Abstract

The red-blue-green (RGB) calibration technique consists in constructing an a priori calibration table of the isochromatic retardation versus the triplet of RGB values obtained with a RGB CCD camera. In this way a lookup table (LUT) is built in which the entry is the corresponding RGB triplet and the output is the given retardation. This calibration (a radiometric quantity) depends on the geometric and chromatic parameters of the setup. Once the calibration is performed, the isochromatic retardation at a given point of the sample is computed as the one that minimizes the Euclidean distance between the measured RGB triplet and the triplets stored in the LUT. We present an enhanced RGB calibration algorithm for isochromatic fringe pattern demodulation. We have improved the standard demodulation algorithm used in RGB calibration by changing the Euclidean cost function to a regularized one in which the fidelity term corresponds to the Euclidean distance between RGB triplets; the regularizing term forces piecewise continuity for the isochromatic retardation. Additionally we have implemented a selective search in the RGB calibration LUT. We have tested the algorithm with simulated as well as real photoelastic data with good results.

© 2002 Optical Society of America

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References

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  1. P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]
  2. C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
    [CrossRef]
  3. Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.1994; September/October1994; pp. 13–17.
  4. J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
    [CrossRef]
  5. S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
    [CrossRef]
  6. A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
    [CrossRef]
  7. J. A. Quiroga, A. Garcia-Botella, “Demodulation of isochromatic RGB fringe patterns by a improved calibration technique,” Proceedings of the Fourth International Workshop on Automatic processing of Fringe Patterns, W. Osten, W. Jüptner, eds. Elsevier, Paris, (2001), pp. 126–133.
  8. M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
    [CrossRef]
  9. J. A. Quiroga, A. Gonzalez-Cano, “Separation of isoclinics and isochromatics from photoelastic data using a regularized phase-tracking technique,” Appl. Opt. 39, 2931–2940 (2000).
    [CrossRef]
  10. J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
    [CrossRef]
  11. W. Liu, Z. Wang, G. Mu, Z. Fang, “Color-coded projection grating method for shape measurement with a single exposure,” Appl. Opt. 39, 3504–3508 (2000).
    [CrossRef]
  12. M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
    [CrossRef]
  13. K. Ramesh, Digital Photoelasticity (Springer-Verlag, Berlin, 2000).
    [CrossRef]

2002 (1)

J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
[CrossRef]

2000 (2)

1998 (2)

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
[CrossRef]

1997 (1)

M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
[CrossRef]

1995 (2)

A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
[CrossRef]

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

1994 (1)

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Ajovalastic, A.

A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
[CrossRef]

Barone, S.

A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
[CrossRef]

Buckberry, C.

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

Carazo-Alvarez, J.

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Ekman, M. J.

M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
[CrossRef]

Fang, Z.

Garcia-Botella, A.

J. A. Quiroga, A. Garcia-Botella, “Demodulation of isochromatic RGB fringe patterns by a improved calibration technique,” Proceedings of the Fourth International Workshop on Automatic processing of Fringe Patterns, W. Osten, W. Jüptner, eds. Elsevier, Paris, (2001), pp. 126–133.

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Gonzalez-Cano, A.

Gotoh, J.

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

Haake, S. J.

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Hartl, M.

M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
[CrossRef]

Hayashi, T.

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.1994; September/October1994; pp. 13–17.

Krupka, I.

M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
[CrossRef]

Liska, M.

M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
[CrossRef]

Liu, W.

Marroquin, J. L.

J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
[CrossRef]

Morimoto, Y.

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.1994; September/October1994; pp. 13–17.

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.1994; September/October1994; pp. 13–17.

Mu, G.

Nurse, A. D.

M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
[CrossRef]

Patterson, E. A.

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

Petrucci, G.

A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
[CrossRef]

Quiroga, J. A.

J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
[CrossRef]

J. A. Quiroga, A. Gonzalez-Cano, “Separation of isoclinics and isochromatics from photoelastic data using a regularized phase-tracking technique,” Appl. Opt. 39, 2931–2940 (2000).
[CrossRef]

J. A. Quiroga, A. Garcia-Botella, “Demodulation of isochromatic RGB fringe patterns by a improved calibration technique,” Proceedings of the Fourth International Workshop on Automatic processing of Fringe Patterns, W. Osten, W. Jüptner, eds. Elsevier, Paris, (2001), pp. 126–133.

Ramesh, K.

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

Servin, M.

J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
[CrossRef]

Shimizu, M.

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

Takashi, M.

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Towers, D.

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

Wang, Z.

Yoneyama, S.

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

Appl. Opt. (2)

Exp. Mech. (1)

A. Ajovalastic, S. Barone, G. Petrucci, “Towards RGB photoelasticity: full-field automated photoelasticity in white light,” Exp. Mech.September1995, pp. 193–200.
[CrossRef]

Exp. Tech. (1)

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.1994; September/October1994; pp. 13–17.

Meas. Sci. Tech. (1)

J. A. Quiroga, M. Servin, J. L. Marroquin, “Regularized phase tracking technique for demodulation of isochromatics from a single tricolour image,” Meas. Sci. Tech. 13, 132–140 (2002).
[CrossRef]

Meas. Sci. Technol. (1)

C. Buckberry, D. Towers, “Automatic analysis of isochromatic and isoclinic fringes in photoelasticity using phase-measuring techniques,” Meas. Sci. Technol. 6, 1227–1235 (1995).
[CrossRef]

Opt. Eng. (2)

M. Hartl, I. Krupka, M. Liska, “Diferential colorimetry: tool for evaluation of chromatic interference patterns,” Opt. Eng. 36, 2384–2391 (1997).
[CrossRef]

M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
[CrossRef]

Opt. Lasers Eng. (2)

J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
[CrossRef]

S. Yoneyama, M. Shimizu, J. Gotoh, M. Takashi, “Photoelastic analysis with a single tricolor image,” Opt. Lasers Eng. 29, 423–435 (1998).
[CrossRef]

Other (3)

P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

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

J. A. Quiroga, A. Garcia-Botella, “Demodulation of isochromatic RGB fringe patterns by a improved calibration technique,” Proceedings of the Fourth International Workshop on Automatic processing of Fringe Patterns, W. Osten, W. Jüptner, eds. Elsevier, Paris, (2001), pp. 126–133.

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Figures (8)

Fig. 1
Fig. 1

Plot of the R, G, and B channels of a experimental RGB LUT. Points (symbols), experimental values; solid curves, interpolation.

Fig. 2
Fig. 2

3D curve described in RGB color space for the RGB LUT depicted in Fig. 1. The points (Faint circles) were obtained experimentally, and the solid curve is the interpolation.

Fig. 3
Fig. 3

RG projection of a simulated RGB LUT curve (solid curve) together with the RG values of a noisy Gaussian spatial retardation distribution (dots), both generated by means of Eqs. (1). It can be seen that the noise increases the width of the RG projection corresponding to the spatial distribution, which leads to greater errors for the standard RGB calibration algorithm.

Fig. 4
Fig. 4

Comparison of the theoretical retardation values with those obtained by the standard and improved RGB calibration algorithms. A significant reduction in the demodulation error can be observed when the improved algorithm is used.

Fig. 5
Fig. 5

Comparison of the theoretical retardation values with those obtained by the standard and improved calibration algorithms with chromatic coordinates rgb instead of RGB values. In this case the high values of the demodulation error obtained with the standard algorithm make it unreliable; however, the improved algorithm gives a good estimation of the retardation values.

Fig. 6
Fig. 6

(a) Retardation obtained for an arcshaped test object under axial traction. The demodulated retardation was calculated with the improved RGB calibration method proposed in this paper. Contour curves have been added for the sake of clarity (b) Distribution of the experimental values of the retardation obtained for an arc-shaped test object under axial traction. The demodulated retardation was calculated with the standard RGB calibration method. In this case, artiefacts that are due to bad demodulation are clearly seen. (c) Plot of the demodulated retardation along the central vertical profile of the object. Solid curve, results obtained with the improved algorithm; dashed curve, results of the standard algorithm.

Fig. 7
Fig. 7

Comparison of the demodulated retardation along the central vertical profile of the arcshaped object obtained with RGB values and chromatic coordinates rgb under the same geometric and chromatic conditions. Solid curve, results obtained with the RGB LUT; dotted curve, obtained with the rgb LUT.

Fig. 8
Fig. 8

Plot of the demodulated retardation along the central vertical profile of the test object for two different geometric conditions, in this case different f-numbers. In the two cases the rgb LUT was used, and it can be seen that the retardation values obtained are very close.

Equations (6)

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R= r¯λSλsin2Δλdλ,G= g¯λSλsin2Δλdλ,B= b¯λSλsin2Δλdλ.
Ur, Δi=Rr-RˆΔi2+Gr-ĜΔi2+Br-BˆΔi2.
r=RR+G+B,g=GR+G+B,b=BR+G+B.
Ûr, Δi=Rr-RˆΔi2+Gr-ĜΔi2+Br-BˆΔi2+β sΔi-δs2ms,
AGV-Z0=cosΔ/λ,
σΔλ=1|tanΔ/λ|σGV|GV-Z0|.

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