Abstract

We discuss the uncertainty limit in distance sensing by laser triangulation. The uncertainty in distance measurement of laser triangulation sensors and other coherent sensors is limited by speckle noise. Speckle arises because of the coherent illumination in combination with rough surfaces. A minimum limit on the distance uncertainty is derived through speckle statistics. This uncertainty is a function of wavelength, observation aperture, and speckle contrast in the spot image. Surprisingly, it is the same distance uncertainty that we obtained from a single-photon experiment and from Heisenberg’s uncertainty principle. Experiments confirm the theory. An uncertainty principle connecting lateral resolution and distance uncertainty is introduced. Design criteria for a sensor with minimum distance uncertainty are determined: small temporal coherence, small spatial coherence, a large observation aperture.

© 1994 Optical Society of America

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  1. S. Parthasarathy, J. Birk, J. Dessimoz, “Laser rangefinder for robot control and inspection,” in Robot Vision, A. Rosenfeld, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 336, 2–10 (1982).
  2. G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
  3. J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).
  4. G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).
  5. G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [CrossRef] [PubMed]
  6. W. Dremel, G. Häusler, M. Maul, “Triangulation with large dynamical range,” in Optical Techniques for Industrial Inspection, P. G. Cielo, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 665, 182–187 (1986).
  7. G. Häusler, J. M. Herrmann, “Range sensing by shearing interferometery: influence of speckle,” Appl. Opt. 27, 4631–4637 (1988).
    [CrossRef] [PubMed]
  8. G. Häusler, J. M. Herrmann, “3-D sensing with a confocal optical ‘macroscope’,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 359 (1990).
  9. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).
  10. R. Baribeau, M. Rioux, “Influence of speckle on laser range finders,” Appl. Opt. 30, 2873–2878 (1991).
    [CrossRef] [PubMed]
  11. G. Häusler, “About fundamental limits of three-dimensional sensing or nature makes no presents,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 352–353 (1990).
  12. G. Häusler, J. M. Herrmann, “Physical limits of 3-D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, O. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1822, 150–158 (1992).
  13. K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
    [CrossRef]
  14. N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
    [CrossRef]
  15. H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
    [CrossRef]
  16. O. Falconi, “Maximum sensitivities of optical direction and twist measuring instruments,” J. Opt. Soc. Am. 54, 1315–1320 (1964).
    [CrossRef]
  17. E. Ingelstam, “An optical uncertainty principle and its application to the amount of information obtainable from multiple-beam interferences,” Ark. Fys. 7, 309–322 (1953).
  18. B. S. Thornton, “An uncertainty relation in interferometry,” Opt. Acta 4, 41–42 (1957).
    [CrossRef]

1991 (1)

1988 (2)

1986 (1)

G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).

1985 (2)

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).

1983 (1)

N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
[CrossRef]

1982 (1)

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

1975 (1)

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

1964 (1)

1957 (1)

B. S. Thornton, “An uncertainty relation in interferometry,” Opt. Acta 4, 41–42 (1957).
[CrossRef]

1953 (1)

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

Asakura, T.

N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
[CrossRef]

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

Baribeau, R.

Bickel, G.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

Birk, J.

S. Parthasarathy, J. Birk, J. Dessimoz, “Laser rangefinder for robot control and inspection,” in Robot Vision, A. Rosenfeld, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 336, 2–10 (1982).

Case, S. K.

J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).

Dessimoz, J.

S. Parthasarathy, J. Birk, J. Dessimoz, “Laser rangefinder for robot control and inspection,” in Robot Vision, A. Rosenfeld, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 336, 2–10 (1982).

Dremel, W.

W. Dremel, G. Häusler, M. Maul, “Triangulation with large dynamical range,” in Optical Techniques for Industrial Inspection, P. G. Cielo, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 665, 182–187 (1986).

Falconi, O.

Häusler, G.

G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
[CrossRef] [PubMed]

G. Häusler, J. M. Herrmann, “Range sensing by shearing interferometery: influence of speckle,” Appl. Opt. 27, 4631–4637 (1988).
[CrossRef] [PubMed]

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

W. Dremel, G. Häusler, M. Maul, “Triangulation with large dynamical range,” in Optical Techniques for Industrial Inspection, P. G. Cielo, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 665, 182–187 (1986).

G. Häusler, J. M. Herrmann, “3-D sensing with a confocal optical ‘macroscope’,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 359 (1990).

G. Häusler, “About fundamental limits of three-dimensional sensing or nature makes no presents,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 352–353 (1990).

G. Häusler, J. M. Herrmann, “Physical limits of 3-D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, O. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1822, 150–158 (1992).

Heckel, W.

Herrmann, J. M.

G. Häusler, J. M. Herrmann, “Range sensing by shearing interferometery: influence of speckle,” Appl. Opt. 27, 4631–4637 (1988).
[CrossRef] [PubMed]

G. Häusler, J. M. Herrmann, “Physical limits of 3-D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, O. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1822, 150–158 (1992).

G. Häusler, J. M. Herrmann, “3-D sensing with a confocal optical ‘macroscope’,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 359 (1990).

Ingelstam, E.

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

Jalkio, J. A.

J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).

Kim, R. C.

J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).

Litschel, R.

G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).

Maul, M.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

W. Dremel, G. Häusler, M. Maul, “Triangulation with large dynamical range,” in Optical Techniques for Industrial Inspection, P. G. Cielo, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 665, 182–187 (1986).

Nagamatsu, N.

N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
[CrossRef]

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

Nakagawa, K.

N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
[CrossRef]

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

Parthasarathy, S.

S. Parthasarathy, J. Birk, J. Dessimoz, “Laser rangefinder for robot control and inspection,” in Robot Vision, A. Rosenfeld, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 336, 2–10 (1982).

Pedersen, H. M.

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

Rioux, M.

Seitz, G.

G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).

Shindo, Y.

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

Thornton, B. S.

B. S. Thornton, “An uncertainty relation in interferometry,” Opt. Acta 4, 41–42 (1957).
[CrossRef]

Tiziani, H.

G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).

Appl. Opt. (3)

Ark. Fys (1)

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

Feinwerktechnik Messtechnik (1)

G. Seitz, H. Tiziani, R. Litschel, “3-D-Koordinatenmessung durch optische Triangulation,” Feinwerktechnik Messtechnik 94, 423–425 (1986).

J. Opt. (Paris) (1)

K. Nakagawa, N. Nagamatsu, T. Asakura, Y. Shindo, “An effect of the extended detecting aperture on the contrast of monochromatic and white-light speckles,” J. Opt. (Paris) 13, 147–153 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

B. S. Thornton, “An uncertainty relation in interferometry,” Opt. Acta 4, 41–42 (1957).
[CrossRef]

H. M. Pedersen, “On the contrast of polychromatic speckle patterns and its dependence on surface roughness,” Opt. Acta 22, 15–24 (1975).
[CrossRef]

Opt. Eng. (2)

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

J. A. Jalkio, R. C. Kim, S. K. Case, “Three-dimensional inspection using multistrip structured light,” Opt. Eng. 24, 966–974 (1985).

Opt. Quantum Electron. (1)

N. Nagamatsu, K. Nakagawa, T. Asakura, “The autocorrelation function of polychromatic laser speckle patterns near the image plane,” Opt. Quantum Electron. 15, 507–512 (1983).
[CrossRef]

Other (6)

W. Dremel, G. Häusler, M. Maul, “Triangulation with large dynamical range,” in Optical Techniques for Industrial Inspection, P. G. Cielo, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 665, 182–187 (1986).

G. Häusler, J. M. Herrmann, “3-D sensing with a confocal optical ‘macroscope’,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 359 (1990).

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975).

G. Häusler, “About fundamental limits of three-dimensional sensing or nature makes no presents,” in Optics in Complex Systems, F. Lanz, H. Preuss, G. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1319, 352–353 (1990).

G. Häusler, J. M. Herrmann, “Physical limits of 3-D sensing,” in Optics, Illumination, and Image Sensing for Machine Vision VII, O. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1822, 150–158 (1992).

S. Parthasarathy, J. Birk, J. Dessimoz, “Laser rangefinder for robot control and inspection,” in Robot Vision, A. Rosenfeld, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 336, 2–10 (1982).

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Figures (10)

Fig. 1
Fig. 1

Principle of laser triangulation.

Fig. 2
Fig. 2

Spot images S′ on the position-sensitive detector for different observation apertures: (a) 0.063, (b) 0.031, (c) 0.016, (d) 0.013. The observation aperture decreases from (a) to (d). The illumination aperture is constant, 0.013.

Fig. 3
Fig. 3

While a planar ground glass is displaced in the lateral direction, the apparent location of the spot image varies, although its true position is constant at the reticle position. The true position is concealed from the observer because of the microtopology that varies locally.

Fig. 4
Fig. 4

Local variation of the apparent position of the spot image versus the lateral shift of the object. The object position is given in units of the Airy disk radius of the observing lens.

Fig. 5
Fig. 5

Basic experiment for the localization of speckled spot images. The spot S is projected onto ground glass and imaged through the aperture sin u into the image plane. The evaluation of the spot image is performed by the CCD camera.

Fig. 6
Fig. 6

Speckled image of a projected axicon ring pattern illuminated with a semiconductor laser.

Fig. 7
Fig. 7

Reduction in speckle contrast when the laser is driven as a LED. (The current is below the threshold.)

Fig. 8
Fig. 8

Uncertainty in the spot localization versus the observation aperture for different light sources (different coherence lengths). The spot size is the same for all measurements.

Fig. 9
Fig. 9

Uncertainty in the spot localization versus the observation aperture and the spot size for different light sources: (a) He–Ne laser, (b) superluminescent diode, (c) LED, (d) Xe arc lamp. The object was ground glass with a roughness of σ z = 3 μm.

Fig. 10
Fig. 10

Experiment as in Fig. 9(a). However, the ground glass is replaced here by a Cu substrate with scratches. The preassumptions of the speckle theory no longer hold, and a strong dependence on the spot size occurs.

Tables (1)

Tables Icon

Table 1 Ratio of Experimental and Theoretical Distance Uncertainty for Fixed Spot Size (100 μm) and Different Observation Apertures (Laser Illumination)

Equations (37)

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Δ z = Δ x sin θ 1 β ,
Δ z = Δ x sin θ .
var ( x cog ) = ( x cog ) 2 - ( x cog ) 2 ,
x cog = - + - + I ( x , y ) x d y d x - + - + I ( x , y ) d y d x .
var ( x cog ) = [ I ( x 1 , y 1 ) x 1 d x 1 d y 1 I ( x 1 , y 1 ) d x 1 d y 1 ] 2 .
r 0 s 0 0.61 λ sin u .
I ( x 1 , y 1 ) d x 1 d y 1 = I spot d x 1 d y 1 .
var ( x cog ) = I ( x 1 , y 1 ) I ( x 2 , y 2 ) x 1 x 2 d x 1 d y 1 d x 2 d y 2 I 2 ( spot d x 1 d y 1 ) 2 .
I ( x 1 , y 1 ) I ( x 2 , y 2 ) = I ( x 1 , y 1 ) 2 [ 1 + μ A ( x 1 , y 1 ; x 2 , y 2 ) 2 ] .
var ( x cog ) = μ A ( Δ x , Δ y ) 2 d Δ x d Δ y × I ( x 1 , y 1 ) 2 x 1 2 d x 1 d y 1 I 2 ( spot d x 1 d y 1 ) 2 .
- + - + μ A ( Δ x , Δ y ) 2 d Δ x d Δ y = 0 2 π 0 + | 2 J 1 ( 2 π a r λ z ) 2 π a r λ z | 2 r d r d φ = 8 π ( λ z 2 π a ) 2 0 + J 1 2 ( x ) x d x = 1 π ( λ z a ) 2
0 + J 1 2 ( x ) x d x = 1 2 .
- + - + I ( x 1 , y 1 ) 2 x 1 2 d x 1 d y 1 I 2 ( spot d x 1 d y 1 ) 2 .
I 2 s p o t x 1 2 d x 1 d y 1 I 2 ( s p o t d x 1 d y 1 ) 2 = 0 2 π 0 r 0 r 3 cos 2 ( φ ) d r d φ ( 0 2 π 0 r 0 r d r d φ ) 2 = 1 4 π
0 2 π cos 2 ( φ ) d φ = π .
var ( x cog ) = 1 4 π 2 ( λ sin u ) 2 ,
δ x cog = 1 2 π λ sin u ,
C = ( SNR ) rms - 1 = ( I 2 - I 2 I ) 1 / 2 = [ C I ( 0 , 0 ) I 2 ] 1 / 2 ,
I ( x , y ) I ( x + Δ x , y + Δ y ) = C I ( Δ x , Δ y ) + I ( x , y ) 2 .
var ( x cog ) = 1 4 π 1 I 2 - + - + C I ( Δ x , Δ y ) d Δ x d Δ y .
var ( x cog ) = 1 4 π C I ( 0 , 0 ) I 2 - + - + C I x , Δ y ) C I ( 0 , 0 ) d Δ x d Δ y .
var ( x cog ) = 1 4 π C 2 - + - + C I ( Δ x , Δ y ) C I ( 0 , 0 ) d Δ x d Δ y .
- + - + C I ( Δ x , Δ y ) C I ( 0 , 0 ) d Δ x d Δ y 0 2 π 0 + | 2 J 1 ( 2 π a r λ ¯ z ) 2 π a r λ ¯ z | 2 d r d φ ,
δ x cog = C 1 2 π λ ¯ sin u .
C 2 = 1 [ 1 + ( 4 σ z / l c ) 2 ] 1 / 2 .
δ x cog = 1 n 1 2 π λ sin u ,
( SNR ) rms - 1 = 1 n .
( SNR ) rms = 1 C .
δ x cog = ( SNR ) rms - 1 1 2 π λ sin u .
δ z f = C 1 2 π λ sin u 1 sin θ .
δ x R = 0.61 λ sin u i .
C = 0.61 λ sin u i 1 Δ x .
δ z = 0.61 λ sin u i 1 Δ x 1 2 π λ sin u 1 sin θ
Δ x δ z = 0.61 λ sin u i 1 2 π λ sin u 1 sin θ .
Δ μ 0.61 λ sin u i .
C 1 N sin u sin u i .
δ z 1 2 π λ sin u i 1 sin θ .

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