Abstract

We show that, for any rotationally symmetric projection with a single virtual viewpoint, it is possible to design a two-mirror rotationally symmetric system that realizes the projection exactly. These mirror pairs are derived from two coupled differential equations. We give examples in which the projections from the sphere at infinity are stereographic, perspective, and equiresolution.

© 2006 Optical Society of America

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References

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  1. R. Kingslake, A History of the Photographic Lens (Academic, 1989).
  2. L. H. Kleinschmidt, "Apparatus for producing topographic views" U.S. patent 994,935 (13 June 1911).
  3. R. A. Hicks, "The page of catadioptric sensor design," http://www.cs.drexel.edu/ahicks/design/design.html (2003).
  4. D. Rees, "Hyperbolic ellipsoidal real time display panoramic viewing installation for vehicles," U.S. patent 3,229,576 (18 January 1966).
  5. Y. Yagi and S. Kawato, "Panoramic scene analysis with conic projection," in Proceedings of the International Conference on Robots and Systems (IEEE, 1990), pp. 1-10.
  6. K. Yamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboidal projection," in Proceedings of the IEEE International Conference on Robots and Systems, (IEEE, 1993), pp 77-86.
  7. A. Bruckstein and T. Richardson, Omniview Cameras with Curved Surface Mirrors (Bell Laboratories Technical Memo, 1996).
  8. A. Bruckstein and T. Richardson, "Method and system for panoramic viewing with curved surface mirrors," U.S. patent 5,920,376 (6 July 1999).
  9. S. Nayar, "Catadioptric omnidirectional camera," in Proc. Computer Vision Pattern Recognition (IEEE Computer Society Press, 1997), pp. 482-488.
  10. M. Bass, ed. Handbook of Optics (McGraw-Hill, 1995), Vol. II.
  11. W. A. Young, "Wide-angle optical system," U.S. patent 2,430,595 (11 November 1947).
  12. S. Nayar and V. Peri, "Folded catadioptric cameras," in Proc. Computer Vision Pattern Recognition (1999), pp. 217-223.
  13. R. A. Hicks, Designing a mirror to realize a given projection.J. Optical Soc. Am. A 22, 323-330 (2005).
    [Crossref]
  14. C. Geyer and K. Daniilidis, "Catadioptric camera calibration," in Proceedings of the Seventh International Conference on Computer Vision (1999), pp. 398-404.
    [Crossref]
  15. C. Geyer and K. Daniilidis, Mirrors in motion: Epipolar geometry and motion estimation, in Proceedings of the Eleventh International Conference on Computer Vision (2003), pp. 766-773.
    [Crossref]
  16. C. Geyer and K. Daniilidis, "Para-cata-dioptric calibration," IEEE Trans. Pattern Anal. Mach. Intell. , 24, 687-694 (2002).
    [Crossref]
  17. R. Hicks and R. Perline, "Equiresolution catadioptric sensors," Appl. Opt. 44, 6108-6114 (2005).
    [Crossref] [PubMed]

2005 (2)

R. A. Hicks, Designing a mirror to realize a given projection.J. Optical Soc. Am. A 22, 323-330 (2005).
[Crossref]

R. Hicks and R. Perline, "Equiresolution catadioptric sensors," Appl. Opt. 44, 6108-6114 (2005).
[Crossref] [PubMed]

2002 (1)

C. Geyer and K. Daniilidis, "Para-cata-dioptric calibration," IEEE Trans. Pattern Anal. Mach. Intell. , 24, 687-694 (2002).
[Crossref]

Bass, M.

M. Bass, ed. Handbook of Optics (McGraw-Hill, 1995), Vol. II.

Bruckstein, A.

A. Bruckstein and T. Richardson, Omniview Cameras with Curved Surface Mirrors (Bell Laboratories Technical Memo, 1996).

A. Bruckstein and T. Richardson, "Method and system for panoramic viewing with curved surface mirrors," U.S. patent 5,920,376 (6 July 1999).

Daniilidis, K.

C. Geyer and K. Daniilidis, "Para-cata-dioptric calibration," IEEE Trans. Pattern Anal. Mach. Intell. , 24, 687-694 (2002).
[Crossref]

C. Geyer and K. Daniilidis, Mirrors in motion: Epipolar geometry and motion estimation, in Proceedings of the Eleventh International Conference on Computer Vision (2003), pp. 766-773.
[Crossref]

C. Geyer and K. Daniilidis, "Catadioptric camera calibration," in Proceedings of the Seventh International Conference on Computer Vision (1999), pp. 398-404.
[Crossref]

Geyer, C.

C. Geyer and K. Daniilidis, "Para-cata-dioptric calibration," IEEE Trans. Pattern Anal. Mach. Intell. , 24, 687-694 (2002).
[Crossref]

C. Geyer and K. Daniilidis, Mirrors in motion: Epipolar geometry and motion estimation, in Proceedings of the Eleventh International Conference on Computer Vision (2003), pp. 766-773.
[Crossref]

C. Geyer and K. Daniilidis, "Catadioptric camera calibration," in Proceedings of the Seventh International Conference on Computer Vision (1999), pp. 398-404.
[Crossref]

Hicks, R.

Hicks, R. A.

R. A. Hicks, Designing a mirror to realize a given projection.J. Optical Soc. Am. A 22, 323-330 (2005).
[Crossref]

R. A. Hicks, "The page of catadioptric sensor design," http://www.cs.drexel.edu/ahicks/design/design.html (2003).

Kawato, S.

Y. Yagi and S. Kawato, "Panoramic scene analysis with conic projection," in Proceedings of the International Conference on Robots and Systems (IEEE, 1990), pp. 1-10.

Kingslake, R.

R. Kingslake, A History of the Photographic Lens (Academic, 1989).

Kleinschmidt, L. H.

L. H. Kleinschmidt, "Apparatus for producing topographic views" U.S. patent 994,935 (13 June 1911).

Nayar, S.

S. Nayar and V. Peri, "Folded catadioptric cameras," in Proc. Computer Vision Pattern Recognition (1999), pp. 217-223.

S. Nayar, "Catadioptric omnidirectional camera," in Proc. Computer Vision Pattern Recognition (IEEE Computer Society Press, 1997), pp. 482-488.

Peri, V.

S. Nayar and V. Peri, "Folded catadioptric cameras," in Proc. Computer Vision Pattern Recognition (1999), pp. 217-223.

Perline, R.

Rees, D.

D. Rees, "Hyperbolic ellipsoidal real time display panoramic viewing installation for vehicles," U.S. patent 3,229,576 (18 January 1966).

Richardson, T.

A. Bruckstein and T. Richardson, "Method and system for panoramic viewing with curved surface mirrors," U.S. patent 5,920,376 (6 July 1999).

A. Bruckstein and T. Richardson, Omniview Cameras with Curved Surface Mirrors (Bell Laboratories Technical Memo, 1996).

Yachida, M.

K. Yamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboidal projection," in Proceedings of the IEEE International Conference on Robots and Systems, (IEEE, 1993), pp 77-86.

Yagi, Y.

K. Yamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboidal projection," in Proceedings of the IEEE International Conference on Robots and Systems, (IEEE, 1993), pp 77-86.

Y. Yagi and S. Kawato, "Panoramic scene analysis with conic projection," in Proceedings of the International Conference on Robots and Systems (IEEE, 1990), pp. 1-10.

Yamazawa, K.

K. Yamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboidal projection," in Proceedings of the IEEE International Conference on Robots and Systems, (IEEE, 1993), pp 77-86.

Young, W. A.

W. A. Young, "Wide-angle optical system," U.S. patent 2,430,595 (11 November 1947).

Appl. Opt. (1)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

C. Geyer and K. Daniilidis, "Para-cata-dioptric calibration," IEEE Trans. Pattern Anal. Mach. Intell. , 24, 687-694 (2002).
[Crossref]

J. Optical Soc. Am. A (1)

R. A. Hicks, Designing a mirror to realize a given projection.J. Optical Soc. Am. A 22, 323-330 (2005).
[Crossref]

Other (14)

C. Geyer and K. Daniilidis, "Catadioptric camera calibration," in Proceedings of the Seventh International Conference on Computer Vision (1999), pp. 398-404.
[Crossref]

C. Geyer and K. Daniilidis, Mirrors in motion: Epipolar geometry and motion estimation, in Proceedings of the Eleventh International Conference on Computer Vision (2003), pp. 766-773.
[Crossref]

R. Kingslake, A History of the Photographic Lens (Academic, 1989).

L. H. Kleinschmidt, "Apparatus for producing topographic views" U.S. patent 994,935 (13 June 1911).

R. A. Hicks, "The page of catadioptric sensor design," http://www.cs.drexel.edu/ahicks/design/design.html (2003).

D. Rees, "Hyperbolic ellipsoidal real time display panoramic viewing installation for vehicles," U.S. patent 3,229,576 (18 January 1966).

Y. Yagi and S. Kawato, "Panoramic scene analysis with conic projection," in Proceedings of the International Conference on Robots and Systems (IEEE, 1990), pp. 1-10.

K. Yamazawa, Y. Yagi, and M. Yachida, "Omnidirectional imaging with hyperboidal projection," in Proceedings of the IEEE International Conference on Robots and Systems, (IEEE, 1993), pp 77-86.

A. Bruckstein and T. Richardson, Omniview Cameras with Curved Surface Mirrors (Bell Laboratories Technical Memo, 1996).

A. Bruckstein and T. Richardson, "Method and system for panoramic viewing with curved surface mirrors," U.S. patent 5,920,376 (6 July 1999).

S. Nayar, "Catadioptric omnidirectional camera," in Proc. Computer Vision Pattern Recognition (IEEE Computer Society Press, 1997), pp. 482-488.

M. Bass, ed. Handbook of Optics (McGraw-Hill, 1995), Vol. II.

W. A. Young, "Wide-angle optical system," U.S. patent 2,430,595 (11 November 1947).

S. Nayar and V. Peri, "Folded catadioptric cameras," in Proc. Computer Vision Pattern Recognition (1999), pp. 217-223.

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Figures (7)

Fig. 1
Fig. 1

Seeking two mirrors such that the point [ a , b ] projects through [ h , 0 ] to [ f ( a / b ) , 0 ] .

Fig. 2
Fig. 2

(Color online) Cross section of a two-mirror system that achieves stereographic projection.

Fig. 3
Fig. 3

(Color online) A ray-tracing simulation of an image formed by the system depicted in Fig. 2.

Fig. 4
Fig. 4

(Color online) Cross section of a sensor that has equiresolution and is central.

Fig. 5
Fig. 5

(Color online) Ray-tracing simulation of an image formed by the system depicted in Fig. 4.

Fig. 6
Fig. 6

(Color online) Cross section of a sensor that achieves a wide-angle perspective projection.

Fig. 7
Fig. 7

(Color online) Ray-tracing simulation of an image formed by the system depicted in Fig. 6.

Equations (9)

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

z ( t ) = - t h f [ x ( t ) y ( t ) ] + h ,
V = [ x ( t ) , y ( t ) ] x ( t ) 2 + y ( t ) 2 + [ t - x ( t ) , z ( t ) - y ( t ) ] [ t - x ( t ) ] 2 + [ z ( t ) - y ( t ) 2 ] .
[ x ( t ) , y ( t ) ] V = 0 ,
[ 1 , z ( t ) ] W = 0 .
x ( t ) = F ( x ( t ) , y ( t ) , t ) , y ( t ) = G ( x ( t ) , y ( t ) , t ) ,
[ x , y ] α x x 2 + y 2 - y .
t x ( t ) 5 x ( t ) 2 + y ( t ) 2 y ( t ) ( t x ( t ) 5 [ x ( t ) ] 2 + [ y ( t ) ] 2 y ( t ) ) 2 + ( 1 [ x ( t ) 2 + y ( t ) 2 y ( t ) ] 5 t x ( t ) ) 2 + t + x ( t ) ( t + x ( t ) ) 2 + ( 5 { [ x ( t ) ] 2 + [ y ( t ) ] 2 y ( t ) } t x ( t ) + y ( t ) 1 ) 2 + [ 5 x ( t ) 2 + y ( t ) 2 - y ( t ) x ( t ) - 5 t ( x ( t ) 2 + y ( t ) 2 - y ( t ) ) d x d t x ( t ) 2 + 5 t x ( t ) ( x ( t ) d x d t + y ( t ) d y d t x ( t ) 2 + y ( t ) 2 - d y d t ) ] × [ 5 x ( t ) 2 + y ( t ) 2 - y ( t ) x ( t ) - 5 t ( x ( t ) 2 + y ( t ) 2 - y ( t ) ) d x d t x ( t ) 2 + 5 t x ( t ) ( x ( t ) d x d t + y ( t ) d y d t x ( t ) 2 + y ( t ) 2 - d y d t ) ] × [ ( - 1 - 5 t [ x ( t ) 2 + y ( t ) 2 - y ( t ) ] x ( t ) ) ( - t - x ( t ) 5 x ( t ) 2 + y ( t ) 2 - 5 y ( t ) ) 2 + ( - 1 - 5 t [ x ( t ) 2 + y ( t ) 2 - y ( t ) ] x ( t ) ) 2 + - 5 t ( x ( t ) 2 + y ( t ) 2 - y ( t ) ) x ( t ) + y ( t ) - 1 ( - t + x ( t ) ) 2 + ( - 5 t ( x ( t ) 2 + y ( t ) 2 - y ( t ) ) x ( t ) + y ( t ) - 1 ) 2 ] = 0.
[ x , y ] α 2 ( 1 cos [ π 2 arctan ( y x ) ] ) .
[ x , y ] α x y .

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