Abstract

The spatial orientation of the fringe has been demonstrated to be a key point in the reliable phase demodulation from a single n-dimensional fringe pattern regardless of the frequency spectrum of the signal. The recently introduced general n-dimensional quadrature transform (GQT) makes explicit the importance of the fringe orientation in the demodulation process. The GQT is a quadrature operator that transforms cos ϕ into -sin ϕ—where ϕ is the modulating phase—and it is composed of two terms: an orientation factor directly related to the fringe’s spatial orientation and an isotropic n-dimensional generalization of the one-dimensional Hilbert transform. We present a method for the determination of the orientation factor in a general n-dimensional case and its application to the demodulation of a single fringe pattern by the GQT. We have tested the algorithm with simulated as well as real photoelastic fringe patterns with good results.

© 2005 Optical Society of America

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References

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  1. T. Kreis, Holographic Interferometry (Akademie Verlag, Berlin, 1996).
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    [CrossRef]
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  8. J. A. Quiroga, M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
    [CrossRef]
  9. K. Ramesh, Digital Photoelasticity (Springer-Verlag, Berlin, 2000).

2003

2002

2001

1999

1998

Arnold, J. F.

Baird, J. P.

Bone, D. J.

Cuevas, F. J.

Kreis, T.

T. Kreis, Holographic Interferometry (Akademie Verlag, Berlin, 1996).

Larkin, K. G.

Marroquin, J. L.

Oldfield, M. A.

Quiroga, J. A.

Ramesh, K.

K. Ramesh, Digital Photoelasticity (Springer-Verlag, Berlin, 2000).

Rodriguez-Vera, R.

Servi´n, M.

Servin, M.

Zhou, X.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

J. A. Quiroga, M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221–227 (2003).
[CrossRef]

Other

K. Ramesh, Digital Photoelasticity (Springer-Verlag, Berlin, 2000).

T. Kreis, Holographic Interferometry (Akademie Verlag, Berlin, 1996).

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Figures (6)

Fig. 1
Fig. 1

Noisy fringe pattern with additive phase noise with an image size of 100 × 100 pixels.

Fig. 2
Fig. 2

(a) Theoretical QS for the fringe pattern depicted in Fig. 1, (b) corresponding QS obtained by the proposed method, (c) theoretical modulating wrapped phase of the fringe pattern shown in Fig. 1, (d) wrapped phase obtained by the GQT with the QS depicted in Fig. 2(b). For the QS images, black corresponds to -1 and white to 1. For the wrapped phase images, black corresponds to 0 and white to 2π.

Fig. 3
Fig. 3

(a)–(c) Circular bright-field isochromatic fringe patterns of a stress-frozen diametrically compressed disk. The increasing load between snapshots is applied in the upper part of the figures.

Fig. 4
Fig. 4

(QS) obtained by the proposed method for each of the three images depicted in Fig. 3. In this figure, black corresponds to -1, white to 1, and gray to nonprocessed.

Fig. 5
Fig. 5

Profiles along line 90 of the demodulated phase corresponding to the three images depicted in Fig. 3. Dotted, dashed, and solid curves correspond to initial, intermediate, and final states, respectively.

Fig. 6
Fig. 6

(a) Shadow-moiré fringe pattern of a 350-μm indentation. (b) Profile along row 216 of the fringe pattern shown in Fig. 6(a). (c) QS obtained by the proposed method. (d) Recovered phase map.

Equations (14)

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I ( r ) = b ( r ) + m ( r ) cos ϕ ( r ) ,
Q n { I HP ( r ) } = ϕ ( r ) | ϕ ( r ) | I HP ( r ) | ϕ ( r ) | = n ϕ ( r ) I HP ( r ) | ϕ ( r ) | .
n ϕ ( r ) = ϕ ( r ) | ϕ ( r ) | = k = 1 n c k ( 2 π ) e k ,
c k ( 2 π ) = cos α k ( 2 π ) = ϕ / x k | ϕ ( r ) | = ω k | ϕ ( r ) | ,
c k = I / x k | I ( r ) | .
c k = - sin ϕ | sin ϕ |   ϕ / x k | ϕ | = - sgn ( sin ϕ )   ϕ / x k | ϕ | = - sgn ( sin ϕ ) c k ( 2 π ) .
n ˜ ϕ ( r ) = I | I | = k = 1 n c k e k = - sgn ( sin ϕ ) n ϕ ( r ) .
QS { I } = - sgn ( sin ϕ ) .
β k ( 2 π ) = arctan - ϕ / x k + 1 ϕ / x k = arctan - ω k + 1 ω k ,
β k ( π ) = arctan - I / x k + 1 I / x k ,
cos β k ( 2 π ) = ω k ( ω k 2 + ω k + 1 2 ) 1 / 2 ,
cos β k ( π ) = I / x k [ ( I / x k ) 2 + ( I / x k + 1 ) 2 ] 1 / 2 .
cos β k ( π ) = - sign [ sin ϕ ] cos β k ( 2 π ) = QS { I } cos β k ( 2 π ) .
QS { I } = sgn ( cos β k ( π ) ) sgn ( cos β k ( 2 π ) ) = sgn ( I / x k ) sgn ( cos β k ( 2 π ) ) ,

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