Abstract

The fringe orientation angle provides useful information for many fringe-pattern-processing techniques. From a single normalized fringe pattern (background suppressed and modulation normalized), the fringe orientation angle can be obtained by computing the irradiance gradient and performing a further arctangent computation. Because of the 180° ambiguity of the fringe direction, the orientation angle computed from the gradient of a single fringe pattern can be determined only modulo π. Recently, several studies have shown that a reliable determination of the fringe orientation angle modulo 2π is a key point for a robust demodulation of the phase from a single fringe pattern. We present an algorithm for the computation of the modulo 2π fringe orientation angle by unwrapping the orientation angle obtained from the gradient computation with a regularized phase tracking method. Simulated as well as experimental results are presented.

© 2002 Optical Society of America

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References

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  1. T. Kreis, Holographic Interferometry (Akademie, Berlin, 1996).
  2. N. Alcalá-Ochoa, J. L. Marroquin, A. Dávila, “Phase recovery using a twin pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
    [CrossRef]
  3. J. A. Quiroga, J. A. Gomez-Pedrero, A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
    [CrossRef]
  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. M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  6. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1544 (1998).
    [CrossRef]
  7. K. G. Larkin, D. J. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  8. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw Hill, New York, 1978).
  9. D. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  10. 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. 38, 1934–1941 (1999).
    [CrossRef]
  11. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]

2001

1999

1998

1996

Alcalá-Ochoa, N.

N. Alcalá-Ochoa, J. L. Marroquin, A. Dávila, “Phase recovery using a twin pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Arnold, J. F.

Baird, J. P.

Bone, D. J.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw Hill, New York, 1978).

Cuevas, F. J.

Dávila, A.

N. Alcalá-Ochoa, J. L. Marroquin, A. Dávila, “Phase recovery using a twin pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

Garcia-Botella, A.

J. A. Quiroga, J. A. Gomez-Pedrero, A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Ghiglia, D.

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

Gomez-Pedrero, J. A.

J. A. Quiroga, J. A. Gomez-Pedrero, A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Kreis, T.

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

Larkin, K. G.

Malacara, D.

Marroquin, J. L.

Oldfield, M. A.

Pritt, M. D.

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

Quiroga, J. A.

J. A. Quiroga, J. A. Gomez-Pedrero, A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Rodriguez-Vera, R.

Servin, M.

Ströbel, B.

Zhou, X.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

N. Alcalá-Ochoa, J. L. Marroquin, A. Dávila, “Phase recovery using a twin pulsed addition fringe pattern in ESPI,” Opt. Commun. 163, 15–19 (1999).
[CrossRef]

J. A. Quiroga, J. A. Gomez-Pedrero, A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Other

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw Hill, New York, 1978).

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

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

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Figures (9)

Fig. 1
Fig. 1

(a) Noisy circular fringe pattern and (b) associated fringe orientation phase map W ( 2 β π ) , where black represents 0 rad and white represents 2π rad. As indicated in (b), if a circular path is followed, encircling the discontinuity sources, a total variation of 4π rad is found. (c) Amplification of the central white square of W ( 2 β π ) . The positions of the two positive discontinuity sources as well as the 4π dislocation are marked.

Fig. 2
Fig. 2

(a) Computer-generated fringe pattern and (b) associated modulo π fringe orientation phase map β π , where black represents-π/2 rad and white represents π/2 rad.

Fig. 3
Fig. 3

(a) f β C and (b) f β S signals [Eqs. (11)] associated with the fringe orientation map β π shown in Fig. 2(b).

Fig. 4
Fig. 4

(a) Modulo 2π fringe orientation angle β 2 π obtained with the modified RPT method for the fringe pattern of Fig. 2(a); in this case the fringe pattern was corrupted with additive noise normally distributed of standard deviation 1. (b) Modulo 2π fringe orientation angle β 2 π ; in this case the fringe pattern was corrupted with phase noise normally distributed of standard deviation 1 rad. (c) Actual β 2 π map corresponding to Fig. 2(a). (d) Histogram of the difference divided by π between the estimated β 2 π and the actual one for the two cases of additive noise (solid curve) and phase noise (dashed curve). In the phase maps of this figure, black represents 0 rad, and white represents 2π rad.

Fig. 5
Fig. 5

(a) Shadow moiré fringe pattern. A clear variation from left to right of the background and the modulation is clearly visible. (b) Associated normalized fringe pattern.

Fig. 6
Fig. 6

(a) Fringe orientation phase map W ( 2 β π ) computed from Fig. 5(b). Low-modulation regions associated with fringe maxima and minima are clearly visible; also, the center of the closed fringes (low-spatial-frequency region) shows a low-modulation area. In this figure black represents 0 rad, and white represents 2π rad.

Fig. 7
Fig. 7

(a) f β C and (b) f β S signals [Eqs. (11)] associated with the fringe orientation phase map W ( 2 β π ) shown in Fig. 6.

Fig. 8
Fig. 8

(a) Modulo 2π fringe orientation angle β 2 π obtained by the modified RPT method for the fringe pattern of Fig. 5, (b) demodulated phase map obtained by using the SPQ transform [relation (6)] applied to the fringe pattern of Fig. 5 with the orientation angle in (a).

Fig. 9
Fig. 9

(a) ESPI fringe pattern obtained in a deformation measurement experiment. (b) Modulo 2π fringe orientation angle β 2 π obtained by the modified RPT method for the fringe pattern of (a). (c) Demodulated phase map obtained by using the SPQ transform [relation (6)] applied to the fringe pattern of (a) with the orientation angle in (b). (d) Cosine of the phase map shown in (c); note the closeness with the fringe pattern shown in (a).

Equations (14)

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I ( r ) = B ( r ) + M ( r ) cos   ϕ ( r ) ,
β 2 π ( r ) = arctan ϕ ( r ) / y ϕ ( r ) / x ;
I N ( r ) = cos   ϕ ( r ) ,
I N ( r ) = [ sin   ϕ ( r ) ] ϕ ( r ) ;
β π ( r ) = arctan I N ( r ) / y I N ( r ) / x ;
SPQ [ I N ( r ) ] i [ sin   ϕ ( r ) ] exp ( i β 2 π ) .
f C ( r ) = cos   W ( ϕ ) = cos   ϕ ;
f S ( r ) = sin   W ( ϕ ) = sin   ϕ ;
U r ( ϕ ,   ω x ,   ω y ) = ( ξ ,   η ) N L [ | f C ( ξ ,   η ) - cos   p ( x ,   y ,   ξ ,   η ) | 2 + | f S ( ξ ,   η ) - sin   p ( x ,   y ,   ξ ,   η ) | 2 + μ | ϕ ( ξ ,   η ) - p ( x ,   y ,   ξ ,   η ) | 2 m ( ξ ,   η ) ] ,
p ( x ,   y ,   ξ ,   η ) = ϕ ( x ,   y ) + ω x ( x ,   y ) ( x - ξ ) + ω y ( x ,   y ) ( y - η ) ,
W ( 2 β π ) = W ( 2 β 2 π + 2 k π ) = W ( 2 β 2 π ) .
f β C = cos   W ( 2 β π ) , f β S = sin   W ( 2 β π ) ,
f β C = cos   2 β 2 π , f β S = sin   2 β 2 π ;
U r ( ϕ ,   ω x ,   ω y ) = ( ξ ,   η ) N L [ | f β C ( ξ ,   η ) - cos   p ( x ,   y ,   ξ ,   η ) | 2 + | f β S ( ξ ,   η ) - sin   p ( x ,   y ,   ξ ,   η ) | 2 + μ | W 4 π [ ϕ ( ξ ,   η ) - p ( x ,   y ,   ξ ,   η ) ] | 2 m ( ξ ,   η ) ] ,

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