## Camera pose and calibration from 4 or 5 known 3D points (1999)

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Venue: | In Proc. 7th Int. Conf. on Computer Vision |

Citations: | 30 - 0 self |

### BibTeX

@INPROCEEDINGS{Triggs99camerapose,

author = {Bill Triggs},

title = {Camera pose and calibration from 4 or 5 known 3D points},

booktitle = {In Proc. 7th Int. Conf. on Computer Vision},

year = {1999},

pages = {278--284},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

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### Abstract

We describe two direct quasilinear methods for camera pose (absolute orientation) and calibration from a single image of 4 or 5 known 3D points. They generalize the 6 point ‘Direct Linear Transform ’ method by incorporating partial prior camera knowledge, while still allowing some unknown calibration parameters to be recovered. Only linear algebra is required, the solution is unique in non-degenerate cases, and additional points can be included for improved stability. Both methods fail for coplanar points, but we give an experimental eigendecomposition based one that handles both planar and nonplanar cases. Our methods use recent polynomial solving technology, and we give a brief summary of this. One of our aims was to try to understand the numerical behaviour of modern polynomial solvers on some relatively simple test cases, with a view to other vision applications.

### Citations

1033 |
A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses
- Tsai
- 1987
(Show Context)
Citation Context ...es) fail for planes. The 4 point quasilinear method should do better, but in fact it also fails owing to an algorithm-specific rank deficiency. In contrast, relatively simple homography-based methods =-=[21, 10, 18, 22]-=- 1 solve the 4 point planar pose + focal length problem rather stably (barring fronto- and other axis parallelisms) . Unfortunately, these methods fail for more than about 5% non-coplanarity, so it wo... |

837 | A flexible new technique for camera calibration
- Zhang
- 1999
(Show Context)
Citation Context ...and frontoparallelism are common in practice. A planar scene gives only two constraints on the calibration ("the images of the plane's two circular points must lie on the image of the absolute co=-=nic" [20, 11, 18, 22]-=-). As there are 5 calibration parameters, at least 3 prior constraints are required to recover from planarity. Our 5 point method has only 2 prior constraints, so it must (and does) fail for planes. T... |

345 |
Using Algebraic Geometry
- Cox, Little, et al.
- 1998
(Show Context)
Citation Context ...sults. There is no space for details here, but the material deserves to be better known in the vision community as large-scale polynomial solving is rapidly becoming a feasible proposition. See, e.g. =-=[4, 14]-=- for references and further reading. A polynomial p(x) = P p ff x ff in variables x = (x 1 ; : : : ; xn ) is a finite sum of coefficients p ff times monomialssx ff j Q n i=1 x ff i i , with integer ex... |

292 |
A theory of self-calibration of a moving camera
- Maybank, Faugeras
- 1992
(Show Context)
Citation Context ...s. By the decomposition P ' K R (I j \Gamma t), the 4 \Theta 4 Euclidean invariant absolute dual quadric matrix\Omega j \Gamma I 0 0 0 \Delta projects to the dual image of the absolute quadric (DIAC) =-=[19, 9, 13]-=- ! j P\Omega P ? ' K K ? (1) We use this to convert constraints on the calibration K into ones on candidate projections P() or their associated 2 DIAC's ! = !() j P()\Omega P() ? . For the 4 point met... |

213 | Autocalibration and the absolute quadric
- Triggs
- 1997
(Show Context)
Citation Context ...s. By the decomposition P ' K R (I j \Gamma t), the 4 \Theta 4 Euclidean invariant absolute dual quadric matrix\Omega j \Gamma I 0 0 0 \Delta projects to the dual image of the absolute quadric (DIAC) =-=[19, 9, 13]-=- ! j P\Omega P ? ' K K ? (1) We use this to convert constraints on the calibration K into ones on candidate projections P() or their associated 2 DIAC's ! = !() j P()\Omega P() ? . For the 4 point met... |

201 | Model-based object pose in 25 lines of code
- Dementhon, Davis
- 1995
(Show Context)
Citation Context ...on problems. This makes it clear 6 that 1=f is a natural expansion parameter, and suggests that pseudo-affine initialization may be a good implementation strategy for pose + calibration methods, c.f. =-=[5, 3]-=-. 6 Summary and Conclusions The 4 point quasilinear pose method performs reasonably well considering how much information it extracts from such a small amount of input data. The 5 point method is less... |

191 |
Analysis and solutions of the three point perspective pose estimation problem
- Haralick, Lee, et al.
- 1991
(Show Context)
Citation Context ...v 0 ) = (1; 0; 0; 0). Rationale & Existing Work: Our methods use some prior calibration knowledge, and are best seen as intermediate between classical 3--4 point pose-with-knowncalibration algorithms =-=[16, 8, 15]-=-, ands6 point DLT-like ones which assume completely unknown calibration [1, 16]. They were motivated mainly by the need for approximate camera pose + calibration to initialize bundle adjustment in clo... |

142 |
The Algebraic Theory of Modular Systems
- Macaulay
- 1916
(Show Context)
Citation Context ...e recovered by the above construction. In particular, for dense homogeneous polynomials (ones whose coefficients of the given degrees are all nonzero and generic), Macaulay's classical multiresultant =-=[12]-=- chooses A to contain all monomials of degree D = 1 + P n+1 i=1 (d i \Gamma 1). Taking all untruncated rows of the multiplication matrices as above generally gives a rectangular matrix R. Macaulay gav... |

127 | A.Zisserman. Metric rectification for perspective images of planes
- Liebowitz
- 1998
(Show Context)
Citation Context ...and frontoparallelism are common in practice. A planar scene gives only two constraints on the calibration ("the images of the plane's two circular points must lie on the image of the absolute co=-=nic" [20, 11, 18, 22]-=-). As there are 5 calibration parameters, at least 3 prior constraints are required to recover from planarity. Our 5 point method has only 2 prior constraints, so it must (and does) fail for planes. T... |

127 | On plane-based camera calibration: a general algorithm, singularities, applications
- Sturm, Maybank
- 1999
(Show Context)
Citation Context ...and frontoparallelism are common in practice. A planar scene gives only two constraints on the calibration ("the images of the plane's two circular points must lie on the image of the absolute co=-=nic" [20, 11, 18, 22]-=-). As there are 5 calibration parameters, at least 3 prior constraints are required to recover from planarity. Our 5 point method has only 2 prior constraints, so it must (and does) fail for planes. T... |

124 | Autocalibration from planar scenes
- Triggs
(Show Context)
Citation Context |

89 |
Direct linear transformation from comparator to object space coordinates in close-range photogrammetry
- Abdel-Aziz, Karara
- 1971
(Show Context)
Citation Context ...a unique solution in non-degenerate cases; (iii) additional points are easily included to improve stability; and (iv) all points are on an equal footing. The classical `Direct Linear Transform' (DLT) =-=[1, 16]-=- recovers the 5 internal and 6 pose parameters of a fully projective camera from the images of 6 known 3D points. The new methods are analogous to the DLT, but adopt more restrictive calibration model... |

71 | Euclidean Reconstruction from Constant Intrinsic Parameters
- Heyden, Aström
- 1996
(Show Context)
Citation Context ...s. By the decomposition P ' K R (I j \Gamma t), the 4 \Theta 4 Euclidean invariant absolute dual quadric matrix\Omega j \Gamma I 0 0 0 \Delta projects to the dual image of the absolute quadric (DIAC) =-=[19, 9, 13]-=- ! j P\Omega P ? ' K K ? (1) We use this to convert constraints on the calibration K into ones on candidate projections P() or their associated 2 DIAC's ! = !() j P()\Omega P() ? . For the 4 point met... |

52 |
EOEcient incremental algorithms for the sparse resultant and the mixed volume
- Emiris, Canny
- 1996
(Show Context)
Citation Context ... appropriate rows of their multiplication matrices to R. If some of the polynomial coefficients vanish the Macaulay construction may fail. Sparse `Newton ' multiresultants are available in such cases =-=[7, 6, 4]-=-. The above is all we need for the quasilinear 4 and 5 point methods, as the P i and hence (2), (3) are usually dense. However, as mentioned above, the 4 point method fails unnecessarily for coplanar ... |

51 | Efficient incremental algorithms for sparse resultant and the mixed volume
- Emiris, Canny
- 1995
(Show Context)
Citation Context ...he appropriate rows of their multiplication matrices to . If some of the polynomial coefficients vanish the Macaulay construction may fail. Sparse ‘Newton’ multiresultants are available in such cases =-=[7, 6, 4]-=-. The above is all we need for the quasilinear 4 and 5 point methods, as the and hence (2), (3) are usually dense. However, as mentioned above, the 4 point method fails unnecessarily for coplanar poin... |

50 | Euclidean shape and motion from multiple perspective views by affine iterations
- Christy, Horaud
- 1996
(Show Context)
Citation Context ...on problems. This makes it clear 6 that 1=f is a natural expansion parameter, and suggests that pseudo-affine initialization may be a good implementation strategy for pose + calibration methods, c.f. =-=[5, 3]-=-. 6 Summary and Conclusions The 4 point quasilinear pose method performs reasonably well considering how much information it extracts from such a small amount of input data. The 5 point method is less... |

25 |
Eigenproblems are at the heart of polynomial system solving
- Stetter
- 1996
(Show Context)
Citation Context ...cator. 2 I.e. , R y 1 R1 = I N 1 \ThetaN 1 . Such R y 1 are easily calculated from most numerical decompositions of R1 . 4 This multiplication matrix approach to numerical rootfinding is quite recent =-=[17, 14, 4]-=-, although its roots go back a century. So far as I know, the observation that it continues to work when A 0 and U span more than the null space of R is new. This is numerically useful, as it allows e... |

11 |
Linear n ≥ 4-point pose determination
- Quan, Lan
- 1998
(Show Context)
Citation Context ...v 0 ) = (1; 0; 0; 0). Rationale & Existing Work: Our methods use some prior calibration knowledge, and are best seen as intermediate between classical 3--4 point pose-with-knowncalibration algorithms =-=[16, 8, 15]-=-, ands6 point DLT-like ones which assume completely unknown calibration [1, 16]. They were motivated mainly by the need for approximate camera pose + calibration to initialize bundle adjustment in clo... |

6 |
An introduction to linear algebra methods for solving polynomial equations
- Mourrain
- 1998
(Show Context)
Citation Context ...sults. There is no space for details here, but the material deserves to be better known in the vision community as large-scale polynomial solving is rapidly becoming a feasible proposition. See, e.g. =-=[4, 14]-=- for references and further reading. A polynomial p(x) = P p ff x ff in variables x = (x 1 ; : : : ; xn ) is a finite sum of coefficients p ff times monomialssx ff j Q n i=1 x ff i i , with integer ex... |

5 |
Close Range Photogrammetry and
- Atkinson
- 1996
(Show Context)
Citation Context ...s distortion) image position. Radial lens distortion is also significant in many close range applications, but we will not consider it here as it is difficult to handle in our DLT-like framework. See =-=[16, 2]-=- for extensions of the DLT which partially account for lens distortion. Degeneracy is a significant problem for all calibration methods using near-minimal data: for certain relative positionings of th... |

3 |
Camera calibration using a planar surface. Unpublished
- Kanatani
- 1998
(Show Context)
Citation Context ...es) fail for planes. The 4 point quasilinear method should do better, but in fact it also fails owing to an algorithm-specific rank deficiency. In contrast, relatively simple homography-based methods =-=[21, 10, 18, 22]-=- 1 solve the 4 point planar pose + focal length problem rather stably (barring fronto- and other axis parallelisms) . Unfortunately, these methods fail for more than about 5% non-coplanarity, so it wo... |

1 |
Linear -point pose determination
- Quan, Lan
- 1998
(Show Context)
Citation Context ...;<=80>?0>@2>8 values . Rationale & Existing Work: Our methods use some prior calibration knowledge, and are best seen as intermediate between classical 3–4 point pose-with-knowncalibration algorithms =-=[16, 8, 15]-=-, and AB point DLT-like ones which assume completely unknown calibration [1, 16]. They were motivated mainly by the need for approximate camera pose + calibration to initialize bundle adjustment in cl... |