Abstract

A new optical profilometry—linearly coded profilometry (LCP)—is presented. It uses a sawtoothlike linear light structure to code the surface to be measured and a phase-shifting technique to decode the profile. Two kinds of coding-light structure, one with right-angle triangle teeth and the other with isosceles triangle teeth, are proposed. For coding with light with right-angle triangle teeth, a general decoding method is given. In addition, an optimum sampling manner and the measurement error are discussed with respect to a special case. For coding with light with isosceles triangle teeth, a decoding method with three samples is given. In our laboratory, an experimental system was established, and experiments that verified the reliability of the proposed methods were performed. Experimental results typical of those obtained are given. We find that LCP is similar to the widely used phase-measuring profilometry but has a faster measuring speed.

© 1997 Optical Society of America

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References

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  1. D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  2. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  3. M. Halioua, H. C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Laser Eng. 11, 185–215 (1989).
    [CrossRef]
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]

1989 (1)

M. Halioua, H. C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Laser Eng. 11, 185–215 (1989).
[CrossRef]

1983 (1)

1982 (1)

1970 (1)

Allen, J. B.

Halioua, M.

M. Halioua, H. C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Laser Eng. 11, 185–215 (1989).
[CrossRef]

Ina, H.

Johnson, W. O.

Kobayashi, S.

Liu, H. C.

M. Halioua, H. C. Liu, “Optical three-dimensional sensing by phase measuring profilometry,” Opt. Laser Eng. 11, 185–215 (1989).
[CrossRef]

Meadows, D. M.

Mutoh, K.

Takeda, M.

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Figures (9)

Fig. 1
Fig. 1

Cross-sectional diagram of the intensity distribution produced by the coding light.

Fig. 2
Fig. 2

General profile distribution of samples I1, I2, and I3, used to decode the profile.

Fig. 3
Fig. 3

Cross-sectional diagram of the isosceles triangle teeth coding-light intensity distribution.

Fig. 4
Fig. 4

Distribution of the samples from δ = 0, δ = T/2, and δ = 3T4 when 0 ≤ ϕm < T/4 and logic expression (17) holds (the first of four possible cases).

Fig. 5
Fig. 5

Distribution of the samples from δ = 0, δ = T/2, and δ = 3T4 when T/4 ≤ ϕm < T/2 and logic expression (20) holds (the second of four possible cases).

Fig. 6
Fig. 6

Distribution of the samples from δ = 0, δ = T/2, and δ = 3T4 when T/2 ≤ ϕm < 3T/4 and logic expression (23) holds (the third of four possible cases).

Fig. 7
Fig. 7

Distribution of the samples from δ = 0, δ = T/2, and δ = 3T4 when 3T/4 ≤ ϕm < T and logic expression (26) holds (fourth of the four cases).

Fig. 8
Fig. 8

Block diagram of the measurement system used in the experiments. The star represents multiplication.

Fig. 9
Fig. 9

Experimental results: (a) Original partial sphere. (b) Sphere coded with light with right-angle triangle teeth. (c) Sphere coded with light with isosceles triangle teeth. (d) Three-dimensional plot showing the measured profile.

Equations (40)

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Ix, y=a+bx/Tm,
Ix, y=Rx, ya+bϕx, y/Tm=bRx, yϕ0+ϕx, yTm=Mx, yϕ0+ϕx, yTm.
Ix, y=a+bx+δTm,
Ix, y=Mx, yϕ0+ϕx, y+δTm.
I0=Mx, yϕmx, y, I1=Mx, yϕmx, y+δ1,  IJ-1=Mx, yϕmx, y+δJ-1, IJ=Mx, yϕmx, y+δJ-T,  if IJ<I0, IJ+1=Mx, yϕmx, y+δJ+1-T,  IN-1=Mx, yϕmx, y+δN-1-T,
1JN-1,
0<δ1<δ2<<δN-1<T,
T-δJϕmT-δJ-1.
EM, ϕm=i=0J-1Ii-Mϕm+δi2+i=JN-1Ii-Mϕm+δi-T2.
c1-NMϕm-c2M+N-JTM=0,
c1ϕm+2c2+N-JTMϕm-NMϕm2+2c6T-N-JT2-c4M+c3-c5T=0,
c1=i=0N-1Ii, c2=i=0N-1δi, c3=i=0N-1Iiδi, c4=i=0N-1δi2, c5=i=JN-1Ii, c6=i=JN-1δi.
ϕm=Tc2c5-c2c3+N-JTc3+c1c4-2Tc1c6+N-JT2c1-c5Nc3-TNc5-c1c2+N-JTc1,
Ix, y=a+bxTm0xTm<T/2a+bT-xTmT/2xTm<T=bϕ0+xTm0xTmT/2bϕ0+T-xTmT/2xTm<T.
Ix, y=Rx, ybϕ0+ϕx, yTm0ϕx, yTm<T/2Rx, ybϕ0+T-ϕx, yTmT/2ϕx, yTm<T.
Ix, y=Mx, yϕ0+ϕx, y+δTm0ϕx, y+δTm<T/2Mx, yϕ0+T-ϕx, y+δTmT/2ϕx, y+δTm<T.
I1<I2I3<12I1+I2,
I1=Mϕ0+ϕm, I2=MT2+ϕ0-ϕm, I3=MT4+ϕ0-ϕm.
ϕm=T8I1+I2-2I3I2-I3, M=4TI2-I3, ϕ0=T8I1+2I3-I2I2-I3.
I2<I112I1+I2.
I1=Mϕ0+ϕm, I2=Mϕ0+T2-ϕm, I3=Mϕ0+ϕm-T4.
ϕm=T83I1-I2-2I3I1-I3, M=4TI1-I3, ϕ0=T8I2-I1+2I3I1-I3.
I2<I1I3>12I1+I2.
I1=Mϕ0+T-ϕm, I2=Mϕ0+ϕm-T2, I3=Mϕ0+ϕm-T4.
ϕm=T86I3-I1-5I2I3-I2, M=4TI3-I2, ϕ0=T8I1+3I2-2I3I3-I2.
I1<I2I3>12I1+I2.
I1=Mϕ0+T-ϕm, I2=Mϕ0+ϕm-T2, I3=Mϕ0+5T4-ϕm.
ϕm=T8I2+6I3-7I1I3-I1, M=4TI3-I1, ϕ0=T83I1+I2-2I3I3-I1.
ϕmx, y=I1x, yδ0I2x, y-I1x, yI1x, yI2x, y,0<ϕmT-δ0I1x, yT-δ0I1x, y-I2x, yI2x, y<I1x, y,T-δ0<ϕmT,
dϕmx, y=δ0I2-I12I2dI1-I1dI20<ϕmT-δ0T-δ0I2-I12I1dI2-I2dI1T-δ0<ϕmT.
dIi=0, dIidIj=σ12i=j0ij,
dϕm2=σI2M2δ022ϕm2+2ϕmδ0+δ02σI2M2T-δ022ϕm2-2ϕmT-δ0+T-δ02,
0<ϕmT-δ0, T-δ0<ϕmT,
dϕm2max=σI2M2δ02T2+T-δ020<ϕm<TσI2M2T-δ02T2+δ02T-δ0<ϕmT.
ϕmx, y=I1x, yT2I2x, y-I1x, y0<ϕmT/2I1x, yT2I1x, y-I2x, yT/2<ϕmT,
dϕm2max=5σI2M2.
dϕm2max=5M2σe2+σq2+σ02+M2σδ2.
bRx, yx0+T<Isat,
bRx, yx0+T=KrIsat.
dϕm2max=5x0+T2Kr2Isat2×σe2+σq2+σ02+Kr2Isat2x0+T2σδ2.

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