Abstract

A well-founded and computationally fast method is presented for filtering and interpolating noisy and discontinuous wrapped phase fields that preserves both the 2π discontinuities that come from the wrapping effect and the true discontinuities that may be present. It also permits the incorporation of an associated quality map, if it is available, in a natural way. Examples of its application to the computation of the isoclinic phase from photoelastic data and to the recovery of discontinuous phase fields from speckle interferometry are presented.

© 1999 Optical Society of America

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References

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  1. N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, 1993).
    [CrossRef]
  2. J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
    [CrossRef]
  3. J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
    [CrossRef]
  4. D. C. Ghiglia and L. A. Romero, J. Opt. Soc. Am. A 11, 107 (1994).
    [CrossRef]
  5. S. Botello, J. L. Marroquin, and M. Rivera, Appl. Opt. 37, 7587 (1998).
    [CrossRef]
  6. P. S. Theocaris and E. E. Gdoutos, Matrix Methods in Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]
  7. J. A. Quiroga and A. Gonzalez-Cano, Proc. SPIE 3744, 328 (1999).
  8. N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).
  9. MathWorks, “Matlab function reference,” on-line version (MathWorks, Inc., 1998; http://www.mathworks.com ).

1999

J. A. Quiroga and A. Gonzalez-Cano, Proc. SPIE 3744, 328 (1999).

N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).

1998

J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
[CrossRef]

J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
[CrossRef]

S. Botello, J. L. Marroquin, and M. Rivera, Appl. Opt. 37, 7587 (1998).
[CrossRef]

1994

Botello, S.

J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
[CrossRef]

S. Botello, J. L. Marroquin, and M. Rivera, Appl. Opt. 37, 7587 (1998).
[CrossRef]

Dupré, J. C.

N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).

Fisher, N. I.

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, 1993).
[CrossRef]

Gdoutos, E. E.

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

Ghiglia, D. C.

Gonzalez-Cano, A.

J. A. Quiroga and A. Gonzalez-Cano, Proc. SPIE 3744, 328 (1999).

Lagarde, A.

N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).

Marroquin, J. L.

S. Botello, J. L. Marroquin, and M. Rivera, Appl. Opt. 37, 7587 (1998).
[CrossRef]

J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
[CrossRef]

J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
[CrossRef]

Plouzenec, N.

N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).

Quiroga, J. A.

J. A. Quiroga and A. Gonzalez-Cano, Proc. SPIE 3744, 328 (1999).

Rivera, M.

S. Botello, J. L. Marroquin, and M. Rivera, Appl. Opt. 37, 7587 (1998).
[CrossRef]

J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
[CrossRef]

Rodriquez-Vera, R.

J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
[CrossRef]

Romero, L. A.

Servin, M.

J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
[CrossRef]

Theocaris, P. S.

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

Appl. Opt.

Exp. Tech.

N. Plouzenec, J. C. Dupré, and A. Lagarde, Exp. Tech. 23(1), 30 (1999).

J. Opt. Soc. Am. A

J. L. Marroquin, R. Rodriquez-Vera, and M. Servin, J. Opt. Soc. Am. A 15, 6 (1998).
[CrossRef]

D. C. Ghiglia and L. A. Romero, J. Opt. Soc. Am. A 11, 107 (1994).
[CrossRef]

Proc. SPIE

J. L. Marroquin, M. Rivera, and S. Botello, Proc. SPIE 3459, 238 (1998).
[CrossRef]

J. A. Quiroga and A. Gonzalez-Cano, Proc. SPIE 3744, 328 (1999).

Other

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, 1993).
[CrossRef]

MathWorks, “Matlab function reference,” on-line version (MathWorks, Inc., 1998; http://www.mathworks.com ).

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

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Figures (2)

Fig. 1
Fig. 1

a, b, Ic and Is [Eqs. (6) and (7), respectively] for a diamentrically compressed disk. c, Modulation map mα. d, Observed phase computed as the arctangent of the quotient of b and a. e, Phase reconstructed with the method presented here. f, Phase reconstructed with a conventional interpolation method (see text).

Fig. 2
Fig. 2

a, Metallic harness. b, Phase map obtained by temporal phase stepping from four speckle interferograms of a subject to thermal deformation. c, Phase field reconstructed with the method presented here. d, Black pixels indicate phase jumps smaller than 2π.

Equations (11)

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ϕˆr=arctanIsrIcr
pˆrqProbe(ϕˆr|ϕr=q)=1Zexpmrcosϕˆr-q,
Up=rLpr-pˆr2+λr,spr-ps2,
Up=k=1MUkp,
Ukp=rLprk-pˆrk2+λr,sprk-psk2.
prk=DCT-1ρkr,
ρku,v=DCTpˆku,v1+2λ2-cosπu/m-cosπu/m,
Icr=2mαrcos4αr=Iα1r-Iα2rI0r,
Isr=2mαrsin4αr=2Iα3r-I0rI0r,
I0r=1/2IBr+Iα1r-Iα2r2+2Iα3r-IBr21/2,
mαr=Iα1r-Iα2r2+2Iα3r-IBr2½2I0r.

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