Abstract

We present a method that allows the reconstruction of smooth phase distributions from their laterally sheared representation. The proposed approach is efficient in the sense that only one sheared distribution is needed to completely restore the signal. A mandatory requirement is that the phase distribution is spatially limited. The method is exemplified by means of a synthetic signal, and in addition a practical algorithm is given. Finally, experimental results are presented. The deformation of a metallic surface is investigated by both speckle shearography and electronic speckle pattern interferometry (ESPI) respectively. To give proof of the proposed technique, the phase distribution reconstructed from the shearographic measurement is shown to match the results obtained by the ESPI.

© 2007 Optical Society of America

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References

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  1. W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. London 59, 940-952 (1947).
    [CrossRef]
  2. V. Ronchi, "Forty years of history of a grating interferometer," Appl. Opt. 3, 437-450 (1964).
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  4. D. L. Fried, "Least-square fitting a wave-front distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. A 67, 370-375 (1977).
  5. R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. A 67, 375-378 (1977).
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    [CrossRef]
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    [CrossRef]
  11. C. Elster and I. Weingärtner, "Solution to the shearing problem," Appl. Opt. 38, 5024-5031 (1999).
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2006 (1)

2000 (1)

1999 (2)

1997 (2)

H. Brug, "Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry," Appl. Opt. 36, 2788-2790 (1997).

S. Loheide, "Innovative evaluation method for shearing interferograms," Opt. Commun. 141, 254-258 (1997).
[CrossRef]

1996 (2)

1986 (1)

1977 (3)

D. L. Fried, "Least-square fitting a wave-front distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. A 67, 370-375 (1977).

R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. A 67, 375-378 (1977).

K. Kondo, Y. Ichioka, and T. Suzuki, "Image restoration by Wiener filtering in the presence of signal-dependent noise," Appl. Opt. 16(9), 2554-2558 (1977).

1964 (1)

1949 (1)

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37(1), 10-21 (1949).

1947 (1)

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

1915 (1)

E. T. Whittaker, "On the functions which are represented by the expansion of the interpolation theory," Proc. Royal Soc. Edinburgh , Sec. A 35, 181-194 (1915).

Bates, W. J.

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

Brug, H.

Caprari, R. S.

Ding, J.

Elster, C.

Freischlad, K. R.

Fried, D. L.

D. L. Fried, "Least-square fitting a wave-front distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. A 67, 370-375 (1977).

Goth, A. S.

Guo, C.

Harbers, G.

Hudgin, R. H.

R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. A 67, 375-378 (1977).

Ichioka, Y.

Jin, Z.

Koliopoulos, C. L.

Kondo, K.

Kunst, P. J.

Leibbrandt, G. W. R.

Liang, P.

Loheide, S.

S. Loheide, "Innovative evaluation method for shearing interferograms," Opt. Commun. 141, 254-258 (1997).
[CrossRef]

Malacara, D.

Marroquin, J. L.

Moffatt, E. K.

Ronchi, V.

Servin, M.

Shannon, C. E.

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37(1), 10-21 (1949).

Suzuki, T.

Wang, H.

Weingärtner, I.

Whittaker, E. T.

E. T. Whittaker, "On the functions which are represented by the expansion of the interpolation theory," Proc. Royal Soc. Edinburgh , Sec. A 35, 181-194 (1915).

Appl. Opt. (8)

J. Opt. Soc. Am. A (3)

K. R. Freischlad and C. L. Koliopoulos, "Modal estimation of a wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861 (1986).

D. L. Fried, "Least-square fitting a wave-front distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. A 67, 370-375 (1977).

R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. A 67, 375-378 (1977).

Opt. Commun. (1)

S. Loheide, "Innovative evaluation method for shearing interferograms," Opt. Commun. 141, 254-258 (1997).
[CrossRef]

Opt. Express (1)

Proc. Inst. Radio Eng. (1)

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37(1), 10-21 (1949).

Proc. Phys. Soc. London (1)

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. London 59, 940-952 (1947).
[CrossRef]

Proc. Royal Soc. Edinburgh (1)

E. T. Whittaker, "On the functions which are represented by the expansion of the interpolation theory," Proc. Royal Soc. Edinburgh , Sec. A 35, 181-194 (1915).

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Figures (8)

Fig. 1
Fig. 1

(Color online) Nonzero parts of Δ g ( x ) are sampled by Δ g n .

Fig. 2
Fig. 2

Shifting the sampling lattice (bright dots) avoids a sampling of undefined values.

Fig. 3
Fig. 3

General principle of the proposed algorithm.

Fig. 4
Fig. 4

Example of the proposed method: (a) synthetic distribution, vertically sheared by eight pixels, (b) spectrum of the Fourier transform of (a), and (c) spectrum after multiplication by T(v, s) and resampling; and (d) reconstructed distribution. To provide a better visualization, the square root of the spectra is shown in (b) and (c). All presented data arrays have a square size of N = 256 (in pixels).

Fig. 5
Fig. 5

(Color online) Experimental setup: (a) setup and (b) metallic membrane under investigation.

Fig. 6
Fig. 6

Reconstruction of the results obtained by the shearographic measurement: (a) measured phase difference Δ g n , (b) corresponding spectrum, (c) resampled spectrum, and (d) reconstructed phase distribution g n . All data arrays have square size of N = 512 × 512 pixels. In (b) and (c) only a small region of 128 × 128 pixels in the center of each spectrum is depicted.

Fig. 7
Fig. 7

Results obtained by the ESPI: (a) measured phase difference and (b) demodulated result.

Fig. 8
Fig. 8

(Color online) Comparison of the data values across the horizontal lines in the centers of both Figs. 6(d) and 7(b) respectively.

Equations (13)

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Δ g ( x ) = g ( x s 2 ) g ( x + s 2 ) = g ( x ) [ δ ( x s 2 ) δ ( x + s 2 ) ] = g ( x ) h ( x , s ) ,
F { Δ g ( x ) } ( v ) = G ( v ) H ( v , s ) .
H ( v , s ) = 2 i   sin ( π v s ) .
g ( x ) = F 1 { F { Δ g ( x ) } ( v ) T ( v , s ) } .
Δ g n = Δ g ( x ) n i , n j = 0 N 1 δ ( x i n i Δ x , y i n j Δ x )
n i , n j = 0 , 1 ,  …  , N 1.
Δ g n = g n s / 2 g n + s / 2 .
F k { Δ g n } = n i , n j = 0 N 1 Δ g n exp ( i 2 π N n k ) .
F { Δ g n ( 0 ) } ( v ) = k = F k { Δ g n } sinc [ π ( v Δ v k ) ] ,
F k { g n } = F { Δ g n ( 0 ) } ( v ) T ( v , s ) k i , k j = 0 N 1 δ ( v i k i Δ v , v j k j Δ v ) .
β s i = N Δ x .
F k + 0.5 { g n } = F { Δ g n ( 0 ) } ( v ) T ( v ) × k i , k j = 0 N 1 δ ( v i + Δ v 2 k i Δ v , v j k j Δ v ) .
g n exp ( i π N n i ) = F 1 { F k + 0.5 { g n } } .

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