Abstract

A prominent characteristic of most catadioptric sensors is their lack of uniformity of resolution. We describe catadioptric sensors whose associated projections from the viewing sphere to the image plane have constant Jacobian determinants and so are equiresolution in the sense that any two equal solid angles are allocated the same number of pixels in the image plane. We show that in the orthographic case the catoptric component must be a surface of revolution of constant Gaussian curvature. We compare these equiresolution sensors in both the perspective and orthographic cases with other sensors that were proposed earlier for treating the uniformity-of-resolution problem.

© 2005 Optical Society of America

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  1. C. Geyer, K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” In Proceedings of the European Conference on Computer Vision (IEEE, 2000), pp. 445–461.
  2. D. Rees, “Panoramic television viewing system,” U.S. patent3,505,465 (7April1970).
  3. P. Greguss, “Panoramic imaging block for three-dimensional space,” U.S. patent4,566,736 (28January1986).
  4. Y. Yagi, S. Kawato, “Panoramic scene analysis with conic projection,” in Proceedings of the International Conference on Robots and Systems (IEEE, 1990), pp. 1–10.
  5. K. Yamazawa, Y. Yagi, M. Yachida, “Omnidirectional imaging with hyperboidal projection,” in Proceedings of the IEEE International Conference on Robots and Systems (IEEE, 1993), pp. 77–86.
  6. A. Bruckstein, T. Richardson, “Omniview cameras with curved surface mirrors,” Bell Lab Tech Memo (Bell Laboratories, 1996).
  7. S. Nayar, “Catadioptric omnidirectional camera,” Comput. Vis. Pattern Recog. 2, 482–488 (1997).
  8. J. S. Chahl, M. V. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997).
    [CrossRef]
  9. M. Ollis, H. Herman, S. Singh, “Analysis and design of panoramic stereo vision using equi-angular pixel cameras,” Tech. Rep. (Robotics Institute, Carnegie Mellon University, 1999).
  10. T. Conroy, J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999), pp. 392–397.
    [CrossRef]
  11. S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Sixth International Conference on Computer Vision (IEEE, 1998), pp. 35–42.
    [CrossRef]
  12. J. H. McDermit, T. E. Horton, “Reflective optics for obtaining prescribed irradiative distributions from collimated sources,” Appl. Opt. 13, 1444–1450 (1974).
    [CrossRef] [PubMed]
  13. G. Rines, J. Kuppenheimer, “Reflective optical element,” U.S. patent4,662,726 (5May1986).
  14. D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” in Laser Beam Shaping, F. M. Dickey, S. C. Holswade, eds., Proc. SPIE4095, 1–15 (2000).
    [CrossRef]
  15. M. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976).

1997

S. Nayar, “Catadioptric omnidirectional camera,” Comput. Vis. Pattern Recog. 2, 482–488 (1997).

J. S. Chahl, M. V. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997).
[CrossRef]

1974

Baker, S.

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Sixth International Conference on Computer Vision (IEEE, 1998), pp. 35–42.
[CrossRef]

Bruckstein, A.

A. Bruckstein, T. Richardson, “Omniview cameras with curved surface mirrors,” Bell Lab Tech Memo (Bell Laboratories, 1996).

Chahl, J. S.

Conroy, T.

T. Conroy, J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999), pp. 392–397.
[CrossRef]

Daniilidis, K.

C. Geyer, K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” In Proceedings of the European Conference on Computer Vision (IEEE, 2000), pp. 445–461.

do Carmo, M.

M. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976).

Geyer, C.

C. Geyer, K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” In Proceedings of the European Conference on Computer Vision (IEEE, 2000), pp. 445–461.

Greguss, P.

P. Greguss, “Panoramic imaging block for three-dimensional space,” U.S. patent4,566,736 (28January1986).

Herman, H.

M. Ollis, H. Herman, S. Singh, “Analysis and design of panoramic stereo vision using equi-angular pixel cameras,” Tech. Rep. (Robotics Institute, Carnegie Mellon University, 1999).

Horton, T. E.

Kawato, S.

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

Kuppenheimer, J.

G. Rines, J. Kuppenheimer, “Reflective optical element,” U.S. patent4,662,726 (5May1986).

McDermit, J. H.

Moore, J.

T. Conroy, J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999), pp. 392–397.
[CrossRef]

Nayar, S.

S. Nayar, “Catadioptric omnidirectional camera,” Comput. Vis. Pattern Recog. 2, 482–488 (1997).

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Sixth International Conference on Computer Vision (IEEE, 1998), pp. 35–42.
[CrossRef]

Ollis, M.

M. Ollis, H. Herman, S. Singh, “Analysis and design of panoramic stereo vision using equi-angular pixel cameras,” Tech. Rep. (Robotics Institute, Carnegie Mellon University, 1999).

Rees, D.

D. Rees, “Panoramic television viewing system,” U.S. patent3,505,465 (7April1970).

Richardson, T.

A. Bruckstein, T. Richardson, “Omniview cameras with curved surface mirrors,” Bell Lab Tech Memo (Bell Laboratories, 1996).

Rines, G.

G. Rines, J. Kuppenheimer, “Reflective optical element,” U.S. patent4,662,726 (5May1986).

Shealy, D. L.

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

Singh, S.

M. Ollis, H. Herman, S. Singh, “Analysis and design of panoramic stereo vision using equi-angular pixel cameras,” Tech. Rep. (Robotics Institute, Carnegie Mellon University, 1999).

Srinivasan, M. V.

Yachida, M.

K. Yamazawa, Y. Yagi, 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.

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

Yamazawa, K.

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

Appl. Opt.

Comput. Vis. Pattern Recog.

S. Nayar, “Catadioptric omnidirectional camera,” Comput. Vis. Pattern Recog. 2, 482–488 (1997).

Other

G. Rines, J. Kuppenheimer, “Reflective optical element,” U.S. patent4,662,726 (5May1986).

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

M. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, 1976).

M. Ollis, H. Herman, S. Singh, “Analysis and design of panoramic stereo vision using equi-angular pixel cameras,” Tech. Rep. (Robotics Institute, Carnegie Mellon University, 1999).

T. Conroy, J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999), pp. 392–397.
[CrossRef]

S. Baker, S. Nayar, “A theory of catadioptric image formation,” in Sixth International Conference on Computer Vision (IEEE, 1998), pp. 35–42.
[CrossRef]

C. Geyer, K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” In Proceedings of the European Conference on Computer Vision (IEEE, 2000), pp. 445–461.

D. Rees, “Panoramic television viewing system,” U.S. patent3,505,465 (7April1970).

P. Greguss, “Panoramic imaging block for three-dimensional space,” U.S. patent4,566,736 (28January1986).

Y. Yagi, 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, 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, T. Richardson, “Omniview cameras with curved surface mirrors,” Bell Lab Tech Memo (Bell Laboratories, 1996).

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Figures (15)

Fig. 1
Fig. 1

(Color online) Generic catadioptric sensor.

Fig. 2
Fig. 2

(Color online) Left, a collection of spheres uniformly placed about a catadioptric sensor. Right, the image formed when the mirror is parabolic and the (camera) projection used is orthographic. The numbers of pixels used to represent the spheres vary greatly.

Fig. 3
Fig. 3

(Color online) If a spherical mirror is observed with an orthographic projection, the projections of the spheres in the scene will now all have the same area.

Fig. 4
Fig. 4

Chahl and Srinivasan considered a mirror in which dϕ/dθ is a constant, K.

Fig. 5
Fig. 5

Conroy and Moore10 considered a mirror for which ϕ is proportional to the area of the corresponding disk in the image plane. Here f is the focal length of the camera.

Fig. 6
Fig. 6

Basic form of a catadioptric sensor whose catadioptric is well modeled by an orthographic projection.

Fig. 7
Fig. 7

Why ϕ = 2 arctan[f′(x)]: The dashed line represents the normal to the mirror’s cross section at [x, f(x)]. Thus ψ = arctan[f′(x)], so ϕ = 2 arctan[f′(x)].

Fig. 8
Fig. 8

It may be that the mirror is not smooth at its lowest point. We let ϕ0 denote the angle between the normal at the lowest point and the vertical axis.

Fig. 9
Fig. 9

(Color online) Surfaces of revolution of constant positive (left) and negative (right) Gaussian curvature. In the orthographic case these are the equiresolution mirrors.

Fig. 10
Fig. 10

Comparison of the magnification factors of three mirrors; each magnification factor has been normalized, i.e., divided by its maximum. The cubic has zero magnification at ϕ = 0 because both of its derivatives are zero there. Note that this does not mean that a portion of the scene is omitted.

Fig. 11
Fig. 11

Notation for deriving the differential equation in the perspective case.

Fig. 12
Fig. 12

Cross section of an equiresolution mirror based on dioptrics that are well modeled by perspective projection.

Fig. 13
Fig. 13

Comparison of normalized magnification factors of the three equiangular mirrors and the magnification factor of an equiresolution mirror.

Fig. 14
Fig. 14

(Color online) Comparison of four different mirror designs in the perspective case. Counterclockwise from the upper left: Chahl and Srinivasan, Conroy and Moore, Ollis et al., and Hicks and Perline.

Fig. 15
Fig. 15

(Color online) Five objects, all representing the same solid angle with respect to a sensor employing a spherical mirror, placed at ϕ = 30°, 45°, 60°, 75°, 90°.

Equations (15)

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sin [ A + θ ( 1 - K ) / 2 ] = ( B r ) ( 1 - K ) / 2 .
d r d θ = r cot { - 1 2 [ 1 + α ( θ ) ] d θ } .
m f ( x ) = lim Δ x 0 2 π [ cos ( ϕ ) - cos ( ϕ + Δ ϕ ) ] π ( x + Δ x ) 2 - π x 2 .
m f ( x ) = lim Δ x 0 2 π { cos ( ϕ ) - cos [ ϕ + ( Δ ϕ / Δ x ) Δ x ] } π ( 2 x Δ x + Δ x 2 ) = sin ( ϕ ) x d ϕ d x .
m f ( x ) = 4 f ( x ) f ( x ) x [ 1 + f ( x ) 2 ] 2
4 f ( x ) f ( x ) x [ 1 + f ( x ) 2 ] 2 = K ,
2 f ( x ) f ( x ) [ 1 + f ( x ) 2 ] 2 = K x 2 ,
C - 1 1 + f ( x ) 2 = K x 2 4 ,
2 π { 1 - cos ( ϕ ) + [ 1 - cos ( ϕ 0 ) ] } = K ˜ π x 2 ,
1 + cos ( ϕ 0 ) - 2 1 + f ( x ) 2 = K x 2 2 .
m = d y d x = d y / d θ d x / d θ = ( d r / d θ ) cos ( θ ) - r sin ( θ ) ( d r / d θ ) sin ( θ ) + r cos ( θ ) .
d r d θ = r { sin ( θ ) - cos ( θ ) tan [ ( θ - ϕ ) / 2 ] } cos ( θ ) - sin ( θ ) tan [ ( θ - ϕ ) / 2 ] .
1 - cos ( ϕ ) - [ 1 - cos ( ϕ 0 ) ] = K π f 2 tan 2 ( θ )
m ϕ ( θ ) = sin ( ϕ ) cos 3 ( θ ) 2 π f 2 sin ( θ ) d ϕ d θ .
cos ( ϕ ) - cos ( ϕ 0 ) = K π f 2 tan 2 ( θ ) .

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