Abstract

A powerful technique for processing fringe-pattern images is based on Bayesian estimation theory with prior Markov random-field models. In this approach the solution of a processing problem is characterized as the minimizer of a cost function with terms that specify that the solution should be compatible with the available observations and terms that impose certain (prior) constraints on the solution. We show that, by the appropriate choice of these terms, one can use this approach in almost every processing step for accurate and robust interferogram demodulation and phase unwrapping.

© 1999 Optical Society of America

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References

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  1. S. Geman, D. Geman, “Stochastic relaxation, gibbs distributions and the bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
    [CrossRef] [PubMed]
  2. J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
    [CrossRef]
  3. J. L. Marroquin, M. Servin, J. E. Figueroa, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
    [CrossRef]
  4. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
    [CrossRef]
  5. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1544 (1998).
    [CrossRef]
  6. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
    [CrossRef]
  7. M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “A fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
    [CrossRef]
  8. J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, M. Servin, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [CrossRef]
  9. J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
    [CrossRef]
  10. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  11. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  12. G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).
  13. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vision Graph. Image Process. 55, 408–417 (1993).

1998 (2)

1997 (3)

1995 (2)

1994 (1)

1993 (1)

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vision Graph. Image Process. 55, 408–417 (1993).

1987 (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, gibbs distributions and the bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

Botello, S.

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

Figueroa, J. E.

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, gibbs distributions and the bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, gibbs distributions and the bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

Ghiglia, D. C.

Gollub, G. H.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Marroquin, J.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Marroquin, J. L.

J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1544 (1998).
[CrossRef]

J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
[CrossRef]

J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
[CrossRef]

J. L. Marroquin, M. Servin, J. E. Figueroa, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
[CrossRef]

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “A fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
[CrossRef]

J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, M. Servin, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
[CrossRef]

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

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vision Graph. Image Process. 55, 408–417 (1993).

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

Mitter, S.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Poggio, T.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Rivera, M.

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “A fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
[CrossRef]

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

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

Rodriguez-Vera, R.

Romero, L. A.

Servin, M.

Tapia, M.

Van Loan, C. F.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Appl. Opt. (1)

Comput. Vision Graph. Image Process. (1)

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vision Graph. Image Process. 55, 408–417 (1993).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, gibbs distributions and the bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

J. Am. Stat. Assoc. (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Other (2)

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) ESPI image of a metallic plate subjected to thermal deformation, (b) demodulated (wrapped) phase, (c) unwrapped phase.

Fig. 2
Fig. 2

(a)–(c) Synthetic multiple phase-stepping images generated with (d) the wrapped phase. (e) Phase recovered with the regularization method. (f) Phase recovered with the standard technique (filtering plus least squares).

Fig. 3
Fig. 3

(a) Unwrapped and (b) wrapped nosy quadratic phase maps. (c) Wrapped phase of (a) unwrapped with the method discussed in Subsection 1.E that corresponds to the cost function given by Eq. (9). (d) Phase map of (c) rewrapped to show the reduced dynamic range. (e) Phase of (b) unwrapped with the method discussed in Subsection 1.E that corresponds to the cost function given by Eq. (10) and (f) rewrapped to show the reconstruction accuracy.

Fig. 4
Fig. 4

(a) Image of a mechanical part obtained by the superposition of three ESPI patterns. (b) Mask obtained by the regularized segmentation procedure described in the text.

Tables (1)

Tables Icon

Table 1 Data and Regularization Terms for the Cost Functions Used for Different Operations in Fringe-Pattern Processing

Equations (24)

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Uf=rLfr-gr2+λ r,sLfr-fs2.
φr=Arcosω0·r+αr.
φsArcosω0·s+αr,
ψrArsinω0·r+αr.
Uf=rL |fr-2gr|2+λ r,sLfrexpi2 ω0·s-r-fsexpi2 ω0·r-s2,
ϕr=arctanImfrRefr-ω0·r.
Uf, ω=rL |fr - 2gr|2+λ r,sLfrexpi2 ωr·s-r-fsexpi2 ωs·r-s2+μ r,s |ωr-ωs|2,
ωr=ρrcos θˆr, ρrsin θˆr.
Uf=rLc2r-cos2θgr2+λ r,sLc2r-c2s2,
Up=r,sL1+crspr-ps2+1-crs×pr+ps-12+μpr0-12,
θˆr=θr,  if pr<0.5=θr+π,  if pr0.5.
ωkr=qk cos θˆr, qk sin θˆr.
rLk=1N |frexpiαk-2gkr|2.
αk=arctan-xLImfxgkxxLRefxgkx.
gkx=axcosϕx+nx+αk,  k=1, 2, 3,
gxx, y=gx, y-gx-1, y+2πkx,gyx, y=gx, y-gx, y-1+2πky,
Dfx, fy=rLfxr-gxr2+fyr-gyr2.
Rfx, fy=rLΔxfyr-Δyfxr2,
Ufx, fy=Dfx, fy+λRfx, fy.
Uz=r,sLgr+2πzr-gs+2πzs2+λzr-zs-Rndzr-zs2,
ϕx, y=a-0.0006128-x2+128-y2+n,
Up=rL |pr-pˆr|2+λ r, s |pr-ps|2,
pˆrk=1Zexp-βgr-qk2,
kprk-1Zexp-γgr-qk22

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