Abstract

Computer image processing techniques for the measurement of vibration amplitude are presented, which utilize phase-shifted time-average holographic interferograms. The calculation of the square root and division with phase-shifted interferograms gives a high contrast fringe pattern which contours the vibration amplitude. The arctangent calculation with phase-shifted fringe patterns also gives the phase distribution proportional to the optical path difference of the measured object before and after vibration. The detection of the phase discontinuity of ±π in the phase image gives exactly the center line of the dark fringe. The distribution of the vibration amplitude can be obtained by the fringe-order determination and its interpolation. The bias deformation of the vibration is obtained by the correction of phase discontinuities of ±2π and ±π in the phase images before and after vibration.

© 1986 Optical Society of America

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References

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  1. K. A. Stetson, “Holographic Vibration Analysis,” in Holographic Nondestructive Testing, R. K. Erf, Ed. (Academic, New York, 1974), pp. 181–220.
    [Crossref]
  2. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  3. K. A. Stetson, R. L. Powell, “Interferometric Hologram Evaluation and Real-Time Vibration Analysis of Diffuse Objects,” J. Opt. Soc. Am. 55, 1694 (1965).
    [Crossref]
  4. K. Biedermann, N.-E. Molin, “Combining Hypersensitization and Rapid in situ Processing for Time-Average Observation in Real-Time Hologram Interferometry,” J. Phys. E 3, 669 (1970).
    [Crossref]
  5. R. Dandliker, “Heterodyne Holographic Interferometry,” Prog. Opt. 17, 3 (1980).
  6. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409–437.
  7. G. E. Sommargren, “Double-Exposure Holographic Interferometry Using Commonpath Reference Waves,” Appl. Opt. 16, 1736 (1977).
    [Crossref] [PubMed]
  8. P. Hariharan, B. F. Oreb, N. Brown, “Real-Time Holographic Interferometry: A Microcomputer System for the Measurement of Vector Displacements,” Appl. Opt. 22, 876 (1983).
    [Crossref] [PubMed]
  9. S. Nakadate, H. Saito, “Fringe Scanning Speckle-Pattern Interferometry,” Appl. Opt. 24, 2172 (1985).
    [Crossref] [PubMed]
  10. 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 (1982).
    [Crossref]
  11. S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
    [Crossref]

1985 (1)

1984 (1)

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[Crossref]

1983 (1)

1982 (1)

1980 (1)

R. Dandliker, “Heterodyne Holographic Interferometry,” Prog. Opt. 17, 3 (1980).

1977 (1)

1970 (1)

K. Biedermann, N.-E. Molin, “Combining Hypersensitization and Rapid in situ Processing for Time-Average Observation in Real-Time Hologram Interferometry,” J. Phys. E 3, 669 (1970).
[Crossref]

1965 (1)

Biedermann, K.

K. Biedermann, N.-E. Molin, “Combining Hypersensitization and Rapid in situ Processing for Time-Average Observation in Real-Time Hologram Interferometry,” J. Phys. E 3, 669 (1970).
[Crossref]

Brown, N.

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409–437.

Dandliker, R.

R. Dandliker, “Heterodyne Holographic Interferometry,” Prog. Opt. 17, 3 (1980).

Hariharan, P.

Ina, H.

Kobayashi, S.

Molin, N.-E.

K. Biedermann, N.-E. Molin, “Combining Hypersensitization and Rapid in situ Processing for Time-Average Observation in Real-Time Hologram Interferometry,” J. Phys. E 3, 669 (1970).
[Crossref]

Nakadate, S.

Oreb, B. F.

Powell, R. L.

Saito, H.

Sommargren, G. E.

Stetson, K. A.

K. A. Stetson, R. L. Powell, “Interferometric Hologram Evaluation and Real-Time Vibration Analysis of Diffuse Objects,” J. Opt. Soc. Am. 55, 1694 (1965).
[Crossref]

K. A. Stetson, “Holographic Vibration Analysis,” in Holographic Nondestructive Testing, R. K. Erf, Ed. (Academic, New York, 1974), pp. 181–220.
[Crossref]

Takeda, M.

Tominaga, M.

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[Crossref]

Toyooka, S.

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[Crossref]

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Phys. E (1)

K. Biedermann, N.-E. Molin, “Combining Hypersensitization and Rapid in situ Processing for Time-Average Observation in Real-Time Hologram Interferometry,” J. Phys. E 3, 669 (1970).
[Crossref]

Opt. Commun. (1)

S. Toyooka, M. Tominaga, “Spatial Fringe Scanning for Optical Phase Measurement,” Opt. Commun. 51, 68 (1984).
[Crossref]

Prog. Opt. (1)

R. Dandliker, “Heterodyne Holographic Interferometry,” Prog. Opt. 17, 3 (1980).

Other (3)

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), pp. 409–437.

K. A. Stetson, “Holographic Vibration Analysis,” in Holographic Nondestructive Testing, R. K. Erf, Ed. (Academic, New York, 1974), pp. 181–220.
[Crossref]

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

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Figures (14)

Fig. 1
Fig. 1

Fringe functions calculated from Eq. (7); line a, |J0(Ka)|. Fringe function in time-average real-time holographic interferometry; line b, [1 + J0(Ka)]/2, where the fringe contrast, β/α, is unity.

Fig. 2
Fig. 2

Fringe functions plotted for (a) χ ≦ 1 and (b) χ > 1, where χ represents the intensity ratio of two reference beams. Fringe function in time-average holographic interferometry, J 0 2 ( K a ), is plotted by line f.

Fig. 3
Fig. 3

Schematic diagram of phase-shifting real-time holographic interferometry.

Fig. 4
Fig. 4

(a) Original holographic interferogram at a frequency of 650 Hz and (b) normalized fringe pattern calculated from Eq. (7). Fringes contour the vibration amplitude. (c) Phase distribution calculated from Eq. (8), where the phases ranging from 0 to 2π are displayed as 0–255 gray levels. The phase discontinuity of ±π occurs at the center lines of dark fringes.

Fig. 5
Fig. 5

Normalized fringe patterns calculated from Eq. (7) for large vibration amplitudes. The numbers of the observed dark fringes are (a) 8, (b) 14, and (c) 16.

Fig. 6
Fig. 6

(a)–(c) Phase distributions calculated from Eq. (8) which correspond to Figs. 5(a)–(c), respectively.

Fig. 7
Fig. 7

Results from (a) detection of the phase discontinuity of ±π, (b) expansion with a 3 × 3 sample area, (c) skeletonizing of the binary (b), and (d) determination of fringe-order numbers for the image phase image shown in Fig. 6(a).

Fig. 8
Fig. 8

Schematic illustration of a 3 × 3 sample area for the detection of a phase discontinuity in a phase image.

Fig. 9
Fig. 9

Vibration amplitude distribution interpolated with a third-order spline function: (a) contour and (b) perspective representations. The interval between contours is 0.1 μm.

Fig. 10
Fig. 10

(a) Phase distribution in the 250th horizontal line from the top in the phase image shown in Fig. 6(a). (b) Result from the correction of phase discontinuities of ±π and ±2π. Lines a and b represent the phase distributions before and after vibration, respectively.

Fig. 11
Fig. 11

Schematic diagram of phase-shifting double-exposure holographic interferometry.

Fig. 12
Fig. 12

(a)–(d) Four phase-shifted interferograms whose phases are 0, π/2, π, and 3π/2, respectively; vibration frequency 1.25 kHz.

Fig. 13
Fig. 13

(a) Normalized fringe and (b) phase images calculated from Eqs. (13) and (8), respectively.

Fig. 14
Fig. 14

Phase distribution proportional to the bias deformation of the vibration in the center horizontal line of the object surface shown in Fig. 13(b). Phase discontinuities of ±π and ±2π are corrected.

Equations (13)

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I f = O 2 [ α + β cos ( ξ + ψ + φ ) ] ,
ξ = 2 π λ ( 1 + cos θ ) a sin ω t + ζ ,
I v = 0 T I f d t = O 2 T [ α + β cos ( ζ + ψ + φ ) J 0 ( K a ) ] ,
a 0 = 1 N i = 1 N I i = O 2 T α ,
a 1 = 2 N i = 1 N I i cos φ i = O 2 T β cos ( ζ + ψ ) J 0 ( K a ) ,
b 1 = 2 N i = 1 N I i sin φ i = O 2 T β sin ( ζ + ψ ) J 0 ( K a ) ,
a 1 2 + b 1 2 a 0 = β α J 0 ( K a ) .
ζ + ψ = tan - 1 ( b 1 a 1 ) .
I = C [ 1 + χ 2 J 0 2 ( K a ) + 2 χ cos ( ζ + φ ) J 0 ( K a ) ] ,
a 0 = C [ 1 + χ 2 J 0 2 ( K a ) ] ,
a 1 = 2 C χ cos ζ J 0 ( K a ) ,
b 1 = 2 C χ sin ζ J 0 ( K a ) .
f ( K a ) = a 1 2 + b 1 2 a 0 = 2 χ J 0 ( K a ) 1 + χ 2 J 0 2 ( K a ) .

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