Abstract

Most of the known optical range sensors require a large amount of two-dimensional raw data from which the three-dimensional (3D) data are decoded and so are associated with considerable cost. The cost arises from expensive hardware as well as from the time necessary to acquire the images. We will address the question of how one can acquire maximum shape information with a minimum amount of image raw data, in terms of information theory. It is shown that one can greatly reduce the amount of raw data needed by proper optical redundancy reduction. Through these considerations, a 3D sensor is introduced, which needs only a single color (red-green-blue) raw image and still delivers data with only approximately 2-µm longitudinal measurement uncertainty.

© 2003 Optical Society of America

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References

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  1. E. Ingelstam, “An optical uncertainty principle and its application to the amount of information obtainable from multiple-beam interference,” Ark. Fys. 7, 309–322 (1953).
  2. R. Dorsch, G. Häusler, J. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
    [CrossRef] [PubMed]
  3. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  4. G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–421 (1997).
    [CrossRef]
  5. G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
    [CrossRef]
  6. X. Laboureux, G. Häusler, “Localization and registration of three-dimensional objects in space—where are the limits?” Appl. Opt. 28, 5206–5216 (2001).
    [CrossRef]
  7. L. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1993), 145–201.
    [CrossRef]
  8. V. Srinivasan, H. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef] [PubMed]
  9. B. L. Ulich, “Method and apparatus for three dimensional range resolving imaging,” U.S. patent5,249,046 (28September1993).
  10. C. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).
  11. R. Fano, Transmission of Information (MIT, Cambridge, Mass., 1961).
  12. J. Wozencraft, I. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).
  13. R. Gallager, Information Theory and Reliable Communications (Wiley, New York, 1968).
  14. A. Viterbi, J. Omura, Principles of Digital Communication and Coding (McGraw Hill, New York, 1979).
  15. R. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).
  16. T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
    [CrossRef]
  17. R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992).
  18. M. Neifeld, “Information, resolution, and space-bandwidth product,” Opt. Lett. 23, 1477–1479 (1998).
    [CrossRef]
  19. R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967).
  20. B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001).
    [CrossRef]
  21. R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), p. 11.
  22. R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), pp. 235–238.
  23. G. Häusler, J. Hermann, “Physical limits of 3D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, D. Svetkoff, ed., Proc. SPIE1822, 150–158 (1992).
    [CrossRef]
  24. B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001), Chap. 8.
    [CrossRef]
  25. R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 160–161.
  26. R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 166–167.
  27. B. Horn, M. Brooks, eds., Shape from Shading (MIT, Cambridge, Mass., 1989).
  28. R. Woodham, “Photometric method for determining surface orientation from multiple images,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 513–532.
  29. G. Häusler, D. Ritter, “Parallel three-dimensional sensing by color-coded triangulation,” Appl. Opt. 32, 7164–7169 (1993).
    [CrossRef] [PubMed]
  30. B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
    [CrossRef]
  31. A. Blake, A. Zisserman, G. Knowles, “Surface description from stereo and shading,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 29–52.

2001 (1)

1998 (1)

1997 (1)

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–421 (1997).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1984 (1)

1953 (1)

E. Ingelstam, “An optical uncertainty principle and its application to the amount of information obtainable from multiple-beam interference,” Ark. Fys. 7, 309–322 (1953).

Blahut, R.

R. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).

Blake, A.

A. Blake, A. Zisserman, G. Knowles, “Surface description from stereo and shading,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 29–52.

Bohn, G.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Cover, T.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
[CrossRef]

DeWeert, M. J.

B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
[CrossRef]

Dorsch, R.

Dresel, T.

Ettl, P.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Fano, R.

R. Fano, Transmission of Information (MIT, Cambridge, Mass., 1961).

Frieden, B.

B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001).
[CrossRef]

B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001), Chap. 8.
[CrossRef]

Gallager, R.

R. Gallager, Information Theory and Reliable Communications (Wiley, New York, 1968).

Halioua, M.

Häusler, G.

X. Laboureux, G. Häusler, “Localization and registration of three-dimensional objects in space—where are the limits?” Appl. Opt. 28, 5206–5216 (2001).
[CrossRef]

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–421 (1997).
[CrossRef]

R. Dorsch, G. Häusler, J. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
[CrossRef] [PubMed]

G. Häusler, D. Ritter, “Parallel three-dimensional sensing by color-coded triangulation,” Appl. Opt. 32, 7164–7169 (1993).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

G. Häusler, J. Hermann, “Physical limits of 3D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, D. Svetkoff, ed., Proc. SPIE1822, 150–158 (1992).
[CrossRef]

Hermann, J.

G. Häusler, J. Hermann, “Physical limits of 3D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, D. Svetkoff, ed., Proc. SPIE1822, 150–158 (1992).
[CrossRef]

Herrmann, J.

Ingelstam, E.

E. Ingelstam, “An optical uncertainty principle and its application to the amount of information obtainable from multiple-beam interference,” Ark. Fys. 7, 309–322 (1953).

Jacobs, I.

J. Wozencraft, I. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

Johannesson, R.

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), p. 11.

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), pp. 235–238.

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992).

Knowles, G.

A. Blake, A. Zisserman, G. Knowles, “Surface description from stereo and shading,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 29–52.

Laboureux, X.

Lacovara, P.

B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
[CrossRef]

Laszlo, I.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Leuchs, G.

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–421 (1997).
[CrossRef]

Liu, H.

Moran, S. E.

B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
[CrossRef]

Neifeld, M.

Omura, J.

A. Viterbi, J. Omura, Principles of Digital Communication and Coding (McGraw Hill, New York, 1979).

Ritter, D.

Röhler, R.

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 160–161.

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 166–167.

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967).

Schenk, M.

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

Shannon, C.

C. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

Srinivasan, V.

Thomas, J.

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
[CrossRef]

Ulich, B. L.

B. L. Ulich, “Method and apparatus for three dimensional range resolving imaging,” U.S. patent5,249,046 (28September1993).

B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
[CrossRef]

Venzke, H.

Viterbi, A.

A. Viterbi, J. Omura, Principles of Digital Communication and Coding (McGraw Hill, New York, 1979).

Weaver, W.

C. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

Woodham, R.

R. Woodham, “Photometric method for determining surface orientation from multiple images,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 513–532.

Wozencraft, J.

J. Wozencraft, I. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

Yaroslavsky, L.

L. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1993), 145–201.
[CrossRef]

Zisserman, A.

A. Blake, A. Zisserman, G. Knowles, “Surface description from stereo and shading,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 29–52.

Appl. Opt. (5)

Ark. Fys. (1)

E. Ingelstam, “An optical uncertainty principle and its application to the amount of information obtainable from multiple-beam interference,” Ark. Fys. 7, 309–322 (1953).

Opt. Lett. (1)

Phys. Bl. (1)

G. Häusler, G. Leuchs, “Physikalische Grenzen der optischen Formerfassung mit Licht,” Phys. Bl. 53, 417–421 (1997).
[CrossRef]

Other (23)

G. Häusler, P. Ettl, M. Schenk, G. Bohn, I. Laszlo, “Limits of optical range sensors and how to exploit them,” in Trends in Optics and Phototonics, Ico IV, T. Asakura, ed., Vol. 74 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1999), pp. 328–342.
[CrossRef]

B. L. Ulich, P. Lacovara, S. E. Moran, M. J. DeWeert, “Recent results in imaging lidar,” in Advances in Laser Remote Sensing for Terrestrial and Oceanographic Applications, R. M. Narayanan, J. E. Kalshoven, eds., Proc. SPIE3059, 95–108 (1997).
[CrossRef]

A. Blake, A. Zisserman, G. Knowles, “Surface description from stereo and shading,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 29–52.

L. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics XXXII, E. Wolf, ed. (Elsevier Science, Amsterdam, 1993), 145–201.
[CrossRef]

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967).

B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001).
[CrossRef]

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), p. 11.

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992), pp. 235–238.

G. Häusler, J. Hermann, “Physical limits of 3D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, D. Svetkoff, ed., Proc. SPIE1822, 150–158 (1992).
[CrossRef]

B. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 2001), Chap. 8.
[CrossRef]

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 160–161.

R. Röhler, Informationstheorie in der Optik (Wissenschaftliche Verlagsgesellschaft, Stuttgart, 1967), pp. 166–167.

B. Horn, M. Brooks, eds., Shape from Shading (MIT, Cambridge, Mass., 1989).

R. Woodham, “Photometric method for determining surface orientation from multiple images,” in Shape from Shading, B. Horn, M. Brooks, eds. (MIT, Cambridge, Mass., 1989), pp. 513–532.

B. L. Ulich, “Method and apparatus for three dimensional range resolving imaging,” U.S. patent5,249,046 (28September1993).

C. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois, Urbana, Ill., 1949).

R. Fano, Transmission of Information (MIT, Cambridge, Mass., 1961).

J. Wozencraft, I. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965).

R. Gallager, Information Theory and Reliable Communications (Wiley, New York, 1968).

A. Viterbi, J. Omura, Principles of Digital Communication and Coding (McGraw Hill, New York, 1979).

R. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).

T. Cover, J. Thomas, Elements of Information Theory (Wiley, New York, 1991).
[CrossRef]

R. Johannesson, Informationstheorie—Grundlagen der (Tele-)Kommunikation (Addison-Wesley Studentlitteratur, Lund, Sweden, 1992).

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Figures (7)

Fig. 1
Fig. 1

Block diagram of (a) an information system and (b) a 3D metrology system.

Fig. 2
Fig. 2

Channel structure.

Fig. 3
Fig. 3

Optical system. For large object distances the aperture ratio is approximately D/ f.

Fig. 4
Fig. 4

Image of a coin illuminated with (a) a laser and (b) incoherent illumination.

Fig. 5
Fig. 5

(a) Typical power spectrum Φ zz of an object shape and (b) typical power spectrum Φ zz of a shape derivative.

Fig. 6
Fig. 6

Intensity-encoded images of (a) the shape z(x, y) and (b) the shape derivative ∂z/∂x. Spectra of (c) the object height z(x, y) and (d) the height derivative ∂z/∂x.

Fig. 7
Fig. 7

(a) Rendered 3D data and (b) a sample profile of successive exposures. (c) Rendered 3D data and (d) a sample profile from a single-shot sensor.

Equations (21)

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

CT=B log21+SinputN,
CT=B log2N+SinputN=B log2SoutputN,
Soutput=N+Sinput.
C=ΔTB log2S/N.
Copt=2ΔXΔYBxBy log2SoptNopt,
BxBy=πD216λ2f2.
SoptNopt=I2¯σI2Ī2σI2=1cspeckle2,
Copt=-4ΔXΔYBxBy log2cspeckle.
Λ=APALf2,
Copt=-1λ2 Λ log2cspeckle.
Cel=2ΔXΔYBx,elBy,el log2SelNel.
Bx,el=Mx2ΔX,
By,el=My2ΔY,
Cel=12 MxMy log2SelNel.
SelNel=n2¯σn2n¯2σn2=n¯.
Cel=12 MxMy log2n¯.
Cdis=MxMy log2m.
Φzzfx, fy=const,
φzzx, y=const δx, y.
xz+n=zx+nx.
zx+n.

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