Abstract

A local frequency estimation approach for the fringe pattern with a spatial carrier by which the 2D spatial frequencies at a certain pixel are estimated from its neighborhood is presented. The applications of this approach in the fringe pattern analyses are also introduced. First, a 2D spatial carrier phase-shifting algorithm is derived. With it the detuning errors induced by frequency mismatching are avoided, and the stronger phase deformations can be successfully coped with. Second, a novel aperture extrapolation method is developed by which the phase accuracies of the Fourier-transform method at the aperture boundaries are effectively improved.

© 2007 Optical Society of America

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  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. 72, 156-160 (1982).
    [CrossRef]
  2. K. A. Nugent, "Interferogram analysis using an accurate fully automatic algorithm," Appl. Opt. 24, 3101-3105 (1985).
    [CrossRef] [PubMed]
  3. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, "Fringe-pattern analysis using a 2D Fourier transform," Appl. Opt. 25, 1653-1660 (1986).
    [CrossRef] [PubMed]
  4. C. Roddier and F. Roddier, "Interferogram analysis using Fourier transform techniques," Appl. Opt. 26, 1668-1673 (1987).
    [CrossRef] [PubMed]
  5. J. H. Massig and J. Heppner, "Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests," Appl. Opt. 40, 2081-2088 (2001).
    [CrossRef]
  6. M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes," Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  7. J.-F. Lin and X.-Y. Su, "Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 34, 3297-3302 (1995).
    [CrossRef]
  8. R. J. Marks II, "Gerchberg's extrapolation algorithm in two dimensions," Appl. Opt. 20, 1816-1820 (1981).
  9. Y. Ichioka and M. Inuiya, "Direct phase detecting system," Appl. Opt. 11, 1507-1514 (1972).
    [CrossRef] [PubMed]
  10. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  11. S. Tang and Y. Y. Hung, "Fast profilometer for the automatic measurement of 3-D object shapes," Appl. Opt. 29, 3012-3018 (1990).
    [CrossRef] [PubMed]
  12. L. Mertz, "Real-time fringe-pattern analysis," Appl. Opt. 22, 1535-1539 (1983).
    [CrossRef] [PubMed]
  13. W. W. Macy, "Two-dimensional fringe-pattern analysis," Appl. Opt. 22, 3898-3901 (1983).
    [CrossRef] [PubMed]
  14. S. Toyooka and Y. Iwaasa, "Automatic profilometry of 3D diffuse by spatial phase detection," Appl. Opt. 25, 1630-1633 (1986).
    [CrossRef] [PubMed]
  15. M. Kujawinska and J. Wójciak, "Spatial-carrier phase-shifting technique of fringe pattern analysis," in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61-67 (1991).
  16. D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
    [CrossRef]
  17. R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
    [CrossRef]
  18. P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
    [CrossRef]
  19. Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
    [CrossRef]
  20. M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
    [CrossRef]
  21. P. L. Ransom and J. V. Kokal, "Interferogram analysis by a modified sinusoid fitting technique," Appl. Opt. 25, 4199-4204 (1986).
    [CrossRef] [PubMed]
  22. M. Pirga and M. Kujawinska, "Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns," Opt. Eng. 34, 2459-2466 (1995).
    [CrossRef]
  23. Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations," Opt. Lasers Eng. 45, 304-317 (2007).
  24. J. Zhong and J. Weng, "Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry," Appl. Opt. 43, 4993-4998 (2004).
    [CrossRef] [PubMed]
  25. H. Guo and M. Chen, "Least-squares algorithm for phase-stepping interferometry with an unknown relative step," Appl. Opt. 44, 4854-4859 (2005).
    [CrossRef] [PubMed]
  26. H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
    [CrossRef]
  27. J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
  28. P. Hariharan, B. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2506 (1987).
    [CrossRef] [PubMed]

2005

H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
[CrossRef]

H. Guo and M. Chen, "Least-squares algorithm for phase-stepping interferometry with an unknown relative step," Appl. Opt. 44, 4854-4859 (2005).
[CrossRef] [PubMed]

2004

2001

1997

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

1995

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

M. Pirga and M. Kujawinska, "Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns," Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

J.-F. Lin and X.-Y. Su, "Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 34, 3297-3302 (1995).
[CrossRef]

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

1992

R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

1991

M. Kujawinska and J. Wójciak, "Spatial-carrier phase-shifting technique of fringe pattern analysis," in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61-67 (1991).

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

1990

1987

1986

1985

1984

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

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

1983

1982

1981

R. J. Marks II, "Gerchberg's extrapolation algorithm in two dimensions," Appl. Opt. 20, 1816-1820 (1981).

1972

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Bachor, H.-A.

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Bone, D. J.

Bryanston-Cross, P. J.

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Chan, P. H.

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Chen, M.

H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
[CrossRef]

H. Guo and M. Chen, "Least-squares algorithm for phase-stepping interferometry with an unknown relative step," Appl. Opt. 44, 4854-4859 (2005).
[CrossRef] [PubMed]

Cuevas, F. J.

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

Eiju, T.

Greivenkamp, J. E.

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

Guo, H.

H. Guo and M. Chen, "Least-squares algorithm for phase-stepping interferometry with an unknown relative step," Appl. Opt. 44, 4854-4859 (2005).
[CrossRef] [PubMed]

H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
[CrossRef]

Hariharan, P.

He, H.

H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
[CrossRef]

Heppner, J.

Hung, Y. Y.

Ichioka, Y.

Ina, H.

Inuiya, M.

Iwaasa, Y.

Józwicki, R.

R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Kemao, Q.

Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations," Opt. Lasers Eng. 45, 304-317 (2007).

Kobayashi, S.

Kokal, J. V.

Kujawinska, M.

M. Pirga and M. Kujawinska, "Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns," Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

M. Kujawinska and J. Wójciak, "Spatial-carrier phase-shifting technique of fringe pattern analysis," in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61-67 (1991).

Lin, J.-F.

J.-F. Lin and X.-Y. Su, "Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 34, 3297-3302 (1995).
[CrossRef]

Macy, W. W.

Marks, R. J.

R. J. Marks II, "Gerchberg's extrapolation algorithm in two dimensions," Appl. Opt. 20, 1816-1820 (1981).

Massig, J. H.

Mertz, L.

Mutoh, K.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Nugent, K. A.

Oreb, B.

Pirga, M.

M. Pirga and M. Kujawinska, "Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns," Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

Ransom, P. L.

Roddier, C.

Roddier, F.

Salbut, L.

R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

Sandeman, R. J.

Servin, M.

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Su, X.-Y.

J.-F. Lin and X.-Y. Su, "Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 34, 3297-3302 (1995).
[CrossRef]

Takeda, M.

Tang, S.

Toyooka, S.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Weng, J.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Wójciak, J.

M. Kujawinska and J. Wójciak, "Spatial-carrier phase-shifting technique of fringe pattern analysis," in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61-67 (1991).

Womack, K. H.

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

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Zhong, J.

Appl. Opt.

R. J. Marks II, "Gerchberg's extrapolation algorithm in two dimensions," Appl. Opt. 20, 1816-1820 (1981).

Y. Ichioka and M. Inuiya, "Direct phase detecting system," Appl. Opt. 11, 1507-1514 (1972).
[CrossRef] [PubMed]

L. Mertz, "Real-time fringe-pattern analysis," Appl. Opt. 22, 1535-1539 (1983).
[CrossRef] [PubMed]

W. W. Macy, "Two-dimensional fringe-pattern analysis," Appl. Opt. 22, 3898-3901 (1983).
[CrossRef] [PubMed]

M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes," Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

K. A. Nugent, "Interferogram analysis using an accurate fully automatic algorithm," Appl. Opt. 24, 3101-3105 (1985).
[CrossRef] [PubMed]

S. Toyooka and Y. Iwaasa, "Automatic profilometry of 3D diffuse by spatial phase detection," Appl. Opt. 25, 1630-1633 (1986).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, and R. J. Sandeman, "Fringe-pattern analysis using a 2D Fourier transform," Appl. Opt. 25, 1653-1660 (1986).
[CrossRef] [PubMed]

P. L. Ransom and J. V. Kokal, "Interferogram analysis by a modified sinusoid fitting technique," Appl. Opt. 25, 4199-4204 (1986).
[CrossRef] [PubMed]

S. Tang and Y. Y. Hung, "Fast profilometer for the automatic measurement of 3-D object shapes," Appl. Opt. 29, 3012-3018 (1990).
[CrossRef] [PubMed]

C. Roddier and F. Roddier, "Interferogram analysis using Fourier transform techniques," Appl. Opt. 26, 1668-1673 (1987).
[CrossRef] [PubMed]

J. H. Massig and J. Heppner, "Fringe-pattern analysis with high accuracy by use of the Fourier-transform method: theory and experimental tests," Appl. Opt. 40, 2081-2088 (2001).
[CrossRef]

J. Zhong and J. Weng, "Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry," Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

H. Guo and M. Chen, "Least-squares algorithm for phase-stepping interferometry with an unknown relative step," Appl. Opt. 44, 4854-4859 (2005).
[CrossRef] [PubMed]

P. Hariharan, B. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2506 (1987).
[CrossRef] [PubMed]

J. Mod. Opt.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

M. Pirga and M. Kujawinska, "Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns," Opt. Eng. 34, 2459-2466 (1995).
[CrossRef]

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

J.-F. Lin and X.-Y. Su, "Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 34, 3297-3302 (1995).
[CrossRef]

R. Józwicki, M. Kujawinska, and L. Salbut, "New contra old wavefront measurement concepts for interferometric optical testing," Opt. Eng. 31, 422-433 (1992).
[CrossRef]

J. E. Greivenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

Opt. Laser Technol.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Technol. 23, 147-150 (1991).
[CrossRef]

Opt. Lasers Eng.

Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations," Opt. Lasers Eng. 45, 304-317 (2007).

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Proc. SPIE

H. Guo, M. Chen, and H. He, "A frequency encoding method for fringe projection profilometry," in Third International Conference on Experimental Mechanics and Third Conference of the Asian Committee on Experimental Mechanics, C. Quan, F. S. Chau, A. Asundi, B. S. Wong, and C. T. Lim, eds., Proc. SPIE 5852, 908-913 (2005).
[CrossRef]

Other

M. Kujawinska and J. Wójciak, "Spatial-carrier phase-shifting technique of fringe pattern analysis," in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61-67 (1991).

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Figures (9)

Fig. 1
Fig. 1

Flow chart of 2D local frequency estimation.

Fig. 2
Fig. 2

(Color online) Simulated phase map (a) without and (b) with a spatial carrier.

Fig. 3
Fig. 3

(Color online) Simulation results. (a) Fringe pattern, (b) and (c) estimated frequencies along the x and y directions, respectively.

Fig. 4
Fig. 4

Local frequencies along the cross section (y = 128) of the fringe pattern where (a) and (b) show the frequencies, u and v, respectively. Solid curves, the theoretical values; dotted curves, the estimated values; dashed lines, the carrier frequencies.

Fig. 5
Fig. 5

(Color online) Experimental results. (a) Fringe pattern, (b) and (c) the estimated frequencies along the x and y directions, respectively.

Fig. 6
Fig. 6

(Color online) Phase maps recovered from the simulated fringe pattern using (a) the proposed 2D SCPS algorithm and (b) the conventional 1D algorithm.

Fig. 7
Fig. 7

(Color online) Phase maps recovered from the practical inetrferogram (a) using the proposed 2D SCPS algorithm and (b) using the conventional 1D algorithm.

Fig. 8
Fig. 8

(Color online) Phase measurement results of the simulated fringe pattern with the Fourier-transform method where the fringe pattern is (a) padded with 0, (b) multiplied by a 2D Hanning window, (c) extrapolated by using the Gerchberg algorithm, and (d) extrapolated by using the proposed method. The columns show, from left to right, the fringe patterns after preprocessing, the wrapped phase maps without carrier, the unwrapped phase maps in 3D views where the data outside the aperture have been stripped off, and the phase error distributions.

Fig. 9
Fig. 9

(Color online) Phase measurement results of the practical interferogram using the Fourier-transform method: (a) fringe pattern extrapolated with the proposed method and (b) the unwrapped phase map.

Equations (91)

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

I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ u C x + v C y + ϕ ( x , y ) ] ,
ϕ ( x , y )
u C
v C
u ( x , y ) = [ u C x + v C y + ϕ ( x , y ) ] / x = u C + ϕ ( x , y ) / x ,
v ( x , y ) = [ u C x + v C y + ϕ ( x , y ) ] / y = v C + ϕ ( x , y ) / y ,
( 2 N + 1 ) × ( 2 N + 1 )
I ( x + s , y + t ) = a ( x + s , y + t ) + b ( x + s , y + t ) cos [ u C ( x + s ) + v C ( y + t ) + ϕ ( x + s , y + t ) ] a ( x , y ) + b ( x , y ) cos [ u C x + v C y + ϕ ( x , y ) + ϕ ( x , y ) x s + ϕ ( x , y ) y t + u C s + v C t ] = a ( x , y ) + b ( x , y ) cos [ u C x + v C y + ϕ ( x , y ) + u ( x , y ) s + v ( x , y ) t ] ,
s ( N s N )
t ( N t N )
I s , t
u = arccos [ B ( t ) B 2 ( t ) + 4 A 2 ( t ) 4 A ( t ) 1 2 ] ,
A ( t ) = s = N + 3 N [ ( I s 2 , t I s 1 , t ) ( I s , t I s 3 , t ) ]
B ( t ) = s = N + 3 N [ ( I s 2 , t I s 1 , t ) 2 ( I s , t I s 3 , t ) 2 ]
u = arccos { 1 2 N + 1 t = N N [ B ( t ) B 2 ( t ) + 4 A 2 ( t ) 4 A ( t ) 1 2 ] } .
v = arccos { 1 2 N + 1 s = N N [ D ( s ) D 2 ( s ) + 4 C 2 ( s ) 4 C ( s ) 1 2 ] } ,
C ( s ) = t = N + 3 N [ ( I s , t 2 I s , t 1 ) ( I s , t I s , t 3 ) ]
D ( s ) = t = N + 3 N [ ( I s , t 2 I s , t 1 ) 2 ( I s , t I s , t 3 ) 2 ]
( 0 , π )
( π , π )
( 0 , π )
I s + 1 , t + 1 I s , t I s + 1 , t I s , t + 1 = sin ( u / 2 + v / 2 ) sin ( u / 2 v / 2 ) = tan u / 2 + tan v / 2 tan u / 2 tan v / 2 .
tan u / 2 tan v / 2 = I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 .
tan ( u / 2 ) / tan ( v / 2 )
| tan u / 2 tan v / 2 | = s = N N 1 t = N N 1 | I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 | s = N N 1 t = N N 1 | I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 | .
E = s = N N 1 t = N N 1 ( | I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 | | I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 | ) .
E 0
| tan ( u / 2 ) / tan ( v / 2 ) | 1
| u | | v |
R u v = tan ( u / 2 ) / tan ( v / 2 )
e s , t = ( I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 ) R u v ( I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 ) .
s = N N 1 t = N N 1 e s , t 2
R u v
R u v
R u v = s = N N 1 t = N N 1 ( I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 ) ( I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 ) s = N N 1 t = N N 1 ( I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 ) 2 .
u = 2 arctan ( R u v tan v 2 ) ,
( π , π )
E < 0
| u | > | v |
R v u = tan ( v / 2 ) / tan ( u / 2 )
R v u = s = N N 1 t = N N 1 ( I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 ) ( I s , t I s , t + 1 + I s + 1 , t I s + 1 , t + 1 ) s = N N 1 t = N N 1 ( I s , t + I s , t + 1 I s + 1 , t I s + 1 , t + 1 ) 2 .
v = 2 arctan ( R v - u tan u 2 ) ,
( π , π )
u C = 0.5 π
v C = 0.3
[ 1 , 1 ]
4 × 4
0.5 π   rad
11 × 11   pixels
( 2 N + 1 ) × ( 2 N + 1 )
c 0 ( x , y ) = a ( x , y )
c 1 ( x , y ) = b ( x , y ) cos [ u C x + v C y + ϕ ( x , y ) ]
c 2 ( x , y ) = b ( x , y ) sin [ u C x + v C y + ϕ ( x , y ) ]
I s , t = c 0 + c 1 cos ( u s + v t ) + c 2 sin ( u s + v t ) .
c 0
c 1
c 2
s = N N t = N N [ 1 cos ( u s + v t ) 0 cos ( u s + v t ) cos 2 ( u s + v t ) 0 0 0 sin 2 ( u s + v t ) ]  × [ c 0 c 1 c 2 ] = s = N N t = N N [ I s , t I s , t cos ( u s + v t ) I s , t sin ( u s + v t ) ] .
c 0
c 1
u C x + v C y + ϕ ( x , y ) = arctan c 2 ( x , y ) c 1 ( x , y )
u C x + v C y
ϕ ( x , y )
π / 2
ϕ = arctan [ 2 ( I 2 I 4 ) / ( 2 I 3 I 1 I 5 ) ]
( 2 N + 1 ) × ( 2 N + 1 )
c 0 ( x , y )
c 1 ( x , y )
c 2 ( x , y )
I ( x , y ) = c 0 ( x , y ) + c 1 ( x , y )
( x + s , y + t )
I ( x + s , y + t ) = c 0 ( x , y ) + c 1 ( x , y ) cos [ u ( x , y ) s + v ( x , y ) t ] + c 2 ( x , y ) sin [ u ( x , y ) s + v ( x , y ) t ] .
| s | > N
| t | > N
( x + s , y + t )
( 2 N + 1 ) × ( 2 N + 1 )
( x + s
y + t )
| s | = N + 1
| v | = N + 1
( x + s , y + t )
( x + s , y + t )
( 2 N + 1 ) × ( 2 N + 1 )
3 × 3
N + 1
N 1
( 2 N + 1 ) × ( 2 N + 1 )
c 0 ( x , y )
c 1 ( x , y )
c 2 ( x , y )
( x + s , y + t )

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