Abstract

An algorithm for accurately extracting the local fringe direction is presented. The algorithm estimates, in the neighborhood of n × n points, the direction of the gradient that points normal to the local fringe direction. The performance of four different derivative kernels is also compared. Since this method is sensitive to noise and variations in background and amplitude, a preprocessing step is used to limit these error sources. The method has been applied to the moiré deflectogram of a spherical and a progressive addition ophthalmic lens, resulting in a map of the refractive power of these lenses. The results are compared with the data obtained with a commercial focimeter. This technique is useful for analyzing the fringe patterns where the fringe direction is variable and must be obtained locally.

© 1998 Optical Society of America

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References

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  1. O. Kafri, Y. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
  2. M. Servin, R. Rodriguez-Vera, M. Carpio, A. Morales, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
    [CrossRef] [PubMed]
  3. Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
    [CrossRef] [PubMed]
  4. Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
    [CrossRef]
  5. Q. Yu, X. Liu, K. Andresen, “New spin filters for interferometric fringe patterns and grating patterns,” Appl. Opt. 33, 3705–3711 (1994).
    [CrossRef] [PubMed]
  6. Q. Yu, K. Andresen, “Fringe-orientation maps and fringe skeleton extraction by the two-dimensional derivative-sign binary-fringe method,” Appl. Opt. 33, 6873–6878 (1994).
    [CrossRef] [PubMed]
  7. Q. Yu, K. Andresen, W. Osten, W. Jueptner, “Noise free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 20, 3783–3790 (1996).
    [CrossRef]
  8. W. Pratt, Digital Image Processing (Wiley, New York, 1991), Chap. 16.
  9. H. Vrooman, A. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  10. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]

1996 (2)

Q. Yu, K. Andresen, W. Osten, W. Jueptner, “Noise free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 20, 3783–3790 (1996).
[CrossRef]

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
[CrossRef] [PubMed]

1994 (2)

1991 (1)

1990 (2)

Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
[CrossRef]

M. Servin, R. Rodriguez-Vera, M. Carpio, A. Morales, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
[CrossRef] [PubMed]

1985 (1)

Andresen, K.

Carpio, M.

Glatt, Y.

O. Kafri, Y. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Jueptner, W.

Q. Yu, K. Andresen, W. Osten, W. Jueptner, “Noise free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 20, 3783–3790 (1996).
[CrossRef]

Kafri, O.

O. Kafri, Y. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Liu, X.

Maas, A.

Morales, A.

Murata, K.

Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
[CrossRef]

Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
[CrossRef] [PubMed]

Nakano, Y.

Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
[CrossRef]

Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
[CrossRef] [PubMed]

Ohmura, R.

Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
[CrossRef]

Osten, W.

Q. Yu, K. Andresen, W. Osten, W. Jueptner, “Noise free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 20, 3783–3790 (1996).
[CrossRef]

Pratt, W.

W. Pratt, Digital Image Processing (Wiley, New York, 1991), Chap. 16.

Rodriguez-Vera, R.

Servin, M.

Ströbel, B.

Vrooman, H.

Yu, Q.

Appl. Opt. (7)

Opt. Laser Technol. (1)

Y. Nakano, R. Ohmura, K. Murata, “Refractive power mapping of progressive power lenses using Talbot interferometry and digital image processing,” Opt. Laser Technol. 22, 195–198 (1990).
[CrossRef]

Other (2)

W. Pratt, Digital Image Processing (Wiley, New York, 1991), Chap. 16.

O. Kafri, Y. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

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Figures (15)

Fig. 1
Fig. 1

Tested derivative kernels with a window size of 5 × 5: K1, Prewitt; K2, truncated pyramid; K3, fit to a plane; K4, square weighted.

Fig. 2
Fig. 2

Simulated circular fringe pattern with a period of 25 pixels with the background and the fringe amplitude Gaussian shaped. A normal random noise of 15 gray levels of amplitude is added.

Fig. 3
Fig. 3

a, Intensity distributions along the central line of Fig. 2. b, Same line after filtering and fringe normalization.

Fig. 4
Fig. 4

a, LFD map of Fig. 2. The noisy points correspond to extreme intensity points in the fringe pattern. b, LFD after filtering the low modulation points by interpolation of valid points.

Fig. 5
Fig. 5

Fringe angle error versus kernel size for the four derivative kernels depicted in Fig. 1.

Fig. 6
Fig. 6

Contour map of the fringe angle error value with respect to the fringe period and the kernel size. Circular dots are the optimal kernel size for each fringe period. The straight line is the least-squares fitting of these points.

Fig. 7
Fig. 7

Refractive power versus fringe angle calculated with grating pitch p = 0.1 mm, separation z is set to the first Talbot plane and for three different angles between the gratings θ.

Fig. 8
Fig. 8

Relative power error versus fringe angle with the same parameters of Fig. 7 and with Δα = 2°. The error is around 7–10% for α between 20° and 70°.

Fig. 9
Fig. 9

Deflectograms of a monofocal lens of +2.00 Dp. Note the fringe curve near the lens border and the reference fringes in the upper left corner. The gratings are, a, along the horizontal axis and, b, along the vertical axis.

Fig. 10
Fig. 10

Map corresponding to the fringe rotation of, a, Fig. 9a and, b, Fig. 9b. To convey high-contrast information, 45° has been subtracted from the original fringe rotation maps.

Fig. 11
Fig. 11

Local refractive power map for the spherical lens tested.

Fig. 12
Fig. 12

Profile of the refractive power shown in Fig. 11 along the central horizontal diameter.

Fig. 13
Fig. 13

Deflectograms of a progressive addition lens of +2.00 Dp. The gratings are, a, along the horizontal axis and, b, along the vertical axis.

Fig. 14
Fig. 14

a, Refractive power map, a, in the y direction and, b, in the x direction, determined from Figs. 12a and 12b, respectively.

Fig. 15
Fig. 15

a, Refractive power map, a, in the y direction and, b, in the x direction, measured with a commercial focimeter.

Equations (2)

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I x ,   y = a x ,   y + b x ,   y cos ϕ x ,   y + n x ,   y ,
P x ,   y = sin   θ   tan   α x ,   y + cos   θ - 1 z .

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