## Autocalibration and the absolute quadric (1997)

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

Citations: | 223 - 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.

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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 ... |

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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... |

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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... |

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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... |

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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 ... |

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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... |

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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... |

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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... |

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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 ... |