I think that it has considerable potential for other constrained problems in vision. Sequential Quadratic Programming [6] is a general num... |

371 | Camera selfcalibration: theory and experiments
- Faugeras, Luong, et al.
- 1992
(Show Context)
Citation Context ...ctivity. Additional constraints are required to `Euclideanize' it. We will focus on the traditional case of a single camera with fixed but unknown intrinsic parameters moving arbitrarily in the scene =-=[13, 4, 7]-=-, but our formalism easily extends to handle multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; hand... |

299 |
Faugeras. A theory of self-calibration of a moving camera
- Maybank, Olivier
- 1992
(Show Context)
Citation Context ...ctivity. Additional constraints are required to `Euclideanize' it. We will focus on the traditional case of a single camera with fixed but unknown intrinsic parameters moving arbitrarily in the scene =-=[13, 4, 7]-=-, but our formalism easily extends to handle multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; hand... |

219 | A Factorization Based Algorithm for Multi-Image Projective Structure and Motion
- Sturm, Triggs
- 1996
(Show Context)
Citation Context ...nding the scene. Several other configurations have also been tried with success. Uniform random noise was added to the image points. The initial projective reconstruction was projective factorization =-=[18, 20]-=- followed by projective bundle adjustment (not indispensable) . The nonlinear method was initialized with a calibration wrong by about 50%. Mean 3D reconstruction error over 10 trials was estimated by... |

179 | Canonic representations for the geometries of multiple projective views
- Luong, Viéville
- 1994
(Show Context)
Citation Context ...hosen coordinates, Zeller & Faugeras [21] apply least squares optimization over many images, and Hartley (reported in [14]) uses a preliminary SVD based simplification. The second approach stratifies =-=[12, 3]-=- the problem into affine and Euclidean parts. Affine structure is encoded in p1 or the absolute homography H1 --- the inter-image mapping defined by projecting pixels up onto p1 . For fixed calibratio... |

156 | Self-calibration from multiple views with a rotating camera
- Hartley
- 1994
(Show Context)
Citation Context ... in the scene [13, 4, 7], but our formalism easily extends to handle multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation =-=[8]-=- or a plane [1]; handle zoom modulo an initial pre-calibration [15, 16]; or assume a rigidly moving stereo head [22]. For practical applications it is important to exploit any constraints that may be ... |

150 | On the geometry and algebra of the point and line correspondances between
- Faugeras, Mourrain
- 1995
(Show Context)
Citation Context ...ble in practical applications, but too `opaque' to provide much theoretical insight. At present it is hard to relate the two approaches, but progress in tensorial and GrassmannCayley -like formalisms =-=[19, 5]-=- and computational nonlinear algebra (e.g. [2]) may soon make this much easier. Many constrained optimization schemes exist [6]. I will give a brief outline of the simple one used here, as I think tha... |

137 | defense of the 8-point algorithm
- Hartley, “In
- 1997
(Show Context)
Citation Context ...ith equations of high degree. It is discussed in every text on numerical methods, but does not seem to have been widely known in vision before Hartley made the point for fundamental matrix estimation =-=[9]-=-. 5 Experiments To give a rough idea of the performance of the algorithm, we briefly report on numerical experiments with synthetic data. Images of random point clouds were taken with identical widean... |

137 | Relative orientation
- HORN
- 1990
(Show Context)
Citation Context ...ions and an origin). If required, the rotational part of each rectified projection K \Gamma1 P i TsR i (I j \Gamma t i ) can be perturbed 3 to be precisely orthonormal (e.g. using quaternions and SVD =-=[11]-=-). As always, a final, close-lying-outlier-insensitive bundle adjustment over all parameters is recommended for precise work. 3.2 Degeneracy Autocalibration has some intrinsic limitations that apply u... |

111 | Euclidean reconstruction from image sequences with varying and unknown focal length and principal point
- Heyden, Åström
- 1997
(Show Context)
Citation Context ...ng this to gives equations relating the unknowns n and . These can be solved iteratively given a reasonable initial guess for or K. [7]. Most other authors start from an approximate prior calibration =-=[12, 10]-=-. Heyden & ˚Aström’s formulation [10] also Hartley pioneered this sort of approach using bounds on partially (but independently) foreshadows ours given below. The modulus constraint [12, 15] — that ) ... |

109 | motion sequences for monocular selfcalibration and uncalibrated Euclidean reconstruction”, CVPR
- Sturm
- 1997
(Show Context)
Citation Context ...t fixes all points at infinity: affine structure follows easily, but\Omega is essentially arbitrary so autocalibration is impossible. Various other types of motion lead to further degeneracies: Sturm =-=[17]-=- gives a detailed catalog. Such ambiguities must typically be handled by imposing further constraints (known skew, aspect ratio, motion. . . ). This can be difficult with algebraic approaches, but is ... |

109 | Factorization methods for projective structure and motion
- Triggs
- 1996
(Show Context)
Citation Context ...nding the scene. Several other configurations have also been tried with success. Uniform random noise was added to the image points. The initial projective reconstruction was projective factorization =-=[18, 20]-=- followed by projective bundle adjustment (not indispensable) . The nonlinear method was initialized with a calibration wrong by about 50%. Mean 3D reconstruction error over 10 trials was estimated by... |

104 | Self-calibration from image triplets
- Armstrong, Zisserman, et al.
- 1996
(Show Context)
Citation Context ...13, 4, 7], but our formalism easily extends to handle multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane =-=[1]-=-; handle zoom modulo an initial pre-calibration [15, 16]; or assume a rigidly moving stereo head [22]. For practical applications it is important to exploit any constraints that may be available, as t... |

73 | Euclidean reconstruction from constant intrinsic parameters
- Heyden, Åström
- 1996
(Show Context)
Citation Context ... can be solved iteratively given a reasonable initial guess for p1 or K. Hartley pioneered this sort of approach using bounds on p1 [7]. Most other authors start from an approximate prior calibration =-=[12, 10]-=-. Heyden &sAstrom's formulation [10] also partially (but independently) foreshadows ours given below. The modulus constraint [12, 15] --- that H1 = K RK \Gamma1 being conjugate to a rotation matrix mu... |

70 | The geometry of projective reconstruction i: Matching constraints and the joint image
- Triggs
- 1995
(Show Context)
Citation Context ...espond via epipolar planes, the above angle-between-visual-planes formula must give the same result for corresponding epipolar lines in either image. A compact derivation applies the closure identity =-=[19]-=- F 21 P 1s2 [e 12 ] \Theta P 2 to either side of\Omega to derive the quadric matching constraint F 21 ! F ? 21s[e 12 ] \Theta ! [e 12 ] ? \Theta . Allowing for symmetry and rank deficiency, this amoun... |

49 | Camera SelfCalibration from Video Sequences
- Zeller, Faugeras
- 1996
(Show Context)
Citation Context ...ations. ! has 5 d.o.f. so at least 3 images are required. Various resolution procedures exist. Maybank, Faugeras & Luong [13, 4] use algebraic elimination in wellchosen coordinates, Zeller & Faugeras =-=[21]-=- apply least squares optimization over many images, and Hartley (reported in [14]) uses a preliminary SVD based simplification. The second approach stratifies [12, 3] the problem into affine and Eucli... |

48 | Stratification of 3-D vision: projective, affine, and metric representations
- Faugeras
- 1995
(Show Context)
Citation Context ...hosen coordinates, Zeller & Faugeras [21] apply least squares optimization over many images, and Hartley (reported in [14]) uses a preliminary SVD based simplification. The second approach stratifies =-=[12, 3]-=- the problem into affine and Euclidean parts. Affine structure is encoded in p1 or the absolute homography H1 --- the inter-image mapping defined by projecting pixels up onto p1 . For fixed calibratio... |

38 | Motion of an uncalibrated stereo rig: Self-calibration and metric reconstruction
- Zhang, Luong, et al.
- 1993
(Show Context)
Citation Context ... or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; handle zoom modulo an initial pre-calibration [15, 16]; or assume a rigidly moving stereo head =-=[22]-=-. For practical applications it is important to exploit any constraints that may be available, as this both increases stability and allows autocalibration from more restricted types of motion. Used on... |

32 | Euclidean 3D reconstruction from image sequences with variable focal lengths
- Pollefeys, Gool, et al.
- 1996
(Show Context)
Citation Context ...le multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; handle zoom modulo an initial pre-calibration =-=[15, 16]-=-; or assume a rigidly moving stereo head [22]. For practical applications it is important to exploit any constraints that may be available, as this both increases stability and allows autocalibration ... |

15 |
Triggs,B.: Projective Geometry for Image Analysis, Tutorial notes
- Mohr
- 1996
(Show Context)
Citation Context ...edures exist. Maybank, Faugeras & Luong [13, 4] use algebraic elimination in wellchosen coordinates, Zeller & Faugeras [21] apply least squares optimization over many images, and Hartley (reported in =-=[14]-=-) uses a preliminary SVD based simplification. The second approach stratifies [12, 3] the problem into affine and Euclidean parts. Affine structure is encoded in p1 or the absolute homography H1 --- t... |

7 |
Euclidean reconstruction from multiple views
- Hartley
- 1993
(Show Context)
Citation Context ...ctivity. Additional constraints are required to `Euclideanize' it. We will focus on the traditional case of a single camera with fixed but unknown intrinsic parameters moving arbitrarily in the scene =-=[13, 4, 7]-=-, but our formalism easily extends to handle multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; hand... |

1 |
A toolkit for nonlinear algebra. Report available from http://http.cs.berkeley.edu/ jfc
- Canny
- 1993
(Show Context)
Citation Context ... provide much theoretical insight. At present it is hard to relate the two approaches, but progress in tensorial and GrassmannCayley -like formalisms [19, 5] and computational nonlinear algebra (e.g. =-=[2]-=-) may soon make this much easier. Many constrained optimization schemes exist [6]. I will give a brief outline of the simple one used here, as I think that it has considerable potential for other cons... |

1 | Self-calibration of a moving camera by precalibration
- Sturm
- 1996
(Show Context)
Citation Context ...le multiple cameras and prior calibration, motion or scene constraints. Alternative approaches restrict the motion to a pure rotation [8] or a plane [1]; handle zoom modulo an initial pre-calibration =-=[15, 16]-=-; or assume a rigidly moving stereo head [22]. For practical applications it is important to exploit any constraints that may be available, as this both increases stability and allows autocalibration ... |