## Calibration of a trinocular system formed with wide angle lens cameras |

Optics Express, Vol. 20, Issue 25, pp. 27691-27696 (2012)

http://dx.doi.org/10.1364/OE.20.027691

Acrobat PDF (4293 KB)

### Abstract

To obtain 3D information of large areas, wide angle lens cameras are used to reduce the number of cameras as much as possible. However, since images are high distorted, errors in point correspondences increase and 3D information could be erroneous. To increase the number of data from images and to improve the 3D information, trinocular sensors are used. In this paper a calibration method for a trinocular sensor formed with wide angle lens cameras is proposed. First pixels locations in the images are corrected using a set of constraints which define the image formation in a trinocular system. When pixels location are corrected, lens distortion and trifocal tensor is computed.

© 2012 OSA

## 1. Introduction

3. R. Hartley, “Lines and points in three views and the trifocal tensor,” Int. J. Comput. Vis. **22**(2), 125–140 (1997). [CrossRef]

4. P. Torr and A. Zisserman, “Robust parameterization and computation of the trifocal tensor,” Image Vis. Comput. **15**(8), 591–605 (1997). [CrossRef]

## 2. Pixel location correction

*p**,*

_{1}

*p**,*

_{2}

*p**,*

_{3}

*p**) is the cross ratio of four collinear points defined when the calibration template is built.*

_{4}

*q**= (*

^{k,l}

*u**,*

^{k,l}

*v**,*

^{k,l}

*w**) represents a point*

^{k,l}*k*in a line

**= (**

*l**a*,

_{l}*b*,

_{l}*c*). In [11

_{l}11. C. Ricolfe-Viala, A. J. Sanchez-Salmeron, and E. Martinez-Berti, “Accurate calibration with highly distorted images,” Appl. Opt. **51**(1), 89–101 (2012). [CrossRef] [PubMed]

*,*

^{i}a_{l}*,*

^{i}b_{l}*represent the set of parameters which defines the straight line*

^{i}c_{l}*in the image*

^{i}l*i*, and

^{i}**is the vector in the direction of the same line,**

*v*_{l}*i*= 1..3. The vector with the direction of the corresponding line in the scene

*l*is called

**. Since the line**

*v*_{l}*l*and the projection in the image

*are in the same plane*

^{i}l*, vectors*

^{i}π_{l}

^{i}**and**

*v*_{l}**have to be perpendicular to the normal vector of plane**

*v*_{l}*. If*

^{i}π_{l}

^{i}

*n**is the normal vector of the plane*

_{l}*following Eq. is true.*

^{i}π_{l}

^{i}

*V**·*

_{l}

^{i}

*n**= 0. Line*

_{l}*l*is projected in the three images

*,*

^{1}l*,*

^{2}l*and a set of*

^{3}l*s*lines exist, where

*s*is the number of lines in the calibration template. If Eq. (5) is satisfied by all straight lines following Eq. arise:

*chessboard*” template.

**,**

*p*_{1}**,**

*p*_{2}**,**

*p*_{3}**) and the set of points in the image or images**

*p*_{4}

*q**. The minimization process is as follows. For a given set of points, straight lines parameters (*

_{d}*a*,

_{l}*b*,

_{l}*c*), vanishing points

_{l}

^{1,j}**,**

*q*_{vp}

^{2,j}**,**

*q*_{vp}

^{3,j}**,**

*q*_{vp}

^{4,j}**, horizons lines**

*q*_{vp}

^{j}l*, and normal vector of planes*

_{h}*are computed. With the computed parameters, function (7) is evaluated and points locations*

^{i}π_{l}

*q**are corrected to minimize (7). This process is repeated until (7) is zero. When (7) is zero, distorted points extracted from the image*

_{d}**have been undistorted to**

*q*_{d}**. To minimize (7) a Levenberg‐Marquardt non linear minimization algorithm is used which starts with the set of points in the image or images**

*q*_{o}**and ends with the set of undistorted points**

*q*_{d}**. Figure 2(a) shows the original distorted points and the corrected ones. To improve the condition of the non-linear searching process points coordinates are referred to the center of the image and not to the left top corner.**

*q*_{o}## 3. Computing the trifocal tensor

**and calibration template points**

*q*_{o}**. Any of existing linear methods can be used since point location in the image has been corrected. With the point location correction errors in point correspondences does not exist and the noise is reduced.**

*p*## 4. Experimental results

12. C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Lens distortion models evaluation,” Appl. Opt. **49**(30), 5914–5928 (2010). [CrossRef] [PubMed]

*non metric calibration of lens distortion*(NMC) proposed by Ahmed in [14

14. M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. **14**(8), 1215–1230 (2005). [CrossRef] [PubMed]

*polynomial fish-eye lens distortion correction*(PFE) presented by Devernay and Faugeras in [15

15. F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. **13**(1), 14–24 (2001). [CrossRef]

**= (**

**q***u*,

**v*) projected from measured world coordinates by the estimated camera matrix and undistorted pixel coordinates

**= (**

**q*_{o}

****u*,

_{o}

****v*) computed with the estimated lens distortion model.We have used three IP cameras Axis 212 PTZ with 2.7 mm lens mounted which gives 85° field of view. Figure 3 shows three acquired images of 640x480 pixels with considerable distortion. Images 3(d),(e),(f) shows the undistorted images with the proposed method and the set of corresponding points which are used to calibrate the trifocal tensor using the linear method described in subsection 3. In this case, corresponding lines and corresponding points are used to calibrate the trifocal tensor. Figure 2(a) shows the points locations of the original distorted and the undistorted image. Corrected points are moved to accomplish all projective geometric and trifocal sensor constraints.

_{o}## 5. Conclusion

## Acknowledgments

## References

1. | R. Hartley and A. Zisserman, |

2. | O. Faugeras and B. Mourrain, “On the geometry and algebra of the point and line correspondences between n images,” In Proc. 5th Int. Conf. Comput. Vision, 951–962 (1995). |

3. | R. Hartley, “Lines and points in three views and the trifocal tensor,” Int. J. Comput. Vis. |

4. | P. Torr and A. Zisserman, “Robust parameterization and computation of the trifocal tensor,” Image Vis. Comput. |

5. | O. Faugeras, Q.-T. Luong, and T. Papadopoulo, |

6. | P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proc. 10th ECCV, 1–13 (2004) |

7. | C. Ressl, “Geometry, constraints and computation of the trifocal tensor,” |

8. | W. Förstner, “On weighting and choosing constraints for optimally reconstructing the geometry of image triplets,” in Proc. 6th ECCV, 669–684 (2000). |

9. | T. Molinier, D. Fofi, and F. Meriaudeau, “Self-calibration of a trinocular sensor with imperceptible structured light and varying intrinsic parameters,” in Proc.7th Int. Conf. Quality Control by Artificial Vision, (2005). |

10. | D. Nistér, “Reconstruction from uncalibrated sequences with a hierarchy of trifocal tensors,” in Proc. 6th ECCV, 649–663 (2000). |

11. | C. Ricolfe-Viala, A. J. Sanchez-Salmeron, and E. Martinez-Berti, “Accurate calibration with highly distorted images,” Appl. Opt. |

12. | C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Lens distortion models evaluation,” Appl. Opt. |

13. | D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” In Proc. CVPR IEEE, 213–219 (2005). |

14. | M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. |

15. | F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl. |

**OCIS Codes**

(150.0155) Machine vision : Machine vision optics

(150.1135) Machine vision : Algorithms

(150.1488) Machine vision : Calibration

**ToC Category:**

Machine Vision

**History**

Original Manuscript: July 27, 2012

Manuscript Accepted: August 22, 2012

Published: November 29, 2012

**Citation**

Carlos Ricolfe-Viala, Antonio-Jose Sanchez-Salmeron, and Angel Valera, "Calibration of a trinocular system formed with wide angle lens cameras," Opt. Express **20**, 27691-27696 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27691

Sort: Year | Journal | Reset

### References

- R. Hartley and A. Zisserman, Multiple view geometry in computer vision. (Cambridge University Press, 2000).
- O. Faugeras and B. Mourrain, “On the geometry and algebra of the point and line correspondences between n images,” In Proc. 5th Int. Conf. Comput. Vision, 951–962 (1995).
- R. Hartley, “Lines and points in three views and the trifocal tensor,” Int. J. Comput. Vis.22(2), 125–140 (1997). [CrossRef]
- P. Torr and A. Zisserman, “Robust parameterization and computation of the trifocal tensor,” Image Vis. Comput.15(8), 591–605 (1997). [CrossRef]
- O. Faugeras, Q.-T. Luong, and T. Papadopoulo, Geometry of multiple images. (Cambridge, MA, USA: MIT Press, 2001).
- P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proc. 10th ECCV, 1–13 (2004)
- C. Ressl, “Geometry, constraints and computation of the trifocal tensor,” PhD Thesis, Instituüt für Photogrammetrie und Fernerkundung, Technische Universität Wien, (2003).
- W. Förstner, “On weighting and choosing constraints for optimally reconstructing the geometry of image triplets,” in Proc. 6th ECCV, 669–684 (2000).
- T. Molinier, D. Fofi, and F. Meriaudeau, “Self-calibration of a trinocular sensor with imperceptible structured light and varying intrinsic parameters,” in Proc.7th Int. Conf. Quality Control by Artificial Vision, (2005).
- D. Nistér, “Reconstruction from uncalibrated sequences with a hierarchy of trifocal tensors,” in Proc. 6th ECCV, 649–663 (2000).
- C. Ricolfe-Viala, A. J. Sanchez-Salmeron, and E. Martinez-Berti, “Accurate calibration with highly distorted images,” Appl. Opt.51(1), 89–101 (2012). [CrossRef] [PubMed]
- C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Lens distortion models evaluation,” Appl. Opt.49(30), 5914–5928 (2010). [CrossRef] [PubMed]
- D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” In Proc. CVPR IEEE, 213–219 (2005).
- M. Ahmed and A. Farag, “Nonmetric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process.14(8), 1215–1230 (2005). [CrossRef] [PubMed]
- F. Devernay and O. Faugeras, “Straight lines have to be straight,” Mach. Vis. Appl.13(1), 14–24 (2001). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.