Abstract

Laser-based flying-spot scanners are strongly affected by speckle that is intrinsic to coherent illumination of diffusing targets. In such systems information is usually extracted by processing the derivative of a photodetector signal that results from collecting over the detector’s aperture the scattered light of a laser beam scanning a bar code. Because the scattered light exhibits a time-varying speckle pattern, the signal is corrupted by speckle noise. In this paper we investigate the power spectral density and total noise power of such signals. We also analyze the influence of speckle noise on edge detection and derive estimates for a signal-to-noise ratio when a laser beam scans different sequences of edges. The theory is illustrated by applying the results to Gaussian scanning beams for which we derive closed form expressions.

© 2003 Optical Society of America

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References

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  1. J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).
  2. J. W. Goodman, “Statistical properties of laser speckle patterns,” See Ref. 1 above, pp. 9–75.
  3. J. C. Dainty, “Recent developments,” see Ref. 1 above, pp. 321–337.
  4. T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 39, E. Wolf, ed. (Elsevier Science B.V., Amsterdam, 1995), pp. 185–248.
  5. E. Marom, S. Krešić-Jurić, L. Bergstein, “Analysis of speckle patterns in bar-code scanning systems,” J. Opt. Soc. Am. A 18, 888–901 (2001).
    [CrossRef]
  6. A. Felipe, J. M. Artigas, A. M. Pans, “Human contrast sensitivity in coherent Maxwellian view: effect of coherent noise and comparison with speckle,” J. Opt. Soc. Am. A 14, 972–983 (1997).
    [CrossRef]
  7. K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Optica Acta 26, 1505–1521 (1979).
    [CrossRef]
  8. K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Optica Acta 26, 1345–1349 (1979).
    [CrossRef]
  9. K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
    [CrossRef]
  10. J. Ohtsubo, “Statistical properties of differentiated partially developed speckle patterns,” J. Opt. Soc. Am. 72, 1249–1252 (1982).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

2001

1997

1982

1980

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

1979

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Optica Acta 26, 1505–1521 (1979).
[CrossRef]

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Optica Acta 26, 1345–1349 (1979).
[CrossRef]

Artigas, J. M.

Asakura, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 39, E. Wolf, ed. (Elsevier Science B.V., Amsterdam, 1995), pp. 185–248.

Bergstein, L.

Ebeling, K. J.

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Optica Acta 26, 1505–1521 (1979).
[CrossRef]

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Optica Acta 26, 1345–1349 (1979).
[CrossRef]

Felipe, A.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Krešic-Juric, S.

Marom, E.

Ohtsubo, J.

Okamoto, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 39, E. Wolf, ed. (Elsevier Science B.V., Amsterdam, 1995), pp. 185–248.

Pans, A. M.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

K. J. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

Optica Acta

K. J. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Optica Acta 26, 1505–1521 (1979).
[CrossRef]

K. J. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Optica Acta 26, 1345–1349 (1979).
[CrossRef]

Other

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).

J. W. Goodman, “Statistical properties of laser speckle patterns,” See Ref. 1 above, pp. 9–75.

J. C. Dainty, “Recent developments,” see Ref. 1 above, pp. 321–337.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, Vol. 39, E. Wolf, ed. (Elsevier Science B.V., Amsterdam, 1995), pp. 185–248.

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Figures (5)

Fig. 1
Fig. 1

Schematic view of a laser scanning system. In most applications the laser beam and photodetector move in unison with respect to a fixed scattering surface.

Fig. 2
Fig. 2

(a) UPCA bar-code symbol with message 012345678905, (b) photodetector signal generated by scanning the bar-code symbol, (c) derivative of the photodetector signal.

Fig. 3
Fig. 3

Typical speckle pattern observed in a photodetector aperture (obtained by computer simulation). The pattern is generated by a circular Gaussian beam with wavelength 670 nm and radius 0.12 mm. The separation between the scattering and photodetector planes is 5 cm.

Fig. 4
Fig. 4

Solid curve represents a noiseless photodetector signal generated by (a) one edge, (b) two edges, (c) square wave pattern. The edges lie on the boundary between regions with reflectances ρ1 and ρ2. The dotted curve represents the signal derivative.

Fig. 5
Fig. 5

Ratio of SNR for the signal derivative and for the raw signal (SNR′/SNR) in the case of (a) two edges, (b) square wave pattern. ω x is the beam size in the scanning direction, D is the separation between edges.

Equations (122)

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pİ=12İİ2exp-2|İ|İİ2.
it=  Ax, yIdx, y; tdxdy,
i=SId,
i=ρΩdP,
Riτ=itit+τ=i21+1NS2×RAΔx, Δy|μΔx, Δy; τ|2dΔxdΔy.
RAΔx, Δy= Ax, yAx+Δx, y+Δydxdy
μΔx, Δy; τ= Uξ, ηU*ξ-Vτ, ηexpi 2πλzξΔx+ηΔydξdη |Uξ, η|2dξdη,
maxτ|μΔx, Δy; τ|0 when |Δx|, |Δy|>,
Riτ=i21+1NSm |μΔx, Δy; τ|2dΔxdΔy,
Sm=S2RA0, 0= Ax, ydxdy2 A2x, ydxdy.
Sc= |μΔx, Δy; 0|2dΔxdΔy=λz2 I2ξ, ηdξdη Iξ, ηdξdη2,
Sif= Riτexpi2πfτdτ.
Sif=i2δf+ScNSm× Iξ, ηIξ-Vτ, ηexpi2πfτdξdηdτ I2ξ, ηdξdη.
Sif=i2δf+VScNSm× IVτ, ηexpi2πfτdτ2dη I2ξ, ηdξdη.
Pi= Sifdf=Ri0.
Pi=i21+ScNSm.
σi2=Pi-i2=i2ScNSm.
Sif=4π2f2Sif.
Sif=4π2i2ScNSm× Iξ, ηIξ-Vτ, ηf2 expi2πfτdξdηdτ I2ξ, ηdξdη.
Sif=4π2i2VScNSm× IVτ, ηf expi2πfτdτ2dη I2ξ, ηdξdη.
Pi=4π2i2ScNSm× Iξ, ηIξ-Vτ, ηf2 expi2πfτdξdηdτdf I2ξ, ηdξdη.
Hξ,ητ=Iξ-Vτ, η
 Iξ-Vτ, ηf2 expi2πfτdτdf= f2Hξ,ητexpi2πfτdτdf= f2Ĥξ,ηfdf.
2Hξ,ητ2=-4π2 f2Ĥξ,ηfexp-i2πfτdf
 Iξ-Vτ, ηf2 expi2πfτdτdf=-14π22Hξ,ητ2τ=0.
2Hξ,ητ2τ=0=V22ξ2 Iξ, η.
 Iξ-Vτ, ηf2 expi2πfτdτdf=-V24π22ξ2 Iξ, η.
 Iξ, ηIξ-Vτf2 expi2πfτdξdηdτdf=-V24π2 Iξ, η2Iξ2dξdη.
ξI Iξ=Iξ2+I 2Iξ2,
Iξ2dξ=- I 2Iξ2dξ
lim|ξ| Iξ, ηξ Iξ, η=0 for all η.
Pi=i2V2ScNSmIξ2dξdη I2ξ, ηdξdη.
Uξ, η=exp-ξ2ωx2-η2ωy2.
|μΔx, Δy; τ|2=exp-Vτωx2exp-πλz2×ωxΔx2+ωyΔy2,
Sc= |μΔx, Δy; 0|2dΔxdΔy=λz2πωxωy.
Riτ=i21+ScNSmexp-Vτωx2.
σi2=1π i2λz2ωxωyNSm.
Sif=i2δf+πScNSmωxVexp-πωxV2f2.
Sif=4π2f2Sif=4π5/2i2ScNSmωxV f2 exp-πωxV2f2.
Pi=2π i2V2λz2ωx3ωyNSm=2Vωx2σi2,
it=  Ax, yIdx, y; tdxdy,
it=Ωd BVt-ξIξ, ηdξdη,
it=VΩd BVt-ξIξ, ηdξdη.
S=12maxtit-mintit.
S=maxt|it|.
SNR=Sσi2, SNR=Sσi2,
σi2=i2λz2NSm I2ξ, ηdξdη Iξ, ηdξdη2,
σi2=i2V2λz2NSmIξ2dξdη Iξ, ηdξdη2.
Bξ=ρ1if ξ<0,ρ2if ξ0.
S1E=12ρ2-ρ1ΩdP.
SNR1E=14ρ2-ρ1ρ22NSmλz2 Iξ, ηdξdη2 I2ξ, ηdξdη.
it=ρ2-ρ1VΩd IVt, ηdη.
LSFξ= Iξ, ηdη.
S1E=ρ2-ρ1VΩd I0, ηdη.
SNR1E=ρ2-ρ1ρ22NSmλz2 I0, ηdη2Iξ2dξdη.
Bξ=ρ2if 0<ξ<D,ρ1otherwise.
maxtit=Ωd BξIξ-D2, ηdξdη
=Ωdρ1 Iξ, ηdξ+ρ2-ρ1D/2D/2 Iξ, ηdξdη.
mintit=ρ1Ωd Iξ, ηdξdη,
S2E=12ρ2-ρ1Ωd-D/2D/2 Iξ, ηdξdη.
SNR2E=14ρ2-ρ1ρ22NSmλz2-D/2D/2 Iξ, ηdξdη2 I2ξ, ηdξdη.
it=ρ2-ρ1VΩdIVt, η-IVt-D, ηdη.
S2E=ρ2-ρ1VΩdI0, η-ID, ηdη.
SNR2E=ρ2-ρ1ρ22NSmλz2I0, η-ID, ηdη2Iξ2dξdη.
maxtit=Ωd BξIξ, ηdξdη.
Bξ=12ρ1+ρ2+1πρ2-ρ1×n=--1n2n+1expi2π 2n+12D ξ.
maxtit=Ωd12ρ1+ρ2P+1πρ2-ρ1×n=--1n2n+1 Iξ, η×expiπ 2n+1D ξdξdη.
maxtit=Ωd12ρ1+ρ2P+2πρ2-ρ1×n=0-1n2n+1 Î2n+12D, 0.
mintit=Ωd12ρ1+ρ2P-2πρ2-ρ1×n=0-1n2n+1 Î2n+12D, 0.
SSW=2πρ2-ρ1Ωdn=0-1n2n+1 Î2n+12D, 0.
SNRSW=16π2ρ2-ρ1ρ2+ρ12NSmλz2×n=0-1n2n+1 Î2n+12D, 02 I2ξ, ηdξdη.
SSW=2ρ2-ρ1ΩdVDn=0 Î2n+12D, 0.
SNRSW=16ρ2-ρ1ρ2+ρ12NSmλz21D2×n=0 Î2n+12D, 02Iξ2dξdη.
Bξ=12ρ1+ρ2+12ρ2-ρ1cosπD ξ.
maxtict=12 Ωdρ1+ρ2+ρ2-ρ1Î12D, 0,
mintict=12 Ωdρ1+ρ2-ρ2-ρ1Î12D, 0.
SSIN=12ρ2-ρ1ΩdÎ12D, 0,
SNRSIN=ρ2-ρ1ρ2+ρ12NSmλz2Î12D, 02 I2ξ, ηdξdη.
SSIN=π2ρ2-ρ1ΩdVD12D, 0,
SNRSIN=π2ρ2-ρ1ρ2+ρ12NSmλz21D2Î12D, 02Iξ2dξdη.
Iξ, η=exp-2ξ2ωx2+η2ωy2.
 Inξ, ηdξdη=π2n ωxωy, n=1, 2, 3,
SNR1E=π4ρ2-ρ1ρ22NSmλz2 ωxωy.
 I0, ηdη=π2 ωy and Iξ2dξdη=π2ωyωx
SNR1E=ρ2-ρ1ρ22NSmλz2 ωxωy.
SNR1ESNR1E=4π.
-D/2D/2 Iξ, ηdξdη=π2 ωxωy erf12Dωx,
erfx=2π0xexp-t2dt.
SNR2E=π4ρ2-ρ1ρ22NSmλz2 ωxωy erf212Dωx.
I0, ηdη-ID, ηdη=π2 ωy1-exp-2Dωx2.
SNR2E=ρ2-ρ1ρ22NSmλz2 ωxωy×1-exp-2Dωx22.
SNR2ESNR2E=4π1-exp-2ωx/D2erf12ωx/D2.
Îfx, fy=π2 ωxωy exp-π22ωx2fx2+ωy2fy2.
SNRSW=16πρ2-ρ1ρ2+ρ12NSmλz2 ωxωy×n=0-1n2n+1exp-π28ωxD22n+122.
SNRSW=8π ρ2-ρ1ρ2-ρ12NSmλz2 ωxωyωxD2×n=0exp-π28ωxD22n+122.
Q=exp-π28ωxD2,
SNRSWSNRSW=π22ωxD2n=0 Q2n+12n=0-1n2n+1 Q2n+122.
QQ9110wheneverωxD1πln10=0.48.
SNRSWSNRSW=π22ωxD21+Q81-13 Q82, ωxD0.48,
SNRSIN=πρ2-ρ1ρ2+ρ12NSmλz2 ωxωy×exp-π24ωxD2,
SNRSIN=π32ρ2-ρ1ρ2+ρ12NSmλz2ωxD2ωxωy×exp-π24ωxD2.
SNRSINSNRSIN=π22ωxD2,
ωxD2π=0.45,
it=  Ax, yIdx, y; tdxdy.
Udx, y; t= bξUξ-Vt, ηexpiϕξ, η×hx-ξ, y-ηdξdη,
hx, y=expikziλzexpi k2zx2+y2,
Idx, y; t= bξb*ξ¯Uξ-Vt, ηU*ξ¯-Vt, η¯hx-ξ, y-ηh*x-ξ¯, y-η¯expiϕξ, η×exp-iϕξ¯, η¯dξdηdξ¯dη¯.
expiϕξ, ηexp-iϕξ¯, η¯=Kδξ-ξ¯δη-η¯,
Idx, y; t=K  |bξ|2|Uξ-Vt, η|2×|hx-ξ, y-η|2dξdη =Kλz2 BξIξ-Vt, ηdξdη,
Idx, y; t=Kλz2 ρP,
it=  Ax, yIdx, y; tdxdy=S Kλz2 BξIξ-Vt, ηdξdη
it=S Kλz2 ρP.
it=ρΩdP,
it=Ωd BξIξ-Vt, ηdξdη.
it=VΩd BVt-ξIξ, ηdξdη,
Bξ=ρ2-ρ1n=-δξ-2n-12D-δξ-2n+12 D.
Bξ=i ρ2-ρ1Dn=--1n expi2π 2n+12D ξ,
it=iρ2-ρ1ΩdVDn=--1n×expiπ2n+1VtD× Iξ, ηexp-i2π 2n+12D ξdξdη.
it=2ρ2-ρ1ΩdVDn=0-1n+1Î2n+12D, 0×sinπ2n+1VtD.
maxtit=2ρ2-ρ1ΩdVDn=0 Î2n+12D, 0.
mintit=-2ρ2-ρ1ΩdVDn=0 Î2n+12D, 0.
SSW=maxt|it|=2ρ2-ρ1Ωd×VDn=0 Î2n+12D, 0.

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