Abstract

Fast Fourier transform (FFT), iteration, and least-squares fit are combined to form an image-processing system for the analysis of a carrier-coded fringe pattern. Only one coded fringe pattern is needed for extracting unambiguous information. The coded fringe pattern is first two-dimensionally FFT filtered to produce an initial coded phase with the carrier phase in it. Several phase iterations are carried out if necessary to improve the coded phase. The least-squares-fit technique is used to obtain a pure carrier phase. Then the carrier is removed by subtracting the pure carrier phase from the coded phase. The algorithm offers an improvement over the Fourier-transform method reported in the literature. A program is designed to execute the algorithm, and the processing is automated by a personal computer with an image board. Theory and applications of speckle interferometry and three-dimensional contouring are presented.

© 1995 Optical Society of America

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1994

1991

1986

1985

1984

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

1983

1982

1976

C. Sciammarella, J. Gilbert, “A holographic-moiré technique to obtain separate patterns for components of displacement,” Exp. Mech. 16, 215–219 (1976).
[CrossRef]

1974

Bachor, H.

Bone, D.

Brangaccio, D.

Bruning, J.

Chen, F.

Creath, K.

Gallagher, J.

Gerhart, G.

P. Plotkowski, Y. Hung, J Hovansian, G. Gerhart, “Improvement fringe carrier technique for unambiguous determination of holographically recorded displacements,” Opt. Eng. 24, 754–756 (1985).
[CrossRef]

Gilbert, J.

C. Sciammarella, J. Gilbert, “A holographic-moiré technique to obtain separate patterns for components of displacement,” Exp. Mech. 16, 215–219 (1976).
[CrossRef]

Gu, Jie

Herriott, D.

Hovansian, J

P. Plotkowski, Y. Hung, J Hovansian, G. Gerhart, “Improvement fringe carrier technique for unambiguous determination of holographically recorded displacements,” Opt. Eng. 24, 754–756 (1985).
[CrossRef]

Hung, Y.

Jie Gu, Y. Hung, F. Chen, “Iteration algorithm for computer aided speckle interferometry,” Appl. Opt. 33, 5308–5317 (1994).
[CrossRef] [PubMed]

P. Plotkowski, Y. Hung, J Hovansian, G. Gerhart, “Improvement fringe carrier technique for unambiguous determination of holographically recorded displacements,” Opt. Eng. 24, 754–756 (1985).
[CrossRef]

Ina, H.

Inaba, S.

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

Kobayashi, S.

Kreis, T.

Kujawinska, M.

Mutoh, K.

Nakadate, S.

Nakano, H.

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

Patorski, K.

Plotkowski, P.

P. Plotkowski, Y. Hung, J Hovansian, G. Gerhart, “Improvement fringe carrier technique for unambiguous determination of holographically recorded displacements,” Opt. Eng. 24, 754–756 (1985).
[CrossRef]

Rosenfeld, D.

Ru, Q.

Saito, H.

Salbut, L

Sandeman, R.

Sciammarella, C.

C. Sciammarella, J. Gilbert, “A holographic-moiré technique to obtain separate patterns for components of displacement,” Exp. Mech. 16, 215–219 (1976).
[CrossRef]

Suzuki, M.

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

Takeda, M.

White, A.

Yatagai, T.

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

Appl. Opt.

Exp. Mech.

C. Sciammarella, J. Gilbert, “A holographic-moiré technique to obtain separate patterns for components of displacement,” Exp. Mech. 16, 215–219 (1976).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

P. Plotkowski, Y. Hung, J Hovansian, G. Gerhart, “Improvement fringe carrier technique for unambiguous determination of holographically recorded displacements,” Opt. Eng. 24, 754–756 (1985).
[CrossRef]

T. Yatagai, S. Inaba, H. Nakano, M. Suzuki, “Automatic flatness tester for very large scale integrated circuit wafers,” Opt. Eng. 23, 401–405 (1984).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Carrier-modulated ESPI fringe pattern of a deformed plate.

Fig. 2
Fig. 2

Projection moiré with a laser-generated nonlinear grating.

Fig. 3
Fig. 3

Flow chart of the programming algorithm.

Fig. 4
Fig. 4

Carrier fringe pattern.

Fig. 5
Fig. 5

Fourier spectrum and the filtering window.

Fig. 6
Fig. 6

Initial phase with the carrier in it.

Fig. 7
Fig. 7

Phase with the carrier in it after two iterations.

Fig. 8
Fig. 8

Selected carrier areas.

Fig. 9
Fig. 9

Information phase.

Fig. 10
Fig. 10

3-D view of the information phase.

Fig. 11
Fig. 11

Information phase after two iterations.

Fig. 12
Fig. 12

3-D view of the iterated information phase.

Fig. 13
Fig. 13

Fringe pattern reconstructed from the phase shown in Fig. 11.

Fig. 14
Fig. 14

Original ESPI fringe pattern of the plate without carrier modulation.

Fig. 15
Fig. 15

Schematic diagram of the projection moiré.

Fig. 16
Fig. 16

Projection moiré pattern of a car model.

Fig. 17
Fig. 17

Height phase distribution of the car model shown in Fig. 16.

Fig. 18
Fig. 18

3-D view of the height phase distribution shown in Fig. 17.

Equations (24)

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I m ( x , y ) = A ( x , y ) + B ( x , y ) cos [ ϕ ( x , y ) + a x + b y + c ] ,
ϕ c ( x , y ) = a x + b y + c .
S ( f x , f y ) = I m ( x , y ) exp [ - j 2 π ( x f x + y f y ) ] d x d y ,
S ( f x , f y ) = F { A ( x , y ) } + exp ( j c ) 2 F { B ( x , y ) × exp [ j ϕ ( x , y ) ] } * δ ( f x - a 2 π , f y - b 2 π ) + exp ( - j c ) 2 F { B ( x , y ) exp [ - j ϕ ( x , y ) ] } * δ ( f x + a 2 π , f y + b 2 π ) ,
ϕ m ( x , y ) = ϕ ( x , y ) + ϕ c ( x , y ) .
I f = B ( x , y ) 2 exp [ j ϕ m ( x , y ) ] .
I R = B ( x , y ) 2 cos ϕ m ( x , y ) , I I = B ( x , y ) 2 sin ϕ m ( x , y ) .
ϕ m ( x , y ) = arctan Im ( I f ) Re ( I f ) .
I m i = 1 + cos k [ ϕ m ( n - 1 ) + π 2 ( i - 1 ) ] ,             i = 1 , 2 , 3 , 4 ,
I m i = w I m i d A .
ϕ m n ( x , y ) = arctan I m 4 - I m 2 I m 1 - I m 3 .
E = ( ϕ m j - ϕ c ) 2 = ( ϕ m j - a x j - b y j - c ) 2 ,
E a = 0 ,             E b = 0 ,             E c = 0
a x j 2 + b x j y j + c x j = x j ϕ m j , a x j y j + b y j 2 + c y j = y j ϕ m j , a x j + b y j + N c = ϕ m j ,
ϕ c ( x , y ) = a x y + b x + c y + d .
E = ( ϕ m j - ϕ c ) 2 ( ϕ m j - a x j y j - b x j - c y j - d ) 2 .
E a = 0 ,             E b = 0 ,             E c = 0 ,             E d = 0
a x j 2 y j 2 + b x j 2 y j + c x j y j 2 + d x j y j = x j y j ϕ m j , a x j 2 y j + b x j 2 + c x j y j + d x j = x j ϕ m j , a x j y j 2 + b x j y j + c y j 2 + d y j = y j ϕ m j , a x j y j + b x j + c y j + N d = ϕ m j .
ϕ ( x , y ) = ϕ m ( x , y ) - ϕ c ( x , y ) .
ϕ c = b x + c y + d e x + f y + g .
ϕ c = a x y + b x + c y + d x y + e x + f y + g ,
x y ϕ c + e x ϕ c + f y ϕ c + g ϕ c - a x y - b x - c y - d = 0.
E = ( x j y j ϕ m j + e x j ϕ m j + f y j ϕ m j + g ϕ m j - a x j y j - b x j - c y j - d ) 2 .
E a = 0 , E b = 0 , E c = 0 , E d = 0 , E e = 0 , E f = 0 , E g = 0

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