Abstract

In this paper, a multiscale monogenic analysis is applied to 2D interference fringe patterns. The monogenic signal was originally developed as a 2D generalization of the well-known analytic signal in the 1D case. The analytic and monogenic tools are both useful to extract phase information, which can then be directly linked with physical quantities. Previous studies have already shown the interest in the monogenic signal in the field of interferometry. This paper presents theoretical and numerical illustrations of the connection between the physical phase information and the phase estimated with the monogenic tool. More specifically, the ideal case of pure cosine waves is deeply studied, and then the complexity of the fringe patterns is progressively increased. One important weakness of the monogenic transform is its singularity at the null frequency, which makes the phase estimations of low-frequency fringes diverge. Moreover, the monogenic transform is originally designed for narrowband signals, and encounters difficulties when dealing with noised signals. These problems can be bypassed by performing a multiscale analysis based on the monogenic wavelet transform. Moreover, this paper proposes a simple strategy to combine the information extracted at different scales in order to get a better estimation of the phase. The numerical tests (synthetic and real signals) show how this approach provides a finer extraction of the geometrical structure of the fringe patterns.

© 2019 Optical Society of America

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References

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  1. W. H. Steel, Interferometry, 2nd ed., Cambridge Studies in Modern Optics 1 (Cambridge University, 1986).
  2. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
    [Crossref]
  3. D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (CRC Press, 1993).
  4. E. Robin, V. Valle, and F. Brémand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44, 7261–7269 (2005).
    [Crossref]
  5. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
    [Crossref]
  6. J. M. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Bragaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [Crossref]
  7. Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
    [Crossref]
  8. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
    [Crossref]
  9. M. Felsberg, “Low-level image processing with the structure multivector,” Ph.D. thesis (Kiel University, 2002).
  10. B. Picinbono, Time-Frequency Analysis (ISTE and Wiley, 2008).
  11. M. Unser, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
    [Crossref]
  12. S. Olhede and G. Metikas, “Multiple monogenic Morse wavelets,” IEEE Transactions on Signal Processing 55, 921–936 (2007).
    [Crossref]
  13. J. Cnops, “The wavelet transform in Clifford analysis,” Comput. Methods Funct. Theory 1, 353–374 (2001).
    [Crossref]
  14. S. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Signal Process. 57, 3426–3441 (2009).
    [Crossref]
  15. R. Soulard and P. Carré, “Elliptical monogenic wavelets for the analysis and processing of color images,” IEEE Trans. Signal Process. 66, 1535–1549 (2015).
    [Crossref]
  16. M. Johansson, “The Hilbert transform,” Ph.D. thesis (Växjö University, 2013).
  17. B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal-part I: fundamentals,” Proc. IEEE 80, 519–538 (1992).
    [Crossref]
  18. W. Burger and M. J. Burge, Digital Image Processing (Springer, 2008).
  19. R. Soulard and P. Carré, “Characterization of color images with multiscale monogenic maxima,” IEEE Trans. Pattern Anal. Mach. Intell. 40, 2289–2302 (2017).
    [Crossref]
  20. P. Carré and R. Soulard, “Color image processing with the elliptical monogenic wavelets representation,” http://xlim-sic.labo.univ-poitiers.fr/projets/colormonogenic/chap1b.php .
  21. J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
    [Crossref]
  22. M. Unser and N. Chenouard, “A unifying parametric framework for 2D steerable wavelet transforms,” SIAM J. Imaging Sci. 6, 102–135 (2013).
    [Crossref]
  23. D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
    [Crossref]
  24. L. Ljung, System Identification Toolbox for use with MATLAB (2011).
  25. M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
  26. K. Polisano, “Modélisation de textures anisotropes par la transformée en ondelettes monogènes et super-résolution de lignes 2-d,” Ph.D. thesis (Université de Grenoble Alpes, 2017).
  27. F. Brémand, “A phase unwrapping technique for object relief determination,” Opt. Laser Eng. 21, 49–60 (1994).
    [Crossref]
  28. M. Arevallilo-Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [Crossref]

2017 (1)

R. Soulard and P. Carré, “Characterization of color images with multiscale monogenic maxima,” IEEE Trans. Pattern Anal. Mach. Intell. 40, 2289–2302 (2017).
[Crossref]

2015 (1)

R. Soulard and P. Carré, “Elliptical monogenic wavelets for the analysis and processing of color images,” IEEE Trans. Signal Process. 66, 1535–1549 (2015).
[Crossref]

2013 (1)

M. Unser and N. Chenouard, “A unifying parametric framework for 2D steerable wavelet transforms,” SIAM J. Imaging Sci. 6, 102–135 (2013).
[Crossref]

2012 (1)

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

2009 (2)

M. Unser, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[Crossref]

S. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Signal Process. 57, 3426–3441 (2009).
[Crossref]

2007 (1)

S. Olhede and G. Metikas, “Multiple monogenic Morse wavelets,” IEEE Transactions on Signal Processing 55, 921–936 (2007).
[Crossref]

2005 (3)

E. Robin, V. Valle, and F. Brémand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44, 7261–7269 (2005).
[Crossref]

D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
[Crossref]

J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

2002 (1)

2001 (2)

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[Crossref]

J. Cnops, “The wavelet transform in Clifford analysis,” Comput. Methods Funct. Theory 1, 353–374 (2001).
[Crossref]

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

1994 (1)

F. Brémand, “A phase unwrapping technique for object relief determination,” Opt. Laser Eng. 21, 49–60 (1994).
[Crossref]

1992 (1)

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal-part I: fundamentals,” Proc. IEEE 80, 519–538 (1992).
[Crossref]

1989 (1)

Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
[Crossref]

1974 (1)

Arevallilo-Herráez, M.

Blu, T.

D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
[Crossref]

Boashash, B.

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal-part I: fundamentals,” Proc. IEEE 80, 519–538 (1992).
[Crossref]

Bone, D. J.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[Crossref]

Bragaccio, D. J.

Brémand, F.

Bruning, J. M.

Burge, M. J.

W. Burger and M. J. Burge, Digital Image Processing (Springer, 2008).

Burger, W.

W. Burger and M. J. Burge, Digital Image Processing (Springer, 2008).

Burton, D. R.

Carré, P.

R. Soulard and P. Carré, “Characterization of color images with multiscale monogenic maxima,” IEEE Trans. Pattern Anal. Mach. Intell. 40, 2289–2302 (2017).
[Crossref]

R. Soulard and P. Carré, “Elliptical monogenic wavelets for the analysis and processing of color images,” IEEE Trans. Signal Process. 66, 1535–1549 (2015).
[Crossref]

Chenouard, N.

M. Unser and N. Chenouard, “A unifying parametric framework for 2D steerable wavelet transforms,” SIAM J. Imaging Sci. 6, 102–135 (2013).
[Crossref]

Cnops, J.

J. Cnops, “The wavelet transform in Clifford analysis,” Comput. Methods Funct. Theory 1, 353–374 (2001).
[Crossref]

Deléchelle, E.

J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Depeursinge, C.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

Felsberg, M.

M. Felsberg, “Low-level image processing with the structure multivector,” Ph.D. thesis (Kiel University, 2002).

Gallagher, J. E.

Gdeisat, M. A.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

Guyot, S.

J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Herriott, D. R.

Higashi, T.

Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
[Crossref]

Johansson, M.

M. Johansson, “The Hilbert transform,” Ph.D. thesis (Växjö University, 2013).

Lalor, M. J.

Larkin, K. G.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[Crossref]

Ljung, L.

L. Ljung, System Identification Toolbox for use with MATLAB (2011).

Metikas, G.

S. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Signal Process. 57, 3426–3441 (2009).
[Crossref]

S. Olhede and G. Metikas, “Multiple monogenic Morse wavelets,” IEEE Transactions on Signal Processing 55, 921–936 (2007).
[Crossref]

Morimoto, Y.

Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
[Crossref]

Nunes, J. C.

J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Oldfield, M. A.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[Crossref]

Olhede, S.

S. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Signal Process. 57, 3426–3441 (2009).
[Crossref]

S. Olhede and G. Metikas, “Multiple monogenic Morse wavelets,” IEEE Transactions on Signal Processing 55, 921–936 (2007).
[Crossref]

Pavillon, N.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

Picinbono, B.

B. Picinbono, Time-Frequency Analysis (ISTE and Wiley, 2008).

Polisano, K.

K. Polisano, “Modélisation de textures anisotropes par la transformée en ondelettes monogènes et super-résolution de lignes 2-d,” Ph.D. thesis (Université de Grenoble Alpes, 2017).

Reid, G. T.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (CRC Press, 1993).

Robin, E.

Robinson, D. W.

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (CRC Press, 1993).

Rosenfeld, D. P.

Seelamantula, C. S.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

Seguchi, Y.

Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
[Crossref]

Soulard, R.

R. Soulard and P. Carré, “Characterization of color images with multiscale monogenic maxima,” IEEE Trans. Pattern Anal. Mach. Intell. 40, 2289–2302 (2017).
[Crossref]

R. Soulard and P. Carré, “Elliptical monogenic wavelets for the analysis and processing of color images,” IEEE Trans. Signal Process. 66, 1535–1549 (2015).
[Crossref]

Steel, W. H.

W. H. Steel, Interferometry, 2nd ed., Cambridge Studies in Modern Optics 1 (Cambridge University, 1986).

Unser, M.

M. Unser and N. Chenouard, “A unifying parametric framework for 2D steerable wavelet transforms,” SIAM J. Imaging Sci. 6, 102–135 (2013).
[Crossref]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

M. Unser, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[Crossref]

D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
[Crossref]

Valle, V.

VandeVille, D.

D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
[Crossref]

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Verhaegen, M.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

White, A. D.

Appl. Opt. (3)

Comput. Methods Funct. Theory (1)

J. Cnops, “The wavelet transform in Clifford analysis,” Comput. Methods Funct. Theory 1, 353–374 (2001).
[Crossref]

Exp. Mech. (1)

Y. Morimoto, Y. Seguchi, and T. Higashi, “Two-dimensional moiré method and grid method using Fourier transform,” Exp. Mech. 29, 399–404 (1989).
[Crossref]

IEEE Trans. Image Process. (1)

M. Unser, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[Crossref]

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

R. Soulard and P. Carré, “Characterization of color images with multiscale monogenic maxima,” IEEE Trans. Pattern Anal. Mach. Intell. 40, 2289–2302 (2017).
[Crossref]

IEEE Trans. Signal Process. (3)

D. VandeVille, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Trans. Signal Process. 14, 1798–7813 (2005).
[Crossref]

S. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Signal Process. 57, 3426–3441 (2009).
[Crossref]

R. Soulard and P. Carré, “Elliptical monogenic wavelets for the analysis and processing of color images,” IEEE Trans. Signal Process. 66, 1535–1549 (2015).
[Crossref]

IEEE Transactions on Signal Processing (1)

S. Olhede and G. Metikas, “Multiple monogenic Morse wavelets,” IEEE Transactions on Signal Processing 55, 921–936 (2007).
[Crossref]

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

J. Opt. Soc. Am. (2)

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[Crossref]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Local demodulation of holograms using the Riesz transform with application to microscopy,” J. Opt. Soc. Am. 29, 2118–2129 (2012).
[Crossref]

Mach. Vis. Appl. (1)

J. C. Nunes, S. Guyot, and E. Deléchelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Opt. Laser Eng. (1)

F. Brémand, “A phase unwrapping technique for object relief determination,” Opt. Laser Eng. 21, 49–60 (1994).
[Crossref]

Proc. IEEE (1)

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal-part I: fundamentals,” Proc. IEEE 80, 519–538 (1992).
[Crossref]

SIAM J. Imaging Sci. (1)

M. Unser and N. Chenouard, “A unifying parametric framework for 2D steerable wavelet transforms,” SIAM J. Imaging Sci. 6, 102–135 (2013).
[Crossref]

Other (10)

P. Carré and R. Soulard, “Color image processing with the elliptical monogenic wavelets representation,” http://xlim-sic.labo.univ-poitiers.fr/projets/colormonogenic/chap1b.php .

L. Ljung, System Identification Toolbox for use with MATLAB (2011).

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

K. Polisano, “Modélisation de textures anisotropes par la transformée en ondelettes monogènes et super-résolution de lignes 2-d,” Ph.D. thesis (Université de Grenoble Alpes, 2017).

W. H. Steel, Interferometry, 2nd ed., Cambridge Studies in Modern Optics 1 (Cambridge University, 1986).

W. Burger and M. J. Burge, Digital Image Processing (Springer, 2008).

M. Johansson, “The Hilbert transform,” Ph.D. thesis (Växjö University, 2013).

M. Felsberg, “Low-level image processing with the structure multivector,” Ph.D. thesis (Kiel University, 2002).

B. Picinbono, Time-Frequency Analysis (ISTE and Wiley, 2008).

D. W. Robinson and G. T. Reid, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (CRC Press, 1993).

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Figures (13)

Fig. 1.
Fig. 1. Phase estimation of a 2D cosine wave.
Fig. 2.
Fig. 2. Phase estimation of a 2D cosine wave at different scales. Phase (left) and amplitude (right).
Fig. 3.
Fig. 3. Influence of the $ a $ parameter on the quality of the estimation.
Fig. 4.
Fig. 4. Phase estimation of a 2D parabolic chirp.
Fig. 5.
Fig. 5. Phase estimation of a parabolic chirp at different scales. Phase (left) and amplitude (right).
Fig. 6.
Fig. 6. Original image (left), straightforward monogenic phase (center), and multiscale monogenic phase (right).
Fig. 7.
Fig. 7. Influence of noise on the quality of the estimation (cosine wave).
Fig. 8.
Fig. 8. Influence of noise on the quality of the estimation (parabolic chirp).
Fig. 9.
Fig. 9. Theoretical (top left), pMPC (top right), straightforward monogenic (bottom left), and multiscale monogenic phase (bottom right).
Fig. 10.
Fig. 10. (top) pMPC, (middle) straightforward monogenic, and (bottom) multiscale monogenic.
Fig. 11.
Fig. 11. Original image (top left), pMPC (top right), straightforward monogenic (bottom left), and multiscale monogenic phase (bottom right).
Fig. 12.
Fig. 12. pMPC (left), straightforward monogenic (center), and multiscale monogenic phase (right).
Fig. 13.
Fig. 13. Original image (left), straightforward monogenic (center), and multiscale monogenic phase (right).

Tables (1)

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Table 1. Similarity between Theoretical and Computed Phases

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

f ( x ) = a ( x ) + b ( x ) cos ( φ ( x ) ) , x = ( x 1 , x 2 ) R 2 ,
f ( t ) = b ( t ) cos ( φ ( t ) ) , t R ,
F { H f } ( ω ) = H ( ω ) F { f } ( ω ) , ω R ,
f ( x ) = b ( x ) cos ( φ ( x ) ) , x R 2 .
F { R s f } ( ω 1 , ω 2 ) = j ω s ω 1 2 + ω 2 2 F { f } ( ω 1 , ω 2 ) ,
f ( x ) = b ~ ( x ) cos ( φ ~ ( x ) ) ,
ψ R F j ω 1 + ω 2 ω ψ ^ ( ω ) .
c i , m = s , ψ i , m = ( ψ i s ) ( 2 ( i + 1 ) m ) ,
d i , m = s , ψ i , m R = { R ( ψ i s ) } ( 2 ( i + 1 ) m ) ,
ϕ j , k = ϕ j , k ( i 0 ) , w i t h i 0 = arg max i = 1 , , L { a j , k ( i ) } .
{ B F T ( φ , ϕ ) = max { 100 ( 1 cos ϕ cos φ 2 cos φ cos φ ¯ 2 ) , 0 } , V A F ( φ , ϕ ) = max { ( 100 ( 1 V ( cos ϕ cos φ ) V ( cos φ ) ) , 0 } ,
f ( t ) = b 0 cos ( ω t ) ,
H f ( t ) = b 0 sin ( ω t ) ,
ϕ ( t ) = φ ( t ) = ω t .
f ( x ) = b 0 cos ( k T x ) ,
{ R 1 f ( x ) = b 0 sin ( k T x ) cos α , R 2 f ( x ) = b 0 sin ( k T x ) sin α .
ϕ ( x ) = φ ( x ) = k T x .
f j , k = b 0 cos [ 2 π f 0 ( j T x cos α + k T y sin α ) ] ,
f ( t ) = cos ( a 2 t 2 ) ,
H f ( t ) = 2 π [ A ( t ) sin ( a 2 t 2 ) + B ( t ) cos ( a 2 t 2 ) ] ,
f ( x ) = b 0 cos [ a 2 ( x 1 2 + x 2 2 ) ] .
Δ ϕ 1 2 a 2 x 2 .
φ ( x ) = 2 π p ( x 1 x 1 c ) 2 + ( x 2 x 2 c ) 2 ,

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