## Linear n-point camera pose determination (1999)

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Venue: | ieee Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 93 - 2 self |

### BibTeX

@ARTICLE{Quan99linearn-point,

author = {Long Quan and Zhongdan Lan},

title = {Linear n-point camera pose determination},

journal = {ieee Transactions on Pattern Analysis and Machine Intelligence},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

AbstractÐThe determination of camera position and orientation from known correspondences of 3D reference points and their images is known as pose estimation in computer vision and space resection in photogrammetry. It is wellknown that from three corresponding points there are at most four algebraic solutions. Less appears to be known about the cases of four and five corresponding points. In this paper, we propose a family of linear methods that yield a unique solution to 4- and 5-point pose determination for generic reference points. We first review the 3-point algebraic method. Then we present our twostep, 4-point and one-step, 5-point linear algorithms. The 5-point method can also be extended to handle more than five points. Finally, we demonstrate our methods on both simulated and real images. We show that they do not degenerate for coplanar configurations and even outperform the special linear algorithm for coplanar configurations in practice. Index TermsÐPose estimation, space resection, 2D-3D image orientation, exterior orientation determination, perspective-n-point-problem, four points, five points. 1

### Citations

2864 |
Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography
- Fischler, Bolles
- 1981
(Show Context)
Citation Context ...togrammetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists [1]. Many variants =-=[2]-=-, [3], [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC paradigm to detect outl... |

488 |
Perceptual Organization and Visual Recognition
- Lowe
- 1985
(Show Context)
Citation Context ...ds for pose estimation using line segments instead of points as image features have also been developed. Dhome et al. [15] and Chen [16], developed algebraic solutions for 3-line algorithms, and Lowe =-=[17]-=- used the NewtonRaphson method for any number of line segments. Liu, et al. [18] combined points and line segments into the same pose estimation procedure. Motivated by the lack of methods which direc... |

323 | Fitting parameterized three-dimensional models to images
- Lowe
- 1991
(Show Context)
Citation Context ... of initialization and convergence, while algebraic methods applied to subsets suffer from poor noise filtering and the difficulty of selecting the common root from the noisy data. For instance, Lowe =-=[7] and-=- Yuan [8] developed techniques based on Newton-Raphson iteration. Dementhon and Davis [9] presented an iterative algorithm for four points or more starting from scaled orthographic �� approximatio... |

247 |
The representation, recognition, and locating of 3-D objects
- Faugeras, Hebert
- 1986
(Show Context)
Citation Context ...ep is the absolute orientation determination, a similarity transformation between two sets of 3D points ~ p i , p i . The best least-square rotation is obtained in closed-form using quaternions [23], =-=[24]-=-. The determination of the translation and the scale follow immediately from the estimate of the rotation. 3 THE LINEAR 4-POINT ALGORITHM The goal is to directly obtain the unique solution from a redu... |

218 |
Review and analysis of solutions of the three point perspective pose estimation problem
- Haralick, Ottenberg, et al.
- 1994
(Show Context)
Citation Context ...uter vision and photogrammetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists =-=[1]-=-. Many variants [2], [3], [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC para... |

150 |
Techniques for calibrating of the scale factor and image center for high accuracy 3D machine vision metrology
- Lenz, Tsai
- 1987
(Show Context)
Citation Context ...f the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras and Toscani [13] using a different constraint on the projection matrix. Lenz and Tsai =-=[14]-=- proposed both linear and nonlinear calibration methods. Although these methods might be applied to pose determination, full calibration is a heavy over-parameterization for the pose problem, giving r... |

120 |
G.: Determination of the attitude of 3-D objects from a single perspective view
- Dhome, Richetin, et al.
- 1989
(Show Context)
Citation Context ...ving reduced stability and requiring more points. Mostly in computer vision, methods for pose estimation using line segments instead of points as image features have also been developed. Dhome et al. =-=[15]-=- and Chen [16], developed algebraic solutions for 3-line algorithms, and Lowe [17] used the NewtonRaphson method for any number of line segments. Liu, et al. [18] combined points and line segments int... |

116 |
Closed form solutions of absolute orientation using unit quaternions
- Horn
- 1987
(Show Context)
Citation Context ...nal step is the absolute orientation determination, a similarity transformation between two sets of 3D points ~ p i , p i . The best least-square rotation is obtained in closed-form using quaternions =-=[23]-=-, [24]. The determination of the translation and the scale follow immediately from the estimate of the rotation. 3 THE LINEAR 4-POINT ALGORITHM The goal is to directly obtain the unique solution from ... |

115 |
Structure from motion using line correspondences
- Spetsakis, Aloimonos
- 1990
(Show Context)
Citation Context ...ation while taking advantage of data redundancy. Developing linear algorithms using redundant data for various vision tasks has always attracted attention of many researchers in computer vision [19], =-=[20]-=-, [21], [22]. The paper is organized as follows. In Section 2, the basic 3-point closed-form solution is reviewed and discussed. Then, the linear 4point, 5-point, and N-point algorithms are successive... |

87 |
Decomposition of transformation matrices for robot vision
- Ganapathy
- 1989
(Show Context)
Citation Context ...ted to pose estimation, but is more general as the calibration simultaneously estimates both pose and the intrinsic parameters of the camera. Abdel-Aziz and Karara [4], Sutherland [11], and Ganapathy =-=[12] pro-=-posed a direct linear method DLT�� for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras and Toscani [13] us... |

68 |
Determination of camera location from 2D to 3D line and point correspondences
- Liu, Faugeras
- 1988
(Show Context)
Citation Context ...ave also been developed. Dhome et al. [15] and Chen [16], developed algebraic solutions for 3-line algorithms, and Lowe [17] used the NewtonRaphson method for any number of line segments. Liu, et al. =-=[18]-=- combined points and line segments into the same pose estimation procedure. Motivated by the lack of methods which directly provide a unique pose solution for the redundant data case, we develop in th... |

60 |
Elements of Photogrammetry
- Wolf
- 1974
(Show Context)
Citation Context ...nstrained by the six independent interpoint distances as in the noncoplanar case. It is also wellknown that the camera pose can be estimated linearly from at least four coplanar reference points [2], =-=[26]-=-. These two methods will be compared in Section 6. 4 THE LINEAR FIVE AND n-POINT ALGORITHMS From ns5 on, there are sufficiently many fourth degree polynomials to directly solve t isx i linearly. For t... |

43 |
A computer program for reconstructing a scene from two projections
- Longuet-Higgins
- 1981
(Show Context)
Citation Context ...d iteration while taking advantage of data redundancy. Developing linear algorithms using redundant data for various vision tasks has always attracted attention of many researchers in computer vision =-=[19]-=-, [20], [21], [22]. The paper is organized as follows. In Section 2, the basic 3-point closed-form solution is reviewed and discussed. Then, the linear 4point, 5-point, and N-point algorithms are succ... |

41 |
Model-based Object Pose in 25
- Dementhon, Davis
- 1995
(Show Context)
Citation Context ...or noise filtering and the difficulty of selecting the common root from the noisy data. For instance, Lowe [7] and Yuan [8] developed techniques based on Newton-Raphson iteration. Dementhon and Davis =-=[9] pre-=-sented an iterative algorithm for four points or more starting from scaled orthographic �� approximation. Horaud et al. [10] convert P4P into a special 3-line problem. Unfortunately, this conversi... |

32 |
Pose determination from line-to-plane correspondences: existence condition and closed-form solutions
- Chen
- 1991
(Show Context)
Citation Context ...tability and requiring more points. Mostly in computer vision, methods for pose estimation using line segments instead of points as image features have also been developed. Dhome et al. [15] and Chen =-=[16]-=-, developed algebraic solutions for 3-line algorithms, and Lowe [17] used the NewtonRaphson method for any number of line segments. Liu, et al. [18] combined points and line segments into the same pos... |

31 |
A non-iterative algorithm for determining all essential matrices corresponding to five point pairs
- Philip
- 1996
(Show Context)
Citation Context ...taking advantage of data redundancy. Developing linear algorithms using redundant data for various vision tasks has always attracted attention of many researchers in computer vision [19], [20], [21], =-=[22]-=-. The paper is organized as follows. In Section 2, the basic 3-point closed-form solution is reviewed and discussed. Then, the linear 4point, 5-point, and N-point algorithms are successively presented... |

30 |
Camera calibration for 3D computer vision
- Faugeras, Toscani
- 1987
(Show Context)
Citation Context ...hy [12] proposed a direct linear method DLT�� for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras and Tosc=-=ani [13]-=- using a different constraint on the projection matrix. Lenz and Tsai [14] proposed both linear and nonlinear calibration methods. Although these methods might be applied to pose determination, full c... |

30 |
A linear algorithm for motion estimation using straight line correspondences
- Liu, Huang
- 1988
(Show Context)
Citation Context ...while taking advantage of data redundancy. Developing linear algorithms using redundant data for various vision tasks has always attracted attention of many researchers in computer vision [19], [20], =-=[21]-=-, [22]. The paper is organized as follows. In Section 2, the basic 3-point closed-form solution is reviewed and discussed. Then, the linear 4point, 5-point, and N-point algorithms are successively pre... |

13 |
An Analytical Solution for the Perspective
- Horaud, Conio, et al.
- 1989
(Show Context)
Citation Context ...eloped techniques based on Newton-Raphson iteration. Dementhon and Davis [9] presented an iterative algorithm for four points or more starting from scaled orthographic �� approximation. Horaud et =-=al. [10]-=- convert P4P into a special 3-line problem. Unfortunately, this conversion uses only partial information and inherits the same fourth degree polynomial equations. Camera calibration is closely related... |

8 |
Minimum Solutions for Orientation
- Wrobel
(Show Context)
Citation Context ...generically suffice for uniqueness outside of certain critical configurations in space. All critical configurations for which multiple distinct or coinciding (unstable) solutions occur are known [5], =-=[6]-=-. The camera pose is ambiguous if the projection center is coplanar with any three of the four reference points or if it goes through the Euclidean horopter curve or one of its degenerate forms (the E... |

5 |
Space Resection: Failure Cases
- Thompson
- 1966
(Show Context)
Citation Context ...ints generically suffice for uniqueness outside of certain critical configurations in space. All critical configurations for which multiple distinct or coinciding (unstable) solutions occur are known =-=[5]-=-, [6]. The camera pose is ambiguous if the projection center is coplanar with any three of the four reference points or if it goes through the Euclidean horopter curve or one of its degenerate forms (... |

5 |
A general phogrammetric solution for the determining object position and orientation
- Yuan
- 1989
(Show Context)
Citation Context ...ation and convergence, while algebraic methods applied to subsets suffer from poor noise filtering and the difficulty of selecting the common root from the noisy data. For instance, Lowe [7] and Yuan =-=[8] dev-=-eloped techniques based on Newton-Raphson iteration. Dementhon and Davis [9] presented an iterative algorithm for four points or more starting from scaled orthographic �� approximation. Horaud et ... |

5 |
Reconstruction tridimensionnelle d'une sce`ne a` partir d'une came'ra en mouvement : de l'influence de la pre'cision
- Brand
- 1995
(Show Context)
Citation Context ... are also demonstrated on real image data from the calibration pattern illustrated in Fig. 3. Five images of the pattern are taken and about 160 target points are accurately extracted from each image =-=[27]-=-. A complete camera calibration is carried out for each image the camera pose from the calibration matrix serves as ground truth. One hundred sets of four and five points are drawn randomly from the e... |

3 |
ªAnalysis and Solutions of the Three Point Perspective Pose Estimation
- Haralick, Lee, et al.
- 1991
(Show Context)
Citation Context ...uter vision and photogrammetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists =-=[1]-=-. Many variants [2], [3], [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC para... |

3 |
ªRandom Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis
- Fischler, Bolles
- 1981
(Show Context)
Citation Context ...togrammetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists [1]. Many variants =-=[2]-=-, [3], [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC paradigm to detect outl... |

3 |
ªDecomposition of Transformation Matrices for Robot Vision,º
- Ganapathy
- 1984
(Show Context)
Citation Context ...ted to pose estimation, but is more general as the calibration simultaneously estimates both pose and the intrinsic parameters of the camera. Abdel-Aziz and Karara [4], Sutherland [11], and Ganapathy =-=[12]-=- proposed a direct linear method ªDLTº for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras and Toscani [13] us... |

2 |
Three-Dimensional Input by Tablet
- Sutherland
- 1974
(Show Context)
Citation Context ...tion is closely related to pose estimation, but is more general as the calibration simultaneously estimates both pose and the intrinsic parameters of the camera. Abdel-Aziz and Karara [4], Sutherland =-=[11], an-=-d Ganapathy [12] proposed a direct linear method DLT�� for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras... |

2 |
Reliability analysis of parameter estimation in linear models with applications to mansuration problems in computer vision
- Foerstner
- 1987
(Show Context)
Citation Context ...mmetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists [1]. Many variants [2], =-=[3]-=-, [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC paradigm to detect outliers ... |

2 |
ªSpace Resection: Failure Cases,º Photogrammetric
- Thompson
- 1966
(Show Context)
Citation Context ...ints generically suffice for uniqueness outside of certain critical configurations in space. All critical configurations for which multiple distinct or coinciding (unstable) solutions occur are known =-=[5]-=-, [6]. The camera pose is ambiguous if the projection center is coplanar with any three of the four reference points or if it goes through the Euclidean horopter curve or one of its degenerate forms (... |

2 |
ªModel-Based Object Pose in 25 Lines of Code,º Int'l
- Dementhon, Davis
- 1995
(Show Context)
Citation Context ...or noise filtering and the difficulty of selecting the common root from the noisy data. For instance, Lowe [7] and Yuan [8] developed techniques based on Newton-Raphson iteration. Dementhon and Davis =-=[9]-=- presented an iterative algorithm for four points or more starting from ªscaled orthographicº approximation. Horaud et al. [10] convert P4P into a special 3-line problem. Unfortunately, this conversio... |

2 |
ªCamera Calibration for 3D Computer Vision,º
- Faugeras, Toscani
- 1987
(Show Context)
Citation Context ...hy [12] proposed a direct linear method ªDLTº for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras and Toscani =-=[13]-=- using a different constraint on the projection matrix. Lenz and Tsai [14] proposed both linear and nonlinear calibration methods. Although these methods might be applied to pose determination, full c... |

1 |
rstner, Reliability Analysis of Parameter Estimation in Linear Models with Applications to Mensuration
- Fo
- 1987
(Show Context)
Citation Context ...mmetry. It is well-known that there are closed-form orientation solutions for three points. The first algebraic solution may be tracked back to [1841] by the photogrammetrists [1]. Many variants [2], =-=[3]-=-, [1] of the basic 3-point algorithm have been developed ever since. Fischler and Bolles [2] give one such method that is popular in computer vision together with a RANSAC paradigm to detect outliers ... |

1 |
ªMinimum Solutions for Orientation,º
- Wrobel
- 1992
(Show Context)
Citation Context ...generically suffice for uniqueness outside of certain critical configurations in space. All critical configurations for which multiple distinct or coinciding (unstable) solutions occur are known [5], =-=[6]-=-. The camera pose is ambiguous if the projection center is coplanar with any three of the four reference points or if it goes through the Euclidean horopter curve or one of its degenerate forms (the E... |

1 |
ªThree-Dimensional Input by Tablet,º Proc
- Sutherland
- 1974
(Show Context)
Citation Context ...tion is closely related to pose estimation, but is more general as the calibration simultaneously estimates both pose and the intrinsic parameters of the camera. Abdel-Aziz and Karara [4], Sutherland =-=[11]-=-, and Ganapathy [12] proposed a direct linear method ªDLTº for solving the 11 entries of the camera projection matrix from at least six corresponding points. The method is further improved by Faugeras... |

1 |
ªStructure from Motion Using Line Correspondences,º Int'l
- Spetsakis, Aloimonos
- 1990
(Show Context)
Citation Context ...ation while taking advantage of data redundancy. Developing linear algorithms using redundant data for various vision tasks has always attracted attention of many researchers in computer vision [19], =-=[20]-=-, [21], [22]. The paper is organized as follows. In Section 2, the basic 3-point closed-form solution is reviewed and discussed. Then, the linear 4point, 5-point, and N-point algorithms are successive... |

1 |
ªReconstruction Tridimensionnelle d'une SceÁne aÁ Partir d'une CameÂra en Mouvement de l'Influence de la PreÂcision,º TheÁses de Doctorat
- Brand
- 1995
(Show Context)
Citation Context ... are also demonstrated on real image data from the calibration pattern illustrated in Fig. 3. Five images of the pattern are taken and about 160 target points are accurately extracted from each image =-=[27]-=-. A complete camera calibration is carried out for each image the camera pose from the calibration matrix serves as ground truth. One hundred sets of four and five points are drawn randomly from the e... |