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

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References

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  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 Proceedings of ACM Multimedia, 179–186 (1999).
  4. 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).
  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 Proceedings of IEEE Visualization, 161–168 (1999).
  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. 23(2), 185–198 (1997).
    [CrossRef]
  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]
  12. M. Emori and H. Saito, “Texture overlay onto deformable surface using HMD,” in Proceedings of IEEE Virtual Reality, 221–222 (2004).
  13. D. G. Lowe, “Object recognition from local scale-invariant features,” in Proceedings of ICCV, 1150–1157 (1999).

2004 (1)

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]

1997 (1)

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

Rao, S.

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]

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]

Raskar, R.

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]

Shashua, A.

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

Toelg, S.

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

van Baar, J.

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]

Willwacher, T.

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]

Comput. Graph. (1)

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]

Int. J. Comput. Vis. (1)

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

Other (11)

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).

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).

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Figures (8)

Fig. 1
Fig. 1

An example of a projector-camera system for the quadric transfer. Projecting a rectangular image onto a curved surface results in a distorted image. To correct the distortion, images can be pre-warped in such a way that it compensates for the curve of the screen. However, the change of the screen results in image distortion.

Fig. 2
Fig. 2

Quadric transfer after screen change.

Fig. 3
Fig. 3

3D plot of projector center (◊), camera center (□), 3D image points (•) and sphere screen. (a) Before screen translation, (b) After screen translation

Fig. 4
Fig. 4

Simulated test patterns on the spherical screen captured by a camera. (a) Compensated image using quadric transfer, (b) Distorted image after shift of the sphere, (c) Corrected image using the proposed method.

Fig. 5
Fig. 5

Simulated mean absolute image position error before and after compensation when the screen moves from –15 to 15.

Fig. 6
Fig. 6

Experimental setup.

Fig. 7
Fig. 7

(a) Real transferred image (b) Camera-captured image.

Fig. 8
Fig. 8

(a) Camera-captured image with distortion after screen change (b) Camera-captured image after compensation using the proposed method.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

x = A x ± ( x T E x ) e
A = B e q T E = q q T Q 3 3
Q = [ a b c d b e f g c f h i d g i 1 ] = [ Q 3 3 q q T 1 ] q T = [ d g i ]
x + Δ x = ( A + Δ A ) x ± ( x T ( E + Δ E ) x ) e
Δ A = e Δ q T Δ E = q Δ q T + Δ q q T + Δ q Δ q T Δ Q 3 3
Δ E Δ E a + Δ E r Δ E a q Δ q T + Δ q q T Δ Q 3 3
m + Δ m x T ( E + Δ E ) x Δ m x T Δ E x = x T ( Δ E a + Δ E r ) x Δ m a + Δ m r Δ m a x T Δ E a x = x T Δ Q 3 3 x + 2 ( q T x ) ( Δ q T x ) Δ m r x T Δ E r x = ( Δ q T x ) 2
x ˜ A x ˜ ± ( x ˜ T E x ˜ ) e
x ^ = A x ± ( x T E x ) e
x ^ c h = ( A + Δ A ) x ^ ± ( x ^ T ( E + Δ E ) x ^ ) e
x ^ c h = h x c h x + α e .
x = A x ^ ± ( x ^ T E x ^ ) e x + Δ x = ( A + Δ A ) x ^ ± ( x ^ T ( E + Δ E ) x ^ ) e
x = A x ± m e
x + Δ x = ( A + Δ A ) x ± ( m + Δ m ) e
Δ x ε e .
ε Δ x e x = Δ y e y = Δ z e z
± 2 m ε = ( x T Δ Q 3 3 x + 2 k Δ q T x ) + Δ m r
± 2 m ε = [ x 2 2 x y 2 x z 2 k x y 2 2 y z 2 k y z 2 2 k z ] Δ φ

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