Abstract

A grid is projected onto an object and focused into a viewing camera which is offset from the angle of illumination. When the image is superimposed by double exposure or by filtering with the image of another object, contour-difference moiré fringes appear. By the use of two offset projectors, measurements of contour sums or contour doubling may be accomplished. The properties, limitations, sensitivity, and procedures are analyzed and discussed. The method lends itself to image enhancement and is extended to the study of vibration analysis in real time. In the latter application, time-average moiré fringes appear according to the zeroth-order Bessel function, which contains the modal amplitude function in its argument.

© 1971 Optical Society of America

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References

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  1. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  2. D. M. Meadows, W. O. Johnson, J. B. Allen, Appl. Opt. 9, 942 (1970).
    [CrossRef] [PubMed]
  3. P. S. Theocaris, Appl. Opt. 10, 1172 (1971).
    [CrossRef] [PubMed]
  4. R. Weller, B. M. Shepard, Proc. Soc. Expt. Stress Anal. 6, 35 (1948).
  5. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, New York, 1969).
  6. R. E. Brooks, L. O. Heflinger, Appl. Opt. 8, 935 (1969).
    [CrossRef] [PubMed]
  7. R. L. Powell, K. A. Stetson, J. Opt. Soc. Am. 55, 1593 (1965).
    [CrossRef]

1971

1970

1969

1965

1948

R. Weller, B. M. Shepard, Proc. Soc. Expt. Stress Anal. 6, 35 (1948).

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Figures (8)

Fig. 1
Fig. 1

Basic elements of contour-difference method. Note that S in figure should read S1.

Fig. 2
Fig. 2

Surface differences between S1 and S2.

Fig. 3
Fig. 3

Moiré phenomenon due to phase shift in carrier fringes.

Fig. 4
Fig. 4

Contour difference between two automobile wheel covers, one perfect and another possessing a small dent.

Fig. 5
Fig. 5

Real-time contour-difference scheme by superposition by the use of a beam splitter.

Fig. 6
Fig. 6

Real-time contour setup by use of two projectors to find absolute, doubled contours or contour sums.

Fig. 7
Fig. 7

Time-average moiré fringes depict constant amplitude of vibration in rubber membrane excited at 100 Hz.

Fig. 8
Fig. 8

Same as Fig. 7 except amplitude of excitation was greater, yielding a greater number of fringes but of same form as Fig. 7.

Equations (16)

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I 1 ( X , Y ) = K { 1 + sin [ ( 2 π / p ) ( X - Z 1 tan α ) ] } ,
I ( X , Y ) = I 1 + I 2 = 2 K ( 1 + { sin ( π / p ) [ 2 X - ( Z 1 + Z 2 ) tan α ] } · cos [ ( π / p ) × ( Z 1 - Z 2 ) tan α ] ) .
cos [ ( π / p ) ( Z 1 - Z 2 ) tan α ] = 0 ,
Z 1 - Z 2 = n p 0 / sin α ,
I ave = 1 Δ A Δ A I ( X , Y ) d A .
I f ( X , Y ) = T 1 T 2 = K K I 1 I 2 ,
I f ( X , Y ) = K K ( sin { ( π / p ) [ 2 X - ( Z 1 + Z 2 ) tan α ] } + cos [ ( π / p ) ( Z 1 - Z 2 ) tan α ] ) 2 .
I ave = 1 Δ A Δ A I f ( X , Y ) d A = K K { ½ + cos 2 [ ( π / p ) ( Z 1 - Z 2 ) tan α ] } .
Z 1 - Z 2 = p 0 n / sin α ,
Z 1 - Z 2 = p 0 n / sin α ,
I A = K [ 1 + sin ( 2 π / p ) ( X - Z 1 tan α ) ] .
I B = K [ 1 + sin ( 2 π / p ) ( X + Z 2 tan α ) ] .
I s = I A + I B = 2 K { 1 + sin [ ( 2 π / p ) X ] · cos [ ( π / p ) ( 2 Z tan α ) ] } .
I s = K I A I B = K K ( sin { ( π / p ) [ 2 X - ( Z 1 - Z 2 ) tan α ] } + cos [ ( π / p ) ( Z 1 + Z 2 ) tan α ] ) 2 .
I T A ( X , Y ) = K { 1 + sin ( 2 π / p ) ( X - Z 0 tan α ) · J 0 [ ( 2 π / p ) tan α A ( X , Y ) ] } ,
I 0 ( X , Y ) = K K { 1 + ½ J 0 [ ( 2 π / p ) tan α A ( X , Y ) ] } .

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