Abstract

I present a design technique for realizing given projections as catadioptric sensors. In general, these problems do not have solutions, but approximate solutions may often be found that are visually acceptable. The method described reduces the problem to solving a linear system. A given transformation from the image plane to an object surface is shown to determine a vector field that is normal to the surface in the case where the vector field is a gradient. For the case when the vector field is not a gradient, several functionals are presented that may be minimized to give approximate solutions. As an application several new designs are described, including a mirror that directly gives a full 360-deg cylindrical projection without the need for any digital processing.

© 2005 Optical Society of America

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  1. R. Kingslake, A History of the Photographic Lens (Academic, Boston, Mass., 1989).
  2. L. H. Kleinschmidt, “Apparatus for producing topographic views,” U.S. patent994,935, June13, 1911.
  3. The page of the catadioptric sensor design is available at http://www.cs.drexel.edu/ahicks/design/design.html , 2003.
  4. J. H. McDermit, T. E. Horton, “Reflective optics for obtaining prescribed irradiative distributions from collimated sources,” Appl. Opt. 13, 1444–1450 (1974).
    [CrossRef] [PubMed]
  5. D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” in Laser Beam Shaping, E. M. Dickey, S. C. Holswade, eds., Proc. SPIE4095, 1–15 (2000).
    [CrossRef]
  6. R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1995).
  7. D. G. Burkhard, D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” J. Solar Energy 17, 221–227 (1975).
    [CrossRef]
  8. H. Ries, J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002).
    [CrossRef]
  9. F. Benford, “Apparatus employed in computing illuminations,” U.S. patent2,371,495, March13, 1945.
  10. S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 35–42.
  11. J. S. Chahl, M. V. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997).
    [CrossRef]
  12. H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).
  13. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).
  14. R. A. Hicks, R. Perline, “Geometric distributions and catadioptric sensor design,” in Proceedings of the Computer Vision Pattern Recognition Conference (IEEE Computer Society Press, Los Alamitos Calif., 2001), pp. 584–589.
  15. R. A. Hicks, “Differential methods in catadioptric sensor design with applications to panoramic imaging,” arXiv preprint cs.CV/0303024, http://www.arxiv.org/abs/cs.CV/0303024 , 2003.
  16. R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in Proceedings of the IEEE International Workshop on Projector–Camera Systems (IEEE Computer Society Press, Los Alamitos, Calif., 2003).
  17. M. Srinivasan, “A new class of mirrors for wide-angle imaging,” in Proceedings of the IEEE Workshop on Omnidirectional Vision (IEEE Computer Society Press, Los Alamitos, Calif., 2003).
  18. M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.
  19. H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic, New York, 1963).
  20. F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer-Verlag, New York, 1983).
  21. A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer-Verlag, New York, 1993).
  22. W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, New York, 2001).

2002 (1)

1997 (1)

1975 (1)

D. G. Burkhard, D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” J. Solar Energy 17, 221–227 (1975).
[CrossRef]

1974 (1)

Baker, S.

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 35–42.

Barsky, B.

M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.

Benford, F.

F. Benford, “Apparatus employed in computing illuminations,” U.S. patent2,371,495, March13, 1945.

Buchdahl, H.

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” J. Solar Energy 17, 221–227 (1975).
[CrossRef]

Chahl, J. S.

Cheney, W.

W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, New York, 2001).

Chorin, A.

A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer-Verlag, New York, 1993).

Flanders, H.

H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic, New York, 1963).

Grossberg, M.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in Proceedings of the IEEE International Workshop on Projector–Camera Systems (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

Halstead, M.

M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.

Hicks, R. A.

R. A. Hicks, “Differential methods in catadioptric sensor design with applications to panoramic imaging,” arXiv preprint cs.CV/0303024, http://www.arxiv.org/abs/cs.CV/0303024 , 2003.

R. A. Hicks, R. Perline, “Geometric distributions and catadioptric sensor design,” in Proceedings of the Computer Vision Pattern Recognition Conference (IEEE Computer Society Press, Los Alamitos Calif., 2001), pp. 584–589.

Horton, T. E.

Kingslake, R.

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

Klein, S.

M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.

Kleinschmidt, L. H.

L. H. Kleinschmidt, “Apparatus for producing topographic views,” U.S. patent994,935, June13, 1911.

Mandell, R.

M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.

Marsden, J.

A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer-Verlag, New York, 1993).

McDermit, J. H.

Muschaweck, J.

Nayar, S.

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 35–42.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in Proceedings of the IEEE International Workshop on Projector–Camera Systems (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

Perline, R.

R. A. Hicks, R. Perline, “Geometric distributions and catadioptric sensor design,” in Proceedings of the Computer Vision Pattern Recognition Conference (IEEE Computer Society Press, Los Alamitos Calif., 2001), pp. 584–589.

Ries, H.

Shealy, D. L.

D. G. Burkhard, D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” J. Solar Energy 17, 221–227 (1975).
[CrossRef]

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” in Laser Beam Shaping, E. M. Dickey, S. C. Holswade, eds., Proc. SPIE4095, 1–15 (2000).
[CrossRef]

Srinivasan, M.

M. Srinivasan, “A new class of mirrors for wide-angle imaging,” in Proceedings of the IEEE Workshop on Omnidirectional Vision (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

Srinivasan, M. V.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Swaminathan, R.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in Proceedings of the IEEE International Workshop on Projector–Camera Systems (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

Warner, F.

F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer-Verlag, New York, 1983).

Winston, R.

R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1995).

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

J. Solar Energy (1)

D. G. Burkhard, D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” J. Solar Energy 17, 221–227 (1975).
[CrossRef]

Other (18)

F. Benford, “Apparatus employed in computing illuminations,” U.S. patent2,371,495, March13, 1945.

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Proceedings of the International Conference on Computer Vision (Narosa, Bombay, India, 1998), pp. 35–42.

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” in Laser Beam Shaping, E. M. Dickey, S. C. Holswade, eds., Proc. SPIE4095, 1–15 (2000).
[CrossRef]

R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1995).

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

L. H. Kleinschmidt, “Apparatus for producing topographic views,” U.S. patent994,935, June13, 1911.

The page of the catadioptric sensor design is available at http://www.cs.drexel.edu/ahicks/design/design.html , 2003.

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, New York, 1993).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

R. A. Hicks, R. Perline, “Geometric distributions and catadioptric sensor design,” in Proceedings of the Computer Vision Pattern Recognition Conference (IEEE Computer Society Press, Los Alamitos Calif., 2001), pp. 584–589.

R. A. Hicks, “Differential methods in catadioptric sensor design with applications to panoramic imaging,” arXiv preprint cs.CV/0303024, http://www.arxiv.org/abs/cs.CV/0303024 , 2003.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in Proceedings of the IEEE International Workshop on Projector–Camera Systems (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

M. Srinivasan, “A new class of mirrors for wide-angle imaging,” in Proceedings of the IEEE Workshop on Omnidirectional Vision (IEEE Computer Society Press, Los Alamitos, Calif., 2003).

M. Halstead, B. Barsky, S. Klein, R. Mandell, “Reconstructing curved surfaces from specular reflection patterns using spline surface fitting of normals,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (Association of Computing Machinery Press, New York, 1996), pp. 335–342.

H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic, New York, 1963).

F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Springer-Verlag, New York, 1983).

A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer-Verlag, New York, 1993).

W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, New York, 2001).

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Figures (16)

Fig. 1
Fig. 1

Panoramic viewed captured with a parabolic mirror.

Fig. 2
Fig. 2

Image from Fig. 1 digitally processed into a strip view.

Fig. 3
Fig. 3

In the planar case, it is easy to see how a given correspondence from the image plane to the object plane determines a field of lines.

Fig. 4
Fig. 4

Statement of a general form of the problem for single-mirror systems: Given a transformation G from an image plane I to a surface S, find a mirror such that the induced optical transformation is as close to the prescribed transformation G as possible.

Fig. 5
Fig. 5

Correspondence between the image plane and an object surface determines a vector field in 3-space, W.

Fig. 6
Fig. 6

To model a passenger-side mirror on a car, we take the object plane to be parallel to the optical axis, which in this case coincides with the x axis.

Fig. 7
Fig. 7

Image formed from a mirror resulting from minimizing the Hodge functional.

Fig. 8
Fig. 8

Image formed from a mirror resulting from minimizing the generalized Hodge functional, for the same design problem as the mirror in Fig. 7.

Fig. 9
Fig. 9

Vector field method for a panoramic mirror. Here we construct a vector field by mapping the image plane onto a cylinder centered about the optical axis.

Fig. 10
Fig. 10

Surface that when used as a mirror gives a panoramic view without any digital unwarping.

Fig. 11
Fig. 11

Panoramic view of a chessboard using the mirror of Fig. 10.

Fig. 12
Fig. 12

Left, experimental setup consisting of a chessboard and a panoramic mirror in its center. Right, this scene imaged with the panoramic mirror.

Fig. 13
Fig. 13

Panoramic “conquistador” mirror, designed with proportions close to 4:3.

Fig. 14
Fig. 14

Chessboard scene imaged with the conquistador mirror.

Fig. 15
Fig. 15

Side-view mirror that expands an 11-deg field of view to a 22-deg field of view.

Fig. 16
Fig. 16

Sample image formed with the mirror of Fig. 15. The increase in the field of view is apparent from the difference in size between the squares viewed in the mirror and the squares in the background.

Equations (20)

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W(r)=q(r)-r|q(r)-r|+G(q(r))-r|G(q(r))-r|,
F=α(x, y, z)W.
(×W)W=0.
W=e1+[λky-x, -k-y, λkz-z][(λky-x)2+(-k-y)2+(λkz-z)2]1/2,
W=e1+[λy, -1, λz](λ2y2+1+λ2z2)1/2=[λy+(λ2y2+1+λ2z2)1/2, -1, λz](λ2y2+1+λ2z2)1/2.
U=1, -1λy+(λ2y2+1+λ2z2)1/2, λzλy+(λ2y2+1+λ2z2)1/2.
fy=1λy+(λ2y2+1+λ2z2)1/2,
fz=-λzλy+(λ2y2+1+λ2z2)1/2.
abcdfy-1λy+(λ2y2+1+λ2z2)1/22+fz--λzλy+(λ2y2+1+λ2z2)1/22dydz
HW(F)=Ω|F-W|2dV
f(x)=f(x),f(0)=1
01[f(x)-f(x)]2dx
815a22+12a1a2-43a2-23a1+1.
f(y, z)=c1z2+c2y+c3z4+c4yz2+c5y2+c6z6+c7yz4+c8y2z2+c9y3.
GHW(F)=Ω|F-αW|2dV.
α=WFWW,
0i,j,kN cijkxiyjzk,
A(1-α)2+(Fy-αW2)2+(Fz-αW3)2dydz.
F(x, y, z)=0i,j,kN cijkxiyjzk,
F=z+c002z2+c011yz+c211x2yz+c010y+c200x2.

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