Abstract

The spatial orientation of fringes has been demonstrated to be a key point in reliable phase demodulation from a single n-dimensional fringe pattern, regardless of the frequency spectrum of the signal. Recent publications have shown a general method for determination of the orientation factor by use of a regularized phase-tracking (RPT) algorithm. We propose a generalization of a RPT algorithm for estimation of the spatial orientation in a general n-dimensional case. The proposed algorithm makes use of a simplified cost function that remains one dimensional regardless of the dimension of the problem. This makes the calculation faster than with a standard RPT algorithm, with which it is necessary to minimize an n + 1-dimensional cost function for each point of the sample space. We have applied the method to the three-dimensional demodulation of a time-evolving fringe pattern, with good results.

© 2004 Optical Society of America

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References

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  1. T. Kreis, Holographic Interferometry (Akademie Verlag, Berlin, 1996).
  2. M. Servin, J. A. Quiroga, J. L. Marroquin, “A general n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
    [CrossRef]
  3. J. A. Quiroga, M. Servin, J. L. Marroquin, D. Crespo, “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” submitted to J. Opt. Soc. Am. A.
  4. X. Zhou, J. P. Baird, J. F. Arnold, “Fringe-orientation estimation by use of a Gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38, 795–804 (1999).
    [CrossRef]
  5. J. A. Quiroga, M. Servin, F. Cuevas, “Modulo 2π fringe-orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
    [CrossRef]
  6. D. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  7. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 35, 2192–2198 (1996).
  8. J. A. Quiroga, M. Servin, J. L. Marroquín, “Robust demodulation of isochromatics from a single tricolour image using an adaptive regularised phase tracking technique,” presented at the international conference, PhotoMechanique 2001, presented at Poitiers, France, 24–26 April, 2001.

2003 (1)

2002 (1)

1999 (1)

1996 (1)

Arnold, J. F.

Baird, J. P.

Crespo, D.

J. A. Quiroga, M. Servin, J. L. Marroquin, D. Crespo, “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” submitted to J. Opt. Soc. Am. A.

Cuevas, F.

Cuevas, F. J.

Ghiglia, D.

D. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Kreis, T.

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

Malacara, D.

Marroquin, J. L.

Marroquín, J. L.

J. A. Quiroga, M. Servin, J. L. Marroquín, “Robust demodulation of isochromatics from a single tricolour image using an adaptive regularised phase tracking technique,” presented at the international conference, PhotoMechanique 2001, presented at Poitiers, France, 24–26 April, 2001.

Pritt, M. D.

D. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Quiroga, J. A.

M. Servin, J. A. Quiroga, J. L. Marroquin, “A general n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[CrossRef]

J. A. Quiroga, M. Servin, F. Cuevas, “Modulo 2π fringe-orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
[CrossRef]

J. A. Quiroga, M. Servin, J. L. Marroquín, “Robust demodulation of isochromatics from a single tricolour image using an adaptive regularised phase tracking technique,” presented at the international conference, PhotoMechanique 2001, presented at Poitiers, France, 24–26 April, 2001.

J. A. Quiroga, M. Servin, J. L. Marroquin, D. Crespo, “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” submitted to J. Opt. Soc. Am. A.

Rodriguez-Vera, R.

Servin, M.

M. Servin, J. A. Quiroga, J. L. Marroquin, “A general n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[CrossRef]

J. A. Quiroga, M. Servin, F. Cuevas, “Modulo 2π fringe-orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
[CrossRef]

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 35, 2192–2198 (1996).

J. A. Quiroga, M. Servin, J. L. Marroquin, D. Crespo, “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” submitted to J. Opt. Soc. Am. A.

J. A. Quiroga, M. Servin, J. L. Marroquín, “Robust demodulation of isochromatics from a single tricolour image using an adaptive regularised phase tracking technique,” presented at the international conference, PhotoMechanique 2001, presented at Poitiers, France, 24–26 April, 2001.

Zhou, X.

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

Other (4)

J. A. Quiroga, M. Servin, J. L. Marroquin, D. Crespo, “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” submitted to J. Opt. Soc. Am. A.

D. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

J. A. Quiroga, M. Servin, J. L. Marroquín, “Robust demodulation of isochromatics from a single tricolour image using an adaptive regularised phase tracking technique,” presented at the international conference, PhotoMechanique 2001, presented at Poitiers, France, 24–26 April, 2001.

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

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Figures (5)

Fig. 1
Fig. 1

(a) Simulated fringe pattern with four isolated circular regions and random noise. (b) β1 π calculated from Eq. (6). (c) Unwrapped orientation map obtained with the algorithm presented in this paper. (d) Comparison of the results obtained with a standard RPT algorithm (dashed curve), our optimized algorithm (dotted curve), and the theoretical value for the orientation (continuous curve).

Fig. 2
Fig. 2

Consecutive snapshots, obtained with a polariscope, of the isochromatics on a disk of a photoelastic material when an increase of the mechanical load is progressively applied on its upper side.

Fig. 3
Fig. 3

β1 π for each of the consecutive snapshots (or planes along the t axis).

Fig. 4
Fig. 4

Orientation term, obtained from the unwrapping of W{2β1 π}, for each of the consecutive snapshots (or planes along the t axis).

Fig. 5
Fig. 5

(a) Final recovered values of the phase for each consecutive snapshot, obtained with the GQT algorithm. (b) Values of the phase shown in (a) that correspond to the region marked by the black rectangle. This is the region where the change in phase between successive snapshots is greatest.

Equations (13)

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Ir=br+mrcos ϕr,
IHPr=mrcos ϕr.
QnIHPr=ϕr|ϕr|IHPr|ϕr|=nϕrIHPr|ϕr|.
ñϕr=I|I|=-sin ϕ|sin ϕ|ϕr|ϕr|=-signsin ϕnϕr.
QSI=-signsin ϕ.
βkπ=arctan-I/xk+1I/xk, βk2π=arctan-ϕ/xk+1ϕ/xk,
cos βkπ=-signsin ϕcos βk2π=QSIcos βk2π.
fCr=cosW2β, fSr=sinW2β
Urθ, ωx, ωy=ξηNL |fCξ, η-cos px, y, ξ, η|2+|fSξ, η-sin px, y, ξ, η|2++μ|W4πθξ, η-px, y, ξ, η|2mξ, η,
px, y, ξ, η=θx, y+ωxx, yx-ξ+ωyx, yy-η,
Urθ, ω=ξNL |fCρ-cos pr, ρ|2+|fSρ-sin pr, ρ|2++μ|W4πθρ-pr, ρ|2mρ,
ωrωˆr=2βr=W{W2βr}.
Urθ=ξNL |fCρ-cos pˆr, ρ|2+|fSρ-sin pˆr, ρ|2++μ|W4πθρ-pˆr, ρ|2mρ,

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