Abstract

We analyze the effect of image noise on the estimation of fringe orientation in principle and interpret the application of a texture-analysis technique to the problem of estimating fringe orientation in interferograms. The gradient of a Gaussian filter and neighboring-direction averaging are shown to meet the requirements of fringe-orientation estimation by reduction of the effects of low-frequency background and contrast variances as well as high-frequency random image noise. The technique also improves inaccurate orientation estimation at low-modulation points, such as fringe centers and broken fringes. Experiments demonstrate that the scales of the Gaussian gradient filter and the direction averaging should be chosen according to the fringe spacings of the interferograms.

© 1999 Optical Society of America

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References

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  1. S. Krishnaswarmy, “Algorithm for computer tracing of interference fringes,” Appl. Opt. 30, 1624–1628 (1991).
    [CrossRef]
  2. A. G. Khadakkar, V. Jyothi, R. Narayanan, “Fringe tracing by image processing,” Opt. Eng. 33, 1872–1875 (1994).
    [CrossRef]
  3. A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
    [CrossRef]
  4. U. Mieth, W. Osten, W. Juptner, “Knowledge assisted fault detection based on line features of skeletons,” in Fringe ’93 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 19 of Akademie Verlag Series in Physical Research (Akademie Verlag, Bremen, Germany, 1993), pp. 365–373.
  5. D. W. Robinson, “Automatic fringe analysis with a computer image-processing system,” Appl. Opt. 22, 2169–2176 (1983).
    [CrossRef] [PubMed]
  6. Q. F. Yu, “Spin filtering processes and automatic extraction of fringe center lines from interferometric patterns,” Appl. Opt. 27, 3782–3784 (1988).
    [CrossRef] [PubMed]
  7. Q. F. Yu, X. L. Liu, A. Klaus, “New spin filter for interferometric fringe patterns and grating patterns,” Appl. Opt. 33, 3705–3711 (1994).
    [CrossRef] [PubMed]
  8. Q. F. Yu, A. Klaus, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33, 6873–6878 (1994).
    [CrossRef] [PubMed]
  9. H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moire deflectometry,” in Fringe ’97 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 3 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Bremen, Germany, 1997), pp. 107–110.
  10. M. Kass, A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graphics Image Process. 37, 362–385 (1987).
    [CrossRef]
  11. A. R. Rao, B. G. Schunck, “Computing oriented texture fields,” CVGIP Graph. Models Image Process. 53, 157–185 (1991).
    [CrossRef]
  12. K. V. Mardia, Statistics of Directional Data (Academic, London, 1972).

1994 (3)

1993 (1)

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

1991 (2)

A. R. Rao, B. G. Schunck, “Computing oriented texture fields,” CVGIP Graph. Models Image Process. 53, 157–185 (1991).
[CrossRef]

S. Krishnaswarmy, “Algorithm for computer tracing of interference fringes,” Appl. Opt. 30, 1624–1628 (1991).
[CrossRef]

1988 (1)

1987 (1)

M. Kass, A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graphics Image Process. 37, 362–385 (1987).
[CrossRef]

1983 (1)

Aparicio, A.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Bernabeu, E.

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moire deflectometry,” in Fringe ’97 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 3 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Bremen, Germany, 1997), pp. 107–110.

Caceres, S.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Canabal, H.

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moire deflectometry,” in Fringe ’97 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 3 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Bremen, Germany, 1997), pp. 107–110.

de Castro, C.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

de Frutos, A. M.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Frechoso, F. A.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Juptner, W.

U. Mieth, W. Osten, W. Juptner, “Knowledge assisted fault detection based on line features of skeletons,” in Fringe ’93 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 19 of Akademie Verlag Series in Physical Research (Akademie Verlag, Bremen, Germany, 1993), pp. 365–373.

Jyothi, V.

A. G. Khadakkar, V. Jyothi, R. Narayanan, “Fringe tracing by image processing,” Opt. Eng. 33, 1872–1875 (1994).
[CrossRef]

Kass, M.

M. Kass, A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graphics Image Process. 37, 362–385 (1987).
[CrossRef]

Khadakkar, A. G.

A. G. Khadakkar, V. Jyothi, R. Narayanan, “Fringe tracing by image processing,” Opt. Eng. 33, 1872–1875 (1994).
[CrossRef]

Klaus, A.

Krishnaswarmy, S.

Liu, X. L.

Mardia, K. V.

K. V. Mardia, Statistics of Directional Data (Academic, London, 1972).

Mieth, U.

U. Mieth, W. Osten, W. Juptner, “Knowledge assisted fault detection based on line features of skeletons,” in Fringe ’93 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 19 of Akademie Verlag Series in Physical Research (Akademie Verlag, Bremen, Germany, 1993), pp. 365–373.

Molpeceres, J. L.

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Narayanan, R.

A. G. Khadakkar, V. Jyothi, R. Narayanan, “Fringe tracing by image processing,” Opt. Eng. 33, 1872–1875 (1994).
[CrossRef]

Osten, W.

U. Mieth, W. Osten, W. Juptner, “Knowledge assisted fault detection based on line features of skeletons,” in Fringe ’93 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 19 of Akademie Verlag Series in Physical Research (Akademie Verlag, Bremen, Germany, 1993), pp. 365–373.

Quiroga, J. A.

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moire deflectometry,” in Fringe ’97 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 3 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Bremen, Germany, 1997), pp. 107–110.

Rao, A. R.

A. R. Rao, B. G. Schunck, “Computing oriented texture fields,” CVGIP Graph. Models Image Process. 53, 157–185 (1991).
[CrossRef]

Robinson, D. W.

Schunck, B. G.

A. R. Rao, B. G. Schunck, “Computing oriented texture fields,” CVGIP Graph. Models Image Process. 53, 157–185 (1991).
[CrossRef]

Witkin, A.

M. Kass, A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graphics Image Process. 37, 362–385 (1987).
[CrossRef]

Yu, Q. F.

Appl. Opt. (5)

Comput. Vis. Graphics Image Process. (1)

M. Kass, A. Witkin, “Analyzing oriented patterns,” Comput. Vis. Graphics Image Process. 37, 362–385 (1987).
[CrossRef]

CVGIP Graph. Models Image Process (1)

A. R. Rao, B. G. Schunck, “Computing oriented texture fields,” CVGIP Graph. Models Image Process. 53, 157–185 (1991).
[CrossRef]

Opt. Eng. (2)

A. G. Khadakkar, V. Jyothi, R. Narayanan, “Fringe tracing by image processing,” Opt. Eng. 33, 1872–1875 (1994).
[CrossRef]

A. Aparicio, J. L. Molpeceres, A. M. de Frutos, C. de Castro, S. Caceres, F. A. Frechoso, “Improved algorithm for the analysis of holographic interferograms,” Opt. Eng. 32, 963–969 (1993).
[CrossRef]

Other (3)

U. Mieth, W. Osten, W. Juptner, “Knowledge assisted fault detection based on line features of skeletons,” in Fringe ’93 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 19 of Akademie Verlag Series in Physical Research (Akademie Verlag, Bremen, Germany, 1993), pp. 365–373.

H. Canabal, J. A. Quiroga, E. Bernabeu, “Local fringe direction calculation and application in moire deflectometry,” in Fringe ’97 Automatic Processing of Fringe Patterns, W. Juptner, W. Osten, eds., Vol. 3 of Akademie Verlag Series in Optical Metrology (Akademie Verlag, Bremen, Germany, 1997), pp. 107–110.

K. V. Mardia, Statistics of Directional Data (Academic, London, 1972).

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Figures (8)

Fig. 1
Fig. 1

Gradient of a Gaussian filter.

Fig. 2
Fig. 2

Bandpass-filtering effect of the Gaussian gradient filter on a noisy sine wave.

Fig. 3
Fig. 3

(a) Simulated noisy interferogram and (b) its theoretical fringe orientations.

Fig. 4
Fig. 4

Fringe-orientation estimation obtained by use of a Gaussian gradient filter at three scales: (a) σ1 = 0.5, (b) σ1 = 2.5, (c) σ1 = 2.5. (d) Orientation errors versus the filter scale σ1 and variances of Gaussian white noise.

Fig. 5
Fig. 5

Filter scale σ1 versus the fringe spacing P according to Table 1.

Fig. 6
Fig. 6

(a) Holographic interferogram with local fringe spacings and its fringe-orientation estimation obtained from the gradient of a Gaussian filter at scales (b) σ1 = 1 and (c) σ1 = 3.

Fig. 7
Fig. 7

Orientation-error reduction by use of neighboring-direction averaging at three scales: (a) σ2 = 1, (b) σ2 = 5, (c) σ2 = 11. (d) Orientation error on the eccentric circles.

Fig. 8
Fig. 8

(a) Discontinuous fringe in a rectangular region and fringe orientations estimated (a) without and (b) with neighboring-direction averaging (scale of σ2 = 15).

Tables (1)

Tables Icon

Table 1 Fringe Spacing P and the Corresponding Optimum Filter Scale σ1M, Width Δσ1 (for Error Tolerances of 0.01 and 0.02), and Minimum Orientation Error Dmin

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

Ix, y=Ax, y+Bx, ycos Φx, y+Nx, y,
Φx, y=2kπ,  k=0, ±1, ±2,,
Φx, y=2k+1π,  k=0, ±1, ±2,,
Φx, y=k+1/2π,  k=0, ±1, ±2,.
cos θ, sin θϕx, y=0.
θx, y=arctanΦx, yyΦx, yx±π2.
Ix, y=B sin Φx, yΦx, y,
θx, y=arctanIx, yyIx, yx±π2.
Ix, y=Ax, y+Bx, ycos Φx, y+Bx, ysin Φx, yΦx, y+Nx, y.
Ix, y=Ax, y±Bx, y+Nx, y
Ix, y=Ax, y±Bx, yΦx, y+Nx, y.
Gθ=cos θ, sin θ··gx, y * Ix, y,
gx, y=exp-x2+y2/2σ12.
Gθ=cos θ, sin θ··gx, y * Ix, y.
hx, y=cos θ, sin θ··gx, y,
θlx, y=arctanGyx, y/Gxx, y,
Rlx, y=Gxx, y2+Gyx, y21/2.
θx, y=½ arctanx-σ2/2x+σ2/2y-σ2/2y+σ2/2 Rl2x, ysin 2θlx, yx-σ2/2x+σ2/2y-σ2/2y+σ2/2 Rl2x, ycos 2θlx, y±π2,
Sθ=x-σ2/2x+σ2/2y-σ2/2y+σ2/2 Rl2x, ycos2θlx, y-θ.
θx, y=12arctanx-σ2/2x+σ2/2y-σ2/2y+σ2/2 2Gxx, yGyx, yx-σ2/2x+σ2/2y-σ2/2y+σ2/2 Gxx, y2+Gyx, y2±π2.
hx=-xσ12exp-x2/2σ12.
|Hfx|=2πσ1fx exp-2π2σ12fx2
|Hfx|=2πσ11-4π2σ12fx2exp-2π2σ12fx2=0,
fM=±12πσ1.
Φx, y=2πx2+y2 /P
D=1Nx,yR |sinθx, y-θTx, y|

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