Abstract

We consider the problem of designing a passenger side mirror for an automobile that does not have a blind spot and that does not distort the image. Our model consists of a coupled pair of partial differential equations that do not have a common solution. Using a best mean-square-error functional, we find approximate solutions using nonlinear optimization. In one case a local minimum provides a mirror that solves the problem, but it does not reverse the image.

© 2005 Optical Society of America

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References

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  1. U.S. Department of Transportation, “Motor vehicle crashes—data analysis and IVI program emphasis,” (Intelligent Transportation System Joint Program Office, 1999).
  2. R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. A 22, 323–330 (2005).
    [CrossRef]
  3. A. Bruckstein, T. Richardson, “Omniview cameras with curved surface mirrors,” Bell Laboratories Tech. Memo (Bell Laboratories, 1996).
  4. S. Nayar, “Catadioptric omnidirectional camera,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE Press, 1997), pp. 482–488.
    [CrossRef]
  5. K. Yamazawa, Y. Yagi, M. Yachida, “Omnidirectional imaging with hyperboidal projection,” in Proceedings of the IEEE International Conference on Robots and Systems (IEEE Press, 1993), pp. 1029–1034.
  6. V. S. Nalwa, “Panoramic projection apparatus,” U.S. patent5,539,483 (23June1996).
  7. J. S. Chahl, M. V. Srinivasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997).
    [CrossRef]
  8. T. Conroy, J. Moore, “Resolution invariant surfaces for panoramic vision systems,” in Proceedings of the 1999 IEEE Computer Society International Conference on Computer Vision (IEEE Press, 1999), pp. 392–397.
  9. R. A. Hicks, R. Bajcsy, “Reflective surfaces as computational sensors,” in Proceedings of the Second Workshop on Perception for Mobile Agents, CVPR 99 (IEEE Press, 1999), pp. 82–86.
  10. 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).
  11. R. A. Hicks, R. Perline, “Geometric distributions and catadioptric sensor design,” in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE Press, 2001), pp. 584–589.
  12. M. Srinivasan, “A new class of mirrors for wide-angle imaging,” in Proceedings of the IEEE Workshop on Omnidirectional Vision (IEEE Press, 2003).
  13. R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in IEEE International Workshop on Projector-Camera Systems (IEEE Press, 2003).
  14. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
    [CrossRef]
  15. F. John, Partial Differential Equations (Springer-Verlag, 1975).
  16. I. Gelfand, S. Fomin, Calculus of Variations (Prentice-Hall, 1963).
  17. W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, 2001).
    [CrossRef]

2005 (1)

1997 (1)

Bajcsy, R.

R. A. Hicks, R. Bajcsy, “Reflective surfaces as computational sensors,” in Proceedings of the Second Workshop on Perception for Mobile Agents, CVPR 99 (IEEE Press, 1999), pp. 82–86.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

Bruckstein, A.

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

Chahl, J. S.

Cheney, W.

W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, 2001).
[CrossRef]

Conroy, T.

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

Fomin, S.

I. Gelfand, S. Fomin, Calculus of Variations (Prentice-Hall, 1963).

Gelfand, I.

I. Gelfand, S. Fomin, Calculus of Variations (Prentice-Hall, 1963).

Grossberg, M.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in IEEE International Workshop on Projector-Camera Systems (IEEE Press, 2003).

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).

Hicks, R. A.

R. A. Hicks, “Designing a mirror to realize a given projection,” J. Opt. Soc. Am. A 22, 323–330 (2005).
[CrossRef]

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

R. A. Hicks, R. Bajcsy, “Reflective surfaces as computational sensors,” in Proceedings of the Second Workshop on Perception for Mobile Agents, CVPR 99 (IEEE Press, 1999), pp. 82–86.

John, F.

F. John, Partial Differential Equations (Springer-Verlag, 1975).

Moore, J.

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

Nalwa, V. S.

V. S. Nalwa, “Panoramic projection apparatus,” U.S. patent5,539,483 (23June1996).

Nayar, S.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in IEEE International Workshop on Projector-Camera Systems (IEEE Press, 2003).

S. Nayar, “Catadioptric omnidirectional camera,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE Press, 1997), pp. 482–488.
[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).

Perline, R.

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

Richardson, T.

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

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.

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

Srinivasan, M. V.

Swaminathan, R.

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in IEEE International Workshop on Projector-Camera Systems (IEEE Press, 2003).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

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 Press, 1993), pp. 1029–1034.

Yagi, Y.

K. Yamazawa, Y. Yagi, M. Yachida, “Omnidirectional imaging with hyperboidal projection,” in Proceedings of the IEEE International Conference on Robots and Systems (IEEE Press, 1993), pp. 1029–1034.

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 Press, 1993), pp. 1029–1034.

Appl. Opt. (1)

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

Other (15)

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

R. A. Hicks, R. Bajcsy, “Reflective surfaces as computational sensors,” in Proceedings of the Second Workshop on Perception for Mobile Agents, CVPR 99 (IEEE Press, 1999), pp. 82–86.

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).

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

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

R. Swaminathan, S. Nayar, M. Grossberg, “Framework for designing catadioptric projection and imaging systems,” in IEEE International Workshop on Projector-Camera Systems (IEEE Press, 2003).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
[CrossRef]

F. John, Partial Differential Equations (Springer-Verlag, 1975).

I. Gelfand, S. Fomin, Calculus of Variations (Prentice-Hall, 1963).

W. Cheney, Analysis for Applied Mathematics (Springer-Verlag, 2001).
[CrossRef]

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

S. Nayar, “Catadioptric omnidirectional camera,” in Proceedings of the 1997 IEEE Computer Society Conference on Computer Vision Pattern Recognition (IEEE Press, 1997), pp. 482–488.
[CrossRef]

K. Yamazawa, Y. Yagi, M. Yachida, “Omnidirectional imaging with hyperboidal projection,” in Proceedings of the IEEE International Conference on Robots and Systems (IEEE Press, 1993), pp. 1029–1034.

V. S. Nalwa, “Panoramic projection apparatus,” U.S. patent5,539,483 (23June1996).

U.S. Department of Transportation, “Motor vehicle crashes—data analysis and IVI program emphasis,” (Intelligent Transportation System Joint Program Office, 1999).

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Figures (8)

Fig. 1
Fig. 1

Diagram corresponding to the derivation of the partial differential equations for our mirror. A polygonal line L, representing a light ray, creates a correspondence between an object plane and an image plane.

Fig. 2
Fig. 2

Mirror surface yielding a 45 deg field of view when imaged orthographically.

Fig. 3
Fig. 3

Simulated view of the test wall with the mirror depicted in Fig. 2.

Fig. 4
Fig. 4

Image of a checkerboard from a mirrored sphere with a 10 deg field of view.

Fig. 5
Fig. 5

Prototype of an approximate minimizer of Eq. (21)— a side-view mirror that images a plane with minimal distortion.

Fig. 6
Fig. 6

Image of a checkerboard on a wall by use of the prototype depicted in Fig. 5.

Fig. 7
Fig. 7

Simulation of two cars, with car 2 in the blind spot of car 1. Car 1 is equipped with a flat side-view passenger mirror and the asymmetric mirror that gives a 45 deg view without distortion.

Fig. 8
Fig. 8

Driver’s view of the two mirrors. The flat mirror is the upper one, in which only the tail of the rear car is visible. Below we can see the asymmetric mirror, in which the rear car is entirely visible and undistorted. Note that the car appears on different sides of the traffic lines—the asymmetric mirror does not reverse images.

Equations (21)

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e 1 = α T + b N ,
b = e 1 , N = 1 d ,
v = a T + b N = e 1 2 a T = e 1 2 ( e 1 1 d N ) = e 1 + 2 d N = ( 1 + 2 d 2 , 2 f y d 2 , 2 f z d 2 ) .
t 2 f y d 2 + y = k ,
[ c , y , z ] [ g 1 ( y , z ) , k , g 2 ( y , z ) ] ,
g 1 ( y , z ) = ( 1 f y 2 f z 2 ) ( y + k ) 2 f y + f ( y , z ) ,
g 2 ( y , z ) = ( y + k ) f z f y + z .
( 1 f y 2 f z 2 ) ( y + k ) 2 f y + f ( y , z ) = α y ,
( y + k ) f z f y + z = α z .
1 f y 2 f z 2 2 f y = λ y ,
f z f y = λ z .
u z 2 + u y 2 = 1 + λ y 2 ,
f y = λ y ± ( λ 2 y 2 + 1 + λ 2 z 2 ) 1 / 2 1 λ 2 z 2 ,
f z = λ z y ± z ( λ 2 y 2 + 1 + λ 2 z 2 ) 1 / 2 1 λ 2 z 2 .
S ( f ) = 1 f y 2 f z 2 2 f y λ y ,
T ( f ) = f z f y λ z .
( f ) = A S ( f ) 2 + T ( f ) 2 d y d z .
h ( y , z ) = a 1 y + a 2 y 2 + a 3 z 2 + a 4 y z 2 .
Ŝ : f f y 2 + f z 2 1 2 y f y ,
T ̂ : f f z z f y .
G ( h ) = A Ŝ ( h ) 2 d y d z .

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