Abstract

This paper presents a pseudo Wigner–Ville-distribution-based method in fringe projection for analyzing temporal behavior of the displacement derivative for a continuously deformed object. In the proposed method, a computer generated fringe pattern is projected on an object undergoing dynamic deformation, and the reflected intensity is recorded in the form of video, i.e., a stack of images are captured sequentially by a CCD camera. Each image represents a recorded fringe pattern at a particular time instant whose phase contains information about the instantaneous out-of-plane displacement or deformation with respect to the undeformed object, and the corresponding spatial phase derivative relates to the displacement derivative. Subsequently, pseudo Wigner–Ville distribution is used for instantaneous phase derivative estimation from the stack of images. Simulation and experimental results are presented to demonstrate the method’s potential.

© 2011 Optical Society of America

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References

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  1. Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61–87 (2005).
    [CrossRef]
  2. J. N. Petzing and J. R. Tyrer, “Recent developments and applications in electronic speckle pattern interferometry,” J. Strain Anal. Eng. Des. 33, 153–169 (1998).
    [CrossRef]
  3. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  4. D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
    [CrossRef]
  5. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
    [CrossRef]
  6. Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
    [CrossRef]
  7. F. Labbe, “Strain-rate measurements by electronic speckle-pattern interferometry (ESPI),” Opt. Lasers Eng. 45, 827–833(2007).
    [CrossRef]
  8. B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
    [CrossRef]
  9. T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
    [CrossRef]
  10. T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
    [CrossRef]
  11. C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
    [CrossRef]
  12. L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).
  13. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
    [CrossRef] [PubMed]
  14. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
    [CrossRef]
  15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]

2010 (1)

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

2009 (3)

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

2008 (3)

D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
[CrossRef]

2007 (1)

F. Labbe, “Strain-rate measurements by electronic speckle-pattern interferometry (ESPI),” Opt. Lasers Eng. 45, 827–833(2007).
[CrossRef]

2006 (2)

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

2005 (1)

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61–87 (2005).
[CrossRef]

2002 (1)

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1998 (1)

J. N. Petzing and J. R. Tyrer, “Recent developments and applications in electronic speckle pattern interferometry,” J. Strain Anal. Eng. Des. 33, 153–169 (1998).
[CrossRef]

1982 (1)

Anaya, T. S.

T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
[CrossRef]

Chen, W.

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
[CrossRef]

Chen, Y. S.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

Cohen, L.

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

Daniel, L.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

De La Torre, M.

T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
[CrossRef]

Francis, D.

D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
[CrossRef]

Francois, M.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Guelorget, B.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

Ho, H. P.

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61–87 (2005).
[CrossRef]

Huang, Y. H.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

Hung, M. Y. Y.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

Hung, Y. Y.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61–87 (2005).
[CrossRef]

Ina, H.

James, S. W.

D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
[CrossRef]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Kadono, H.

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Kobayashi, S.

Labbe, F.

F. Labbe, “Strain-rate measurements by electronic speckle-pattern interferometry (ESPI),” Opt. Lasers Eng. 45, 827–833(2007).
[CrossRef]

Li, C. L.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Liu, L.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

Lu, J.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

Montay, G.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

Ng, S. P.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

Petzing, J. N.

J. N. Petzing and J. R. Tyrer, “Recent developments and applications in electronic speckle pattern interferometry,” J. Strain Anal. Eng. Des. 33, 153–169 (1998).
[CrossRef]

Ping, S.

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Quan, C.

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Rastogi, P.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Saito, T.

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Santoyo, F. M.

T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Shiraishi, T.

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Takeda, M.

Tatam, R. P.

D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
[CrossRef]

Tay, C. J.

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
[CrossRef]

Toyooka, S.

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Tyrer, J. R.

J. N. Petzing and J. R. Tyrer, “Recent developments and applications in electronic speckle pattern interferometry,” J. Strain Anal. Eng. Des. 33, 153–169 (1998).
[CrossRef]

Vial-Edwards, C.

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Strain Anal. Eng. Des. (1)

J. N. Petzing and J. R. Tyrer, “Recent developments and applications in electronic speckle pattern interferometry,” J. Strain Anal. Eng. Des. 33, 153–169 (1998).
[CrossRef]

Key Eng. Mater. (1)

T. Shiraishi, S. Toyooka, H. Kadono, T. Saito, and S. Ping, “Dynamic ESPI system for spatio-temporal strain analysis of a deforming solid object,” Key Eng. Mater. 326–328, 95–98 (2006).
[CrossRef]

Mater. Sci. Eng. A (1)

B. Guelorget, M. Francois, C. Vial-Edwards, G. Montay, L. Daniel, and J. Lu, “Strain rate measurement by electronic speckle pattern interferometry: a new look at the strain localization onset,” Mater. Sci. Eng. A 415, 234–241 (2006).
[CrossRef]

Mater. Sci. Eng. R (1)

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61–87 (2005).
[CrossRef]

Meas. Sci. Technol. (2)

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

D. Francis, S. W. James, and R. P. Tatam, “Surface strain measurement of rotating objects using pulsed laser shearography with coherent fibre-optic imaging bundles,” Meas. Sci. Technol. 19, 105301 (2008).
[CrossRef]

Opt. Eng. (2)

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47, 054301 (2008).
[CrossRef]

T. S. Anaya, M. De La Torre, and F. M. Santoyo, “Microstrain detection using simultaneous endoscopic pulsed digital holography,” Opt. Eng. 47, 073601 (2008).
[CrossRef]

Opt. Laser Technol. (1)

C. J. Tay, C. Quan, and W. Chen, “Dynamic measurement by digital holographic interferometry based on complex phasor method,” Opt. Laser Technol. 41, 172–180 (2009).
[CrossRef]

Opt. Lasers Eng. (3)

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47, 774–781 (2009).
[CrossRef]

F. Labbe, “Strain-rate measurements by electronic speckle-pattern interferometry (ESPI),” Opt. Lasers Eng. 45, 827–833(2007).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Other (1)

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

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Figures (4)

Fig. 1
Fig. 1

(a) Reference fringe pattern. Original phase in radians for frames (b) k = 5 , (c) k = 15 , and (d) k = 25 . Corresponding fringe patterns for frames (e) k = 5 , (f) k = 15 , and (g) k = 25 .

Fig. 2
Fig. 2

Estimated phase derivative in radians per pixel for frames (a) k = 5 , (b) k = 15 , and (c) k = 25 . Corresponding estimation error in radians per pixel for frames (d) k = 5 , (e) k = 15 , and (f) k = 25 . Cosine fringes of the estimated phase derivative for frames (g) k = 5 , (h) k = 15 , and (i) k = 25 .

Fig. 3
Fig. 3

(a) Membrane. (b) Experimental setup.

Fig. 4
Fig. 4

Experimental fringe patterns for frames (a) k = 5 , (b) k = 15 , and (c) k = 30 . Estimated phase derivative in radians per pixel for frames (d) k = 5 , (e) k = 15 , and (f) k = 30 . Cosine fringes of the estimated phase derivative for frames (g) k = 5 , (h) k = 15 , and (i) k = 30 .

Equations (15)

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

I 1 ( x , y ) = a 1 ( x , y ) + b 1 ( x , y ) cos [ ω 0 x + ϕ R ( x , y ) ] + η 1 ( x , y ) ,
I 1 ( x , y ) = a 1 ( x , y ) + 1 2 b 1 ( x , y ) exp [ j ( ω 0 x + ϕ R ( x , y ) ) ] + 1 2 b 1 ( x , y ) exp [ j ( ω 0 x + ϕ R ( x , y ) ) ] + η 1 ( x , y ) .
I 2 ( x , y , k ) = a 2 ( x , y , k ) + b 2 ( x , y , k ) cos [ ω 0 x + ϕ D ( x , y , k ) ] + η 2 ( x , y , k ) ,
I 2 ( x , y , k ) = a 2 ( x , y , k ) + 1 2 b 2 ( x , y , k ) exp [ j ( ω 0 x + ϕ D ( x , y , k ) ) ] + 1 2 b 2 ( x , y , k ) exp [ j ( ω 0 x + ϕ D ( x , y , k ) ) ] + η 2 ( x , y , k ) .
I 1 ( x , y ) = b 1 ( x , y ) exp [ j ( ω 0 x + ϕ R ( x , y ) ) ] + η 1 ( x , y ) ,
I 2 ( x , y , k ) = b 2 ( x , y , k ) exp [ j ( ω 0 x + ϕ D ( x , y , k ) ) ] + η 2 ( x , y , k ) .
A ( x , y , k ) = I 2 ( x , y , k ) I 1 * ( x , y ) ,
A ( x , y , k ) = b ( x , y , k ) exp [ j ϕ ( x , y , k ) ] + η ( x , y , k ) .
b ( x , y , k ) = b 1 ( x , y ) b 2 ( x , y , k ) ,
ϕ ( x , y , k ) = ϕ D ( x , y , k ) ϕ R ( x , y ) .
ω x ( x , y , k ) = ϕ ( x , y , k ) x
A ( x , k ) = b ( x , k ) exp [ j ϕ ( x , k ) ] + η ( x , k ) .
W ( x , Ω , k ) = τ = w ( τ ) A ( x + τ , k ) A * ( x τ , k ) exp ( - j 2 Ω τ ) ,
w ( x ) = 1 ( 2 π σ 2 ) 1 / 2 exp [ x 2 2 σ 2 ] x [ - σ / 2 , σ / 2 ] ,
ω x ( x , k ) = argmax Ω W ( x , Ω , k ) .

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