Abstract

We propose a robust method for computing discontinuous phase maps from a fringe pattern with carrier frequency. Our algorithm is based on the minimization of an edge-preserving regularized cost function, specifically, on a robust regularized potential that uses a paradigm called the plate with adaptive rest condition, i.e., a second-order edge-preserving potential. Given that the proposed cost function is not convex, our method uses as its initial point an overly smoothed phase computed with a standard fringe analysis method and then reconstructs the phase discontinuities. Although the method is general purpose, it is introduced in the context of interferometric gauge-block calibration. The performance of the algorithm is demonstrated by numerical experiments with both synthetic and real data.

© 2006 Optical Society of America

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  1. J. L. Marroquin and M. Rivera, "Quadratic regularization functionals for phase unwrapping," J. Opt. Soc. Am. A 12, 2393-2400 (1995).
    [CrossRef]
  2. M. Rivera, J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, "Fast algorithm for integrating inconsistent gradient fields," Appl. Opt. 36, 8381-8390 (1997).
    [CrossRef]
  3. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
    [CrossRef]
  4. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  5. M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A 22, 1170-1175 (2005).
    [CrossRef]
  6. D. J. Pugh and K. Jackson, "Automatic gauge block measurement using multiple wavelength interferometer," in Contemporary Optical Instrument Design, Fabrication and Testing, H. Beckmann, J. Briers, and P. R. Yoder, eds., Proc. SPIE 656, 233-250 (1986).
  7. G. Boensch, "Gauge blocks as length standards measured by interferometry or comparison: length definition, traceability chain, and limitations," in Recent Developments in Optical Gauge Block Metrology, J. E. Decker and N. Brown, eds., Proc. SPIE 3477, 199-210 (1998).
    [CrossRef]
  8. G. Boensch, "Automatic gauge block measurement by phase stepping interferometry with three laser wavelengths," in Recent Developments in Traceable Dimensional Measurements, J. E. Decker and N. Brown, eds., Proc. SPIE 4401, 1-10 (2001).
    [CrossRef]
  9. J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, and M. Servin, "Regularization methods for processing fringe-pattern images," Appl. Opt. 38, 788-795 (1999).
    [CrossRef]
  10. J. L. Marroquin, J. E. Figueroa, and M. Servin, "Robust quadrature filter," J. Opt. Soc. Am. A 14, 779-791 (1997).
    [CrossRef]
  11. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, "Adaptive quadrature filters for multiphase stepping images," Opt. Lett. 23, 238-240 (1998).
    [CrossRef]
  12. M. Rivera and J. L. Marroquin, "Adaptive rest condition potentials: first and second order edge-preserving regularization," J. Comput. Vision Image Understand. 88, 76-93 (2002).
    [CrossRef]
  13. J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
    [CrossRef]
  14. J. Villa, J. A. Quiroga, and M. Servin, "Improved regularized phase tracking technique for the processing of squared-grating deflectograms," Appl. Opt. 39, 502-508 (2000).
    [CrossRef]
  15. R. Legarda-Saenz, W. Osten, and W. Juptner, "Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns," Appl. Opt. 41, 5519-5526 (2002).
    [CrossRef] [PubMed]
  16. M. Rivera and J. L. Marroquin, "Half-quadratic cost functions for phase unwrapping," Opt. Lett. 29, 504-506 (2004).
    [CrossRef] [PubMed]
  17. M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, "Phase unwrapping using a regularized phase tracking system," Appl. Opt. 37, 1917-1923 (1998).
    [CrossRef]
  18. M. Rivera, J. L. Marroquin, S. Botello, and M. Servin, "A robust spatio-temporal quadrature filter for multi-phase stepping," Appl. Opt. 39, 284-292 (2000).
    [CrossRef]
  19. S. Geman and D. E. McClure, "Bayesian image analysis methods: An application to single photon emission tomography," in Proceedings of the Statistical Computation Section (American Statistical Association, 1985), pp. 12-18.
  20. D. Geman and G. Reynolds, "Constrained restoration and recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1992).
    [CrossRef]
  21. D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Process. 4, 932-946, (1995).
    [CrossRef] [PubMed]
  22. M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vision 19, 57-91 (1996).
    [CrossRef]
  23. M. Rivera and J. L. Marroquin, "Efficient half-quadratic regularization with granularity control," J. Image Vision Comput. 21, 345-357 (2003).
    [CrossRef]
  24. M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.
  25. T. Tasdizen and R. Withaker, "Feature preserving variational smoothing of terrain data," in Proceedings of the Second IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM'03) (Institute of Electrical and Electronics Engineers, 2003), pp. 121-128.
  26. J. M. Huntley, "An image processing system for the analysis of speckle photographs," J. Phys. E 19, 43-49 (1986).
    [CrossRef]
  27. K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).
    [CrossRef]
  28. J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operational Research (Springer-Verlag, 1999).
    [CrossRef]

2005 (1)

2004 (1)

2003 (2)

T. Tasdizen and R. Withaker, "Feature preserving variational smoothing of terrain data," in Proceedings of the Second IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM'03) (Institute of Electrical and Electronics Engineers, 2003), pp. 121-128.

M. Rivera and J. L. Marroquin, "Efficient half-quadratic regularization with granularity control," J. Image Vision Comput. 21, 345-357 (2003).
[CrossRef]

2002 (3)

K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).
[CrossRef]

M. Rivera and J. L. Marroquin, "Adaptive rest condition potentials: first and second order edge-preserving regularization," J. Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

R. Legarda-Saenz, W. Osten, and W. Juptner, "Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns," Appl. Opt. 41, 5519-5526 (2002).
[CrossRef] [PubMed]

2001 (1)

G. Boensch, "Automatic gauge block measurement by phase stepping interferometry with three laser wavelengths," in Recent Developments in Traceable Dimensional Measurements, J. E. Decker and N. Brown, eds., Proc. SPIE 4401, 1-10 (2001).
[CrossRef]

2000 (2)

1999 (2)

1998 (3)

G. Boensch, "Gauge blocks as length standards measured by interferometry or comparison: length definition, traceability chain, and limitations," in Recent Developments in Optical Gauge Block Metrology, J. E. Decker and N. Brown, eds., Proc. SPIE 3477, 199-210 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, "Phase unwrapping using a regularized phase tracking system," Appl. Opt. 37, 1917-1923 (1998).
[CrossRef]

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, "Adaptive quadrature filters for multiphase stepping images," Opt. Lett. 23, 238-240 (1998).
[CrossRef]

1997 (2)

1996 (1)

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vision 19, 57-91 (1996).
[CrossRef]

1995 (2)

D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Process. 4, 932-946, (1995).
[CrossRef] [PubMed]

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

1994 (1)

M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.

1992 (1)

D. Geman and G. Reynolds, "Constrained restoration and recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1992).
[CrossRef]

1987 (1)

J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
[CrossRef]

1986 (2)

D. J. Pugh and K. Jackson, "Automatic gauge block measurement using multiple wavelength interferometer," in Contemporary Optical Instrument Design, Fabrication and Testing, H. Beckmann, J. Briers, and P. R. Yoder, eds., Proc. SPIE 656, 233-250 (1986).

J. M. Huntley, "An image processing system for the analysis of speckle photographs," J. Phys. E 19, 43-49 (1986).
[CrossRef]

1984 (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
[CrossRef]

Black, M. J.

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vision 19, 57-91 (1996).
[CrossRef]

Boensch, G.

G. Boensch, "Automatic gauge block measurement by phase stepping interferometry with three laser wavelengths," in Recent Developments in Traceable Dimensional Measurements, J. E. Decker and N. Brown, eds., Proc. SPIE 4401, 1-10 (2001).
[CrossRef]

G. Boensch, "Gauge blocks as length standards measured by interferometry or comparison: length definition, traceability chain, and limitations," in Recent Developments in Optical Gauge Block Metrology, J. E. Decker and N. Brown, eds., Proc. SPIE 3477, 199-210 (1998).
[CrossRef]

Botello, S.

Cuevas, F. J.

Figueroa, J. E.

Gasvik, K. J.

K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).
[CrossRef]

Geman, D.

D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Process. 4, 932-946, (1995).
[CrossRef] [PubMed]

D. Geman and G. Reynolds, "Constrained restoration and recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1992).
[CrossRef]

Geman, S.

S. Geman and D. E. McClure, "Bayesian image analysis methods: An application to single photon emission tomography," in Proceedings of the Statistical Computation Section (American Statistical Association, 1985), pp. 12-18.

Huntley, J. M.

J. M. Huntley, "An image processing system for the analysis of speckle photographs," J. Phys. E 19, 43-49 (1986).
[CrossRef]

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
[CrossRef]

Jackson, K.

D. J. Pugh and K. Jackson, "Automatic gauge block measurement using multiple wavelength interferometer," in Contemporary Optical Instrument Design, Fabrication and Testing, H. Beckmann, J. Briers, and P. R. Yoder, eds., Proc. SPIE 656, 233-250 (1986).

Juptner, W.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
[CrossRef]

Legarda-Saenz, R.

Malacara, D.

Marroquin, J. L.

M. Rivera and J. L. Marroquin, "Half-quadratic cost functions for phase unwrapping," Opt. Lett. 29, 504-506 (2004).
[CrossRef] [PubMed]

M. Rivera and J. L. Marroquin, "Efficient half-quadratic regularization with granularity control," J. Image Vision Comput. 21, 345-357 (2003).
[CrossRef]

M. Rivera and J. L. Marroquin, "Adaptive rest condition potentials: first and second order edge-preserving regularization," J. Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

M. Rivera, J. L. Marroquin, S. Botello, and M. Servin, "A robust spatio-temporal quadrature filter for multi-phase stepping," Appl. Opt. 39, 284-292 (2000).
[CrossRef]

J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, and M. Servin, "Regularization methods for processing fringe-pattern images," Appl. Opt. 38, 788-795 (1999).
[CrossRef]

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, "Adaptive quadrature filters for multiphase stepping images," Opt. Lett. 23, 238-240 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, "Phase unwrapping using a regularized phase tracking system," Appl. Opt. 37, 1917-1923 (1998).
[CrossRef]

J. L. Marroquin, J. E. Figueroa, and M. Servin, "Robust quadrature filter," J. Opt. Soc. Am. A 14, 779-791 (1997).
[CrossRef]

M. Rivera, J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, "Fast algorithm for integrating inconsistent gradient fields," Appl. Opt. 36, 8381-8390 (1997).
[CrossRef]

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

J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
[CrossRef]

McClure, D. E.

S. Geman and D. E. McClure, "Bayesian image analysis methods: An application to single photon emission tomography," in Proceedings of the Statistical Computation Section (American Statistical Association, 1985), pp. 12-18.

Mitter, S.

J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
[CrossRef]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operational Research (Springer-Verlag, 1999).
[CrossRef]

Osten, W.

Poggio, T.

J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
[CrossRef]

Pouwels, A. E.

M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.

Proesmans, M.

M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.

Pugh, D. J.

D. J. Pugh and K. Jackson, "Automatic gauge block measurement using multiple wavelength interferometer," in Contemporary Optical Instrument Design, Fabrication and Testing, H. Beckmann, J. Briers, and P. R. Yoder, eds., Proc. SPIE 656, 233-250 (1986).

Quiroga, J. A.

Rangarajan, A.

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vision 19, 57-91 (1996).
[CrossRef]

Reynolds, G.

D. Geman and G. Reynolds, "Constrained restoration and recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1992).
[CrossRef]

Rivera, M.

Rodriguez-Vera, R.

Servin, M.

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
[CrossRef]

Tasdizen, T.

T. Tasdizen and R. Withaker, "Feature preserving variational smoothing of terrain data," in Proceedings of the Second IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM'03) (Institute of Electrical and Electronics Engineers, 2003), pp. 121-128.

Van Gool, L.

M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.

Villa, J.

Withaker, R.

T. Tasdizen and R. Withaker, "Feature preserving variational smoothing of terrain data," in Proceedings of the Second IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM'03) (Institute of Electrical and Electronics Engineers, 2003), pp. 121-128.

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Wright, S.

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operational Research (Springer-Verlag, 1999).
[CrossRef]

Yang, C.

D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Process. 4, 932-946, (1995).
[CrossRef] [PubMed]

Appl. Opt. (6)

IEEE Trans. Image Process. (1)

D. Geman and C. Yang, "Nonlinear image recovery with half-quadratic regularization," IEEE Trans. Image Process. 4, 932-946, (1995).
[CrossRef] [PubMed]

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

D. Geman and G. Reynolds, "Constrained restoration and recovery of discontinuities," IEEE Trans. Pattern Anal. Mach. Intell. 14, 367-383 (1992).
[CrossRef]

Int. J. Comput. Vision (1)

M. J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vision 19, 57-91 (1996).
[CrossRef]

J. Am. Statist. Assoc. (1)

J. L. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision," J. Am. Statist. Assoc. 82, 76-89 (1987).
[CrossRef]

J. Comput. Vision Image Understand. (1)

M. Rivera and J. L. Marroquin, "Adaptive rest condition potentials: first and second order edge-preserving regularization," J. Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

J. Image Vision Comput. (1)

M. Rivera and J. L. Marroquin, "Efficient half-quadratic regularization with granularity control," J. Image Vision Comput. 21, 345-357 (2003).
[CrossRef]

J. Opt. Soc. Am (1)

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am 4, 156-160 (1982).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Phys. E (1)

J. M. Huntley, "An image processing system for the analysis of speckle photographs," J. Phys. E 19, 43-49 (1986).
[CrossRef]

Opt. Eng. (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Opt. Lett. (2)

Proc. SPIE (3)

D. J. Pugh and K. Jackson, "Automatic gauge block measurement using multiple wavelength interferometer," in Contemporary Optical Instrument Design, Fabrication and Testing, H. Beckmann, J. Briers, and P. R. Yoder, eds., Proc. SPIE 656, 233-250 (1986).

G. Boensch, "Gauge blocks as length standards measured by interferometry or comparison: length definition, traceability chain, and limitations," in Recent Developments in Optical Gauge Block Metrology, J. E. Decker and N. Brown, eds., Proc. SPIE 3477, 199-210 (1998).
[CrossRef]

G. Boensch, "Automatic gauge block measurement by phase stepping interferometry with three laser wavelengths," in Recent Developments in Traceable Dimensional Measurements, J. E. Decker and N. Brown, eds., Proc. SPIE 4401, 1-10 (2001).
[CrossRef]

Other (5)

S. Geman and D. E. McClure, "Bayesian image analysis methods: An application to single photon emission tomography," in Proceedings of the Statistical Computation Section (American Statistical Association, 1985), pp. 12-18.

K. J. Gasvik, Optical Metrology, 3rd ed. (Wiley, 2002).
[CrossRef]

J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operational Research (Springer-Verlag, 1999).
[CrossRef]

M. Proesmans, A. E. Pouwels, and L. Van Gool, "Couple geometry-driven diffusion equations for low level vision," in Geometry-Driven Diffusion in Computer Vision, B.M.ter Haar-Romeny, ed. (Kluwer Academic, 1994), pp. 191-228.

T. Tasdizen and R. Withaker, "Feature preserving variational smoothing of terrain data," in Proceedings of the Second IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM'03) (Institute of Electrical and Electronics Engineers, 2003), pp. 121-128.

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Figures (5)

Fig. 1
Fig. 1

Interferograms of a GB on a steel plate: (a) tungsten carbide, (b) steel.

Fig. 2
Fig. 2

Cliques of pixels in the following positions: (a) horizontal, (b) vertical, and (c), (d) diagonal.

Fig. 3
Fig. 3

(a) Synthetic one-dimensional data: phase (continuous curve), background (dashed curve), and contrast (dotted curve). Results computed with (b) a quadratic thin-plate potential, (c) a robust membrane potential, (d) a PARC potential. The original data were corrupted with a white-noise signal with a mean equal to zero and a standard deviation of 0.5 rad.

Fig. 4
Fig. 4

Real data from the TESA interferometer at CENAM. (a) Central column of the interferogram in Fig. 1. (b) Results computed with the proposed ADFP algorithm: phase (dotted curve), illumination components (solid curves), and edges (markers).

Fig. 5
Fig. 5

Two-dimensional fringe-pattern analysis with the proposed ADFP algorithm: (a) phase map, (b) background illumination, (c) contrast component, (d) detected edge map.

Tables (1)

Tables Icon

Table 1 L 2 Error in Results and Computational Times Computed with Three Potentials

Equations (18)

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g r = a ^ r + b ^ r cos ( ω ^ T r + ϕ ^ r ) + η r ,
U ( a , b , ϕ , l ; g , ω ) = D ( a , b , ϕ ;     g , ω ) + λ R ( a , b , ϕ , l ) ,
D ( a , b , ϕ ;    g , ω ) = r L [ g r a r b r cos ( ω T r + ϕ r ) ] 2 .
R ( ϕ , l ) = r s [ ( ϕ r ϕ s ) 2 l r s       2 + Φ ( l r s ) ] ,
R ( f ) = q r s ( ϕ q 2 ϕ r + ϕ s ) 2 .
R ( ϕ , l ) = q r s [ ( ϕ q 2 ϕ r + ϕ s ) 2 l q r s           2 + Φ ( l q r s ) ] ,
R 3 ( ϕ , l ) = r q s ρ q r s ( ϕ , l ) ,
ρ q r s ( ϕ , l ) = ( l q r Δ ϕ q r l r s Δ ϕ r s ) 2 + μ [ ( 1 l q r ) 2 + ( 1 l r s ) 2 ] ,
U ( a , b , ϕ , l ) = r [ g r a r b r cos ( ω T r + ϕ r ) ] 2 + q , r , s { λ a ( l q r Δ a q r l r s Δ a r s ) 2 + λ b ( l q r Δ b q r l r s Δ b r s ) 2 + λ c ( l q r Δ ϕ q r l r s Δ ϕ r s ) 2 + μ [ ( 1 l q r ) 2 + ( 1 l r s ) 2 ] } ,
ϕ i + 1 k + 1 = ϕ i k + 1 α ϕ U ( a k , b k , ϕ i     k + 1 , l k ) ,
U ( a k , b k , ϕ k , l k ) U ( a k , b k , ϕ k + 1 , l k ) U ( a k + 1 , b k , ϕ k + 1 , l k ) U ( a k + 1 , b k + 1 , ϕ k + 1 , l k ) U ( a k + 1 , b k + 1 , ϕ k + 1 , l k + 1 ) 0
a r = { 0.6 i f 97 r 165 ;       1.0 o t h e r w i s e } ,
b r = { 0.5 i f 97 r 165 ;       0.9 o t h e r w i s e } ,
ϕ r = { 1.2 + 0.005 r i f 97 r 165 ,
0.2 + 0.005 r       o t h e r w i s e } .
U 1 d ( a , b , ϕ , l ) = Q ( a , b , ϕ ) + i = 2 N 1 { λ a ( l i Δ a i l i 1 Δ a i 1 ) 2 + λ b ( l i Δ b i l i 1 Δ b i 1 ) 2 + λ c ( l i Δ ϕ i l i 1 Δ ϕ i 1 ) 2 + μ [ ( 1 l i ) 2 + ( 1 l i 1 ) 2 ] } ,
1 2 U 1 d ( a , b , ϕ , l ) l i = 0 ,
λ a Δ a i [ ( l i Δ a i l i + 1 Δ a i + 1 ) W ( i + 1 ) + λ a ( l i Δ a i l i 1 Δ a i 1 ) ] + λ b Δ b i [ ( l i Δ b i l i + 1 Δ b i + 1 ) W ( i + 1 ) + λ b ( l i Δ b i l i 1 Δ b i 1 ) ] + λ c Δ ϕ i [ ( l i Δ ϕ i l i + 1 Δ ϕ i + 1 ) W ( i + 1 ) + λ c ( l i Δ ϕ i l i 1 Δ ϕ i 1 ) ] + μ ( l i 1 ) = 0 ,

Metrics