Abstract

We present the application of a regularization algorithm to the processing of photoelastic fringe patterns. The method used is a modified regularized phase-tracking (RPT) algorithm applied to phase-shifted images. In particular, we present an algorithm for isoclinic–isochromatic separation that uses only five images. In the case of isoclinics the method can deal with problems associated with modulation of isochromatics and with isotropic points by means of a modified cost functional. With respect to the isochromatics the problems associated with regions of high fringe density are solved in a robust way by the unmodified RPT algorithm by use of the modulation information. The performance of the method is discussed, and experimental results are presented.

© 2000 Optical Society of America

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References

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  1. E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–53 (1991).
    [CrossRef]
  2. A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
    [CrossRef]
  3. J. Carazo-Alvarez, S. J. Haake, E. A. Patterson, “Completely automated photoelastic fringe analysis,” Opt. Lasers Eng. 21, 133–149 (1994).
    [CrossRef]
  4. Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).
  5. 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]
  6. A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
    [CrossRef] [PubMed]
  7. G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
    [CrossRef]
  8. J. A. Quiroga, A. González-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
    [CrossRef]
  9. N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).
  10. M. J. Ekman, A. D. Nurse, “Completely automated determination of two-dimensional photoelastic parameters using load stepping,” Opt. Eng. 37, 1845–1851 (1998).
    [CrossRef]
  11. N. Plouzenec, J. C. Dupré, A. Lagarde, “Visualisation of photoelastic fringes within three dimensional specimens using an optical slicing method,” in Proceedings of the IUTAM Symposium on Advanced Optical Methods and Applications in Solids Mechanics, A. Lagarde, ed. (Université de Poitiers, Poitiers, France, 1998), pp. P1–P8.
  12. J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, M. Servin, “Regularization methods for processing fringe-patterns images,” Appl. Opt. 38, 788–794 (1999).
    [CrossRef]
  13. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters for the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
    [CrossRef]
  14. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  15. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
    [CrossRef]
  16. M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
    [CrossRef] [PubMed]
  17. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]
  18. P. S. Theocaris, E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]
  19. J. A. Quiroga, A. González-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
    [CrossRef]
  20. Matlab Optimization Toolbox, User’s Guide, version 5 (MathWorks, New York, 1997).

1999 (2)

N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).

J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, M. Servin, “Regularization methods for processing fringe-patterns images,” Appl. Opt. 38, 788–794 (1999).
[CrossRef]

1998 (3)

J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (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]

J. A. Quiroga, A. González-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[CrossRef]

1997 (5)

1996 (1)

1995 (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]

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

1994 (2)

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

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).

1993 (1)

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

1991 (1)

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–53 (1991).
[CrossRef]

Asundi, A.

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

Botello, S.

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]

Cuevas, F. J.

Dupré, J. C.

N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).

N. Plouzenec, J. C. Dupré, A. Lagarde, “Visualisation of photoelastic fringes within three dimensional specimens using an optical slicing method,” in Proceedings of the IUTAM Symposium on Advanced Optical Methods and Applications in Solids Mechanics, A. Lagarde, ed. (Université de Poitiers, Poitiers, France, 1998), pp. P1–P8.

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]

Gdoutos, E. E.

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

González-Cano, A.

J. A. Quiroga, A. González-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[CrossRef]

J. A. Quiroga, A. González-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
[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]

Hayashi, T.

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).

Lagarde, A.

N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).

N. Plouzenec, J. C. Dupré, A. Lagarde, “Visualisation of photoelastic fringes within three dimensional specimens using an optical slicing method,” in Proceedings of the IUTAM Symposium on Advanced Optical Methods and Applications in Solids Mechanics, A. Lagarde, ed. (Université de Poitiers, Poitiers, France, 1998), pp. P1–P8.

Marroquin, J. L.

Morimoto, Y.

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).

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]

A. D. Nurse, “Full-field automated photoelasticity by use of a three-wavelength approach to phase stepping,” Appl. Opt. 36, 5781–5786 (1997).
[CrossRef] [PubMed]

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]

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–53 (1991).
[CrossRef]

Petrucci, G.

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

Plouzenec, N.

N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).

N. Plouzenec, J. C. Dupré, A. Lagarde, “Visualisation of photoelastic fringes within three dimensional specimens using an optical slicing method,” in Proceedings of the IUTAM Symposium on Advanced Optical Methods and Applications in Solids Mechanics, A. Lagarde, ed. (Université de Poitiers, Poitiers, France, 1998), pp. P1–P8.

Quiroga, J. A.

J. A. Quiroga, A. González-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[CrossRef]

J. A. Quiroga, A. González-Cano, “Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns,” Appl. Opt. 36, 8397–8402 (1997).
[CrossRef]

Rivera, M.

Rodriguez-Vera, R.

Servin, M.

Ströbel, B.

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. F.

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–53 (1991).
[CrossRef]

Appl. Opt. (5)

Exp. Mech. (1)

G. Petrucci, “Full-field automatic evaluation of an isoclinic parameter in white light,” Exp. Mech. 37, 420–426 (1997).
[CrossRef]

Exp. Tech. (3)

N. Plouzenec, J. C. Dupré, A. Lagarde, “Whole field determination of isoclinic and isochromatic parameters,” Exp. Tech.30–33 (Jan./Feb. 1999).

A. Asundi, “Phase shifting in photoelasticity,” Exp. Tech. 17, 19–23 (1993).
[CrossRef]

Y. Morimoto, Y. Morimoto, T. Hayashi, “Separation of isochromatics and isoclinics using Fourier transform,” Exp. Tech.13–17 (Sept./Oct. 1994).

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

Meas. Sci. Technol. (2)

J. A. Quiroga, A. González-Cano, “Stress separation from photoelastic data by a multigrid method,” Meas. Sci. Technol. 9, 1204–1210 (1998).
[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]

Opt. Eng. (1)

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. (1)

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

Opt. Lett. (1)

Strain (1)

E. A. Patterson, Z. F. Wang, “Towards full field automated photoelastic analysis of complex components,” Strain 27, 49–53 (1991).
[CrossRef]

Other (3)

Matlab Optimization Toolbox, User’s Guide, version 5 (MathWorks, New York, 1997).

N. Plouzenec, J. C. Dupré, A. Lagarde, “Visualisation of photoelastic fringes within three dimensional specimens using an optical slicing method,” in Proceedings of the IUTAM Symposium on Advanced Optical Methods and Applications in Solids Mechanics, A. Lagarde, ed. (Université de Poitiers, Poitiers, France, 1998), pp. P1–P8.

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

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Figures (14)

Fig. 1
Fig. 1

Scheme of the first sample. Arrows indicate the sense and the direction of the applied force.

Fig. 2
Fig. 2

Images corresponding to the cosine and the sine terms of the isoclinics, Eqs. (15) and (16).

Fig. 3
Fig. 3

Modulation map, m α, associated with Figs. 2(a) and 2(b).

Fig. 4
Fig. 4

(a) Isoclinic phase map obtained by the PS algorithm (see text), (b) isoclinic map obtained by the RPT algorithm. In these images black corresponds to -π rad and white to π rad.

Fig. 5
Fig. 5

Profiles of the isoclinic angle obtained by the PS and the RPT algorithms. The profiles have been taken along the dashed lines of Figs. 3 and 4(a). (a) Profile along line 60; the areas of low modulation are visible as fluctuation in the profile of the PS isoclinic angle. (b) Profile along line 70. Because the profile does not cross any low-modulation area, the results of the PS and the RPT algorithms almost coincide.

Fig. 6
Fig. 6

Sine term of the isochromatics, Eq. (21), of the sample of Fig. 1.

Fig. 7
Fig. 7

Isochromatic phase map computed by the RPT technique. In this image black corresponds to -π rad and white to π rad.

Fig. 8
Fig. 8

(a) and (b) Images corresponding to the cosine and sine terms of the isoclinics, Eqs. (15) and (16), respectively, of a disk under diametral compression. In these images we observe the detail of the bottom part of the disk.

Fig. 9
Fig. 9

Modulation map, m α, associated with images 8(a) and 8(b).

Fig. 10
Fig. 10

(a) Isoclinic phase map obtained by the PS algorithm (see text), (b) isoclinic map obtained by the RPT algorithm. In these images black corresponds to -π rad and white to π rad.

Fig. 11
Fig. 11

Profile of the isoclinic angle obtained by the PS and the RPT algorithms. The profiles have been taken along the dashed lines of Figs. 9 and 10(a), which corresponds to line 50.

Fig. 12
Fig. 12

Sine term of the isochromatics, Eq. (21), of the disk under diametral compression.

Fig. 13
Fig. 13

Modulation map, m δ Eq. (22), corresponding to the cosine and the sine terms shown in Figs. 9 and 12.

Fig. 14
Fig. 14

(a) Isochromatic phase map computed by the RPT technique. In this image black corresponds to -π rad and white to π rad. (b) Continuous isochromatic map computed by the RPT technique.

Equations (22)

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sinKα= Ii,
cosKα= Ij,
WKα=arctan Ii/ Ij,
δ=2π/λCσ1-σ2,
UT=x,yL Ux,yϕ, ωx, ωy,
Uxyϕ, ωx, ωy=ξηNL|fCξ, η-cosϕex, y, ξ, η|2+|fSξ, η-sinϕex, y, ξ, η|2+λ|ϕξ, η-ϕex, y, ξ, η|2mξ, η,
ϕex, y, ξ, η=ϕx, y+ωxx, yx-ξ+ωyx, yy-η,
Iα=IB+I0mα cos4α-β,
IB=I01-12 sin2δ2,
mα=12 sin2δ2,
IB=½Iα1+Iα2,
mI=Iα1-Iα22+2Iα3-IB21/2,
I0=IB+½ mI,
mα=½ mII0.
fαC=2mα cos4α=Iα1-Iα2I0,
fαS=2mα sin4α=2Iα3-I0I0.
Uxyϕ, ωx, ωy=ξηNL|fαCξ, η-cosϕex, y, ξ, η|2+|fαSξ, η-sinϕex, y, ξ, η|2+λ|W4πϕξ, η-ϕex, y, ξ, η|2mξ, η,
P90Q45Q45A-45: Iδ1=I021+cos 2α sin δ,
P-45Q90Q90A0: Iδ2=I021+sin 2α sin δ.
fδC=cos δ=1-4mα,
fδS=sin δ=2Iδ1-I0cos 2α+2Iδ2-I0sin 2αI0.
mδ=fδS2+fδC21/2.

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