## Perturbation of quadric transfer due to deformation of curved screen displays |

Optics Express, Vol. 19, Issue 17, pp. 16236-16243 (2011)

http://dx.doi.org/10.1364/OE.19.016236

Acrobat PDF (1254 KB)

### Abstract

Non-planar screens are increasingly used in mobile projectors and virtual reality environments. When the screen is modeled as a second order polynomial, a quadric transfer method can be employed to compensate for image distortion. This method uses the quadric matrix that models 3D surface information of a quadric screen. However, if the shape of the screen changes or the screen is moved, the 3D shape of the screen must be measured again to update the quadric matrix. We propose a new method of compensating for image distortion resulting from variation of the quadric screen. The proposed method is simpler and faster than remeasuring the 3D screen matrix.

© 2011 OSA

## 1. Introduction

10. A. Shashua and S. Toelg, “The quadric reference surface: theory and applications,” Int. J. Comput. Vis. **23**(2), 185–198 (1997). [CrossRef]

*et al*. proposed the quadric transfer, a geometric compensation method for the projected image on the quadric curved screen. They also used GPU vertex shader for real-time implementation of the quadric transfer [11

11. R. Raskar, J. van Baar, S. Rao, T. Willwacher, and S. Rao, “Quadric transfer for immersive curved screen displays,” Comput. Graph. **23**(3), 451–460 (2004). [CrossRef]

## 2. Quadric Transfer and Screen Change

*et al*[11

11. R. Raskar, J. van Baar, S. Rao, T. Willwacher, and S. Rao, “Quadric transfer for immersive curved screen displays,” Comput. Graph. **23**(3), 451–460 (2004). [CrossRef]

**x**represents the 3D coordinates of the first view,

**A**and

**E**are defined as follows:where

**B**is a 3 × 3 homography matrix between the two coordinates, and

**Q**is the quadric matrix of the screen.

**q**are submatrices of

**Q**as defined in the following equations:If a point

### 2.1 Quadric Transfer

*z*= 1 plane, which is the projection of

*z*= 1 camera image plane. Then we define

*z*= 1 plane.

### 2.2 Quadric Transfer after Screen Change

**X**to

*z*= 1 plane. Then using the scale factor

*h*,

*α*is defined as

**e**passing

*h*can be calculated using the minimum mean square error criterion.

## 3. Change of Quadric Matrix

*m*and

**A**and

**E**, are corrected using the linear solution of the change of the quadric matrix. With the corrected quadric transfer, we can compensate for the change and the movement of the screen.

## 4. Simulations and Experimental Results

### 4.1. Simulations

*x*-axis. When

### 4.2. Experimental Results

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. van Baar, T. Willwacher, S. Rao, and R. Raskar, “Seamless multi-projector display on curved screens,” Eurographics Workshop on Virtual Environments, 281–286 (2003). |

2. | R. Raskar, G. Welch, M. Cutts, A. Lake, L. Stesin and H. Fuchs, “The office of the future: a unified approach to image-based modeling and spatially immersive displays,” SIGGRAPH, 179–188 (1998). |

3. | R. Yang, M. S. Brown, W. B. Seales, and H. Fuchs, “Geometrically correct imagery for teleconferencing,” in |

4. | R. Yang and G. Welch, “Automatic and continuous projector display surface calibration using every-day imagery,” in |

5. | S. Webb and C. Jaynes, “The DOME: a portable multi-projector visualization system for digital artifacts,” IEEE Workshop on Emerging Display Technologies (2005). |

6. | Y. Oyamada and H. Saito, “Focal pre-correction of projected image for deblurring screen image,” IEEE Int. Workshop on Projector-Camera systems (2007). |

7. | R. Raskar, M. Brown, R. Yang, W. Chen, G. Welch, H. Towels, B. Seales, and H. Fuchs, “Multi-projector displays using camera-based registration,” in |

8. | S. Zollmann, T. Langlotz, and O. Bimber, “Passive-active geometric calibration for view-dependent projections onto arbitrary surfaces,” Workshop on Virtual and Augmented Reality of the GI-Fachgruppe AR/VR (2006). |

9. | S. Jordan and M. Greenspan, “Projector optical distortion calibration using gray code patterns,” IEEE Int. Workshop on Projector-Camera systems (2010). |

10. | A. Shashua and S. Toelg, “The quadric reference surface: theory and applications,” Int. J. Comput. Vis. |

11. | R. Raskar, J. van Baar, S. Rao, T. Willwacher, and S. Rao, “Quadric transfer for immersive curved screen displays,” Comput. Graph. |

12. | M. Emori and H. Saito, “Texture overlay onto deformable surface using HMD,” in |

13. | D. G. Lowe, “Object recognition from local scale-invariant features,” in |

**OCIS Codes**

(100.2980) Image processing : Image enhancement

(150.1488) Machine vision : Calibration

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: June 16, 2011

Revised Manuscript: July 21, 2011

Manuscript Accepted: August 5, 2011

Published: August 9, 2011

**Citation**

Junhee Park, Kyung-Mi Lee, and Byung-Uk Lee, "Perturbation of quadric transfer due to deformation of curved screen displays," Opt. Express **19**, 16236-16243 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16236

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### References

- J. van Baar, T. Willwacher, S. Rao, and R. Raskar, “Seamless multi-projector display on curved screens,” Eurographics Workshop on Virtual Environments, 281–286 (2003).
- R. Raskar, G. Welch, M. Cutts, A. Lake, L. Stesin and H. Fuchs, “The office of the future: a unified approach to image-based modeling and spatially immersive displays,” SIGGRAPH, 179–188 (1998).
- R. Yang, M. S. Brown, W. B. Seales, and H. Fuchs, “Geometrically correct imagery for teleconferencing,” in Proceedings of ACM Multimedia, 179–186 (1999).
- R. Yang and G. Welch, “Automatic and continuous projector display surface calibration using every-day imagery,” in Proceedings of 9th Int. Conf. in Central Europe in Computer Graphics, Visualization, and Computer Vision (2001).
- S. Webb and C. Jaynes, “The DOME: a portable multi-projector visualization system for digital artifacts,” IEEE Workshop on Emerging Display Technologies (2005).
- Y. Oyamada and H. Saito, “Focal pre-correction of projected image for deblurring screen image,” IEEE Int. Workshop on Projector-Camera systems (2007).
- R. Raskar, M. Brown, R. Yang, W. Chen, G. Welch, H. Towels, B. Seales, and H. Fuchs, “Multi-projector displays using camera-based registration,” in Proceedings of IEEE Visualization, 161–168 (1999).
- S. Zollmann, T. Langlotz, and O. Bimber, “Passive-active geometric calibration for view-dependent projections onto arbitrary surfaces,” Workshop on Virtual and Augmented Reality of the GI-Fachgruppe AR/VR (2006).
- S. Jordan and M. Greenspan, “Projector optical distortion calibration using gray code patterns,” IEEE Int. Workshop on Projector-Camera systems (2010).
- A. Shashua and S. Toelg, “The quadric reference surface: theory and applications,” Int. J. Comput. Vis. 23(2), 185–198 (1997). [CrossRef]
- R. Raskar, J. van Baar, S. Rao, T. Willwacher, and S. Rao, “Quadric transfer for immersive curved screen displays,” Comput. Graph. 23(3), 451–460 (2004). [CrossRef]
- M. Emori and H. Saito, “Texture overlay onto deformable surface using HMD,” in Proceedings of IEEE Virtual Reality, 221–222 (2004).
- D. G. Lowe, “Object recognition from local scale-invariant features,” in Proceedings of ICCV, 1150–1157 (1999).

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