Abstract

A 3D windowed Fourier transform is proposed for fringe sequence analysis, which processes the joint spatial and temporal information of the fringe sequence simultaneously. The 2D windowed Fourier transform in the spatial domain and the 1D windowed Fourier transform in the temporal domain are two special cases of the proposed method. The principles of windowed Fourier filtering and windowed Fourier ridges are developed. Experimental verification shows encouraging results despite a longer processing time.

© 2006 Optical Society of America

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References

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2005

2004

2003

2001

1992

1982

Ang, K. T.

Argentini, G.

Bone, D. J.

Crespo, D.

Fu, Y.

Ina, H.

Kitoh, M.

Kobayashi, S.

Kujawinska, M.

M. Servin and M. Kujawinska, in Handbook of Optical Engineering, D.Malacara and B.J.Thompson, eds. (Marcel Dekker, 2001).

Larkin, K. G.

Marroquin, J. L.

Miao, H.

Ng, T. W.

Oldfield, M. A.

Qian, K.

Quan, C.

Quiroga, J. A.

Reid, G. T.

D. W. Robinson and G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

Robinson, D. W.

D. W. Robinson and G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, 1993).

Servin, M.

Takeda, M.

Tay, C. J.

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Figures (2)

Fig. 1
Fig. 1

Strategies for fringe analysis. (a) Processing in x y plane (spatial domain), one frame at a time; (b) processing along t axis (temporal domain), one pixel at a time; (c) processing in x y t space (joint spatial-temporal domain).

Fig. 2
Fig. 2

Fringe sequence analysis by WFT and Fourier transform (FT). First row, four frames from a fringe sequence; second row, extracted phase using WFT; third row, cosine value of the extracted phase in the second row; fourth row, extracted phase using FT; last row, cosine value of the extracted phase in the fourth row.

Equations (12)

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f ( r ) = a ( r ) + b ( r ) cos [ φ ( r ) ] ,
φ ( r ) φ ( r 0 ) + [ φ ( r 0 ) ] T ( r r 0 ) ,
φ ( r 0 ) = [ φ ( r ) x r = r 0 φ ( r ) y r = r 0 φ ( r ) t r = r 0 ] T .
cos [ φ ( r ) ] cos { φ ( r 0 ) + [ φ ( r 0 ) ] T ( r r 0 ) } = 1 2 exp { j φ ( r 0 ) + j [ φ ( r 0 ) ] T ( r r 0 ) } + 1 2 exp { j φ ( r 0 ) j [ φ ( r 0 ) ] T ( r r 0 ) } ,
h ( r ; ξ ) = g ( r ) exp ( j ξ T r ) ,
h ( r ; ξ ) = exp [ 1 2 r T K 1 r + j ξ T r ] ,
K 1 = [ 1 σ x 2 0 0 0 1 σ y 2 0 0 0 1 σ t 2 ] ;
f ( r ) = 1 8 π 3 { [ f ( r ) h ( r ; ξ ) ] h ( r ; ξ ) } d ξ x d ξ y d ξ t ,
a ( r ) b ( r ) = a ( u , ν , τ ) b ( x u , y ν , t τ ) d u d ν d τ .
f ¯ ( r ) = 1 8 π 3 ξ t l ξ t h ξ y l ξ y h ξ x l ξ x h [ f ( r ) h ( r ; ξ ) ¯ ] h ( r ; ξ ) d ξ x d ξ y d ξ t .
φ ( r ) = arg max ξ f ( r ) h ( r ; ξ ) ,
φ ( r ) = angle { f ( r ) h [ r ; φ ( r ) ] } .

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