Abstract

We present an intensity-ratio error-compensation method to decrease the measurement error caused by projector gamma nonlinearity and image defocus in triangular-pattern phase-shifting profilometry. The intensity-ratio measurement error is first determined by simulating the measurement with the triangular-pattern phase-shifting method with ideal and real captured triangular-pattern images based on the ideal and real gamma nonlinearity functions. A lookup table that stores the intensity-ratio measurement error corresponding to the measured intensity ratio is constructed and used for intensity-ratio error compensation. Experiments demonstrated that the intensity-ratio error compensation method significantly reduced the measurement error in the triangular-pattern phase-shifting method by 28.5%.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349-393.
    [CrossRef]
  2. M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
    [CrossRef]
  3. J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing (Wiley, 1992), pp. 501-598.
  4. X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
    [CrossRef]
  5. Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. (Bellingham) 37, 1005-1010 (1998).
    [CrossRef]
  6. G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
    [CrossRef]
  7. H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
    [CrossRef] [PubMed]
  8. B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.
    [CrossRef]
  9. T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Photogrammetric Computer Vision (PCV02), Proceedings of the ISPRS Commission III (ISPRS, 2002), Vol. XXXIV, Part 3B, pp. 181-185.
  10. G. Chazan and N. Kiryati, "Pyramidal intensity-ratio depth sensor," Tech. Rep. 121, Department of Electrical Engineering (Center for Communication and Information Technologies, Technion, Haifa, Israel, 1995).
  11. P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
    [CrossRef]
  12. P. Jia, J. Kofman, and C. English, "Multiple-step triangular-pattern phase-shifting and the influence of number of steps and pitch on measurement accuracy," Appl. Opt. 46, 3253-3262 (2007).
    [CrossRef] [PubMed]
  13. P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
    [CrossRef]
  14. Q. Fang, "Linearly coded profilometry with a coding light that has isosceles triangle teeth: even-number-sample decoding method," Appl. Opt. 36, 1615-1620 (1997).
    [CrossRef] [PubMed]
  15. Q. Fang and S. Zheng, "Linearly coded profilometry," Appl. Opt. 36, 2401-2407 (1997).
    [CrossRef] [PubMed]
  16. J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
    [CrossRef]
  17. S. Zhang and P. S. Huang, "Phase error compensation for a 3-D shape measurement system based on the phase-shifting method," Proc. SPIE 6000, E1-E10 (2005).
    [CrossRef]
  18. P. S. Huang, Q. J. Hu, and F. P. Chiang, "Double three-step phase-shifting algorithm," Appl. Opt. 41, 4503-4509 (2002).
    [CrossRef] [PubMed]
  19. P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
    [CrossRef]
  20. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).
  21. P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
    [CrossRef]
  22. C. Zhang, P. S. Huang, and F. P. Chiang, "Microscopic phase-shifting profilometry based on digital micromirror device technology," Appl. Opt. 41, 5896-5904 (2002).
    [CrossRef] [PubMed]

2007

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Multiple-step triangular-pattern phase-shifting and the influence of number of steps and pitch on measurement accuracy," Appl. Opt. 46, 3253-3262 (2007).
[CrossRef] [PubMed]

P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
[CrossRef]

2006

P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
[CrossRef]

2005

S. Zhang and P. S. Huang, "Phase error compensation for a 3-D shape measurement system based on the phase-shifting method," Proc. SPIE 6000, E1-E10 (2005).
[CrossRef]

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
[CrossRef]

2004

H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
[CrossRef]

2002

2000

G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
[CrossRef]

1998

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. (Bellingham) 37, 1005-1010 (1998).
[CrossRef]

1997

1989

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

Araki, K.

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Photogrammetric Computer Vision (PCV02), Proceedings of the ISPRS Commission III (ISPRS, 2002), Vol. XXXIV, Part 3B, pp. 181-185.

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing (Wiley, 1992), pp. 501-598.

Carrihill, B.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.
[CrossRef]

Chazan, G.

G. Chazan and N. Kiryati, "Pyramidal intensity-ratio depth sensor," Tech. Rep. 121, Department of Electrical Engineering (Center for Communication and Information Technologies, Technion, Haifa, Israel, 1995).

Chen, M.

H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
[CrossRef]

Chiang, F.

J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
[CrossRef]

Chiang, F. P.

Chiang, F.-P.

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
[CrossRef]

Choi, Y. B.

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. (Bellingham) 37, 1005-1010 (1998).
[CrossRef]

Creath, K.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349-393.
[CrossRef]

English, C.

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Multiple-step triangular-pattern phase-shifting and the influence of number of steps and pitch on measurement accuracy," Appl. Opt. 46, 3253-3262 (2007).
[CrossRef] [PubMed]

P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
[CrossRef]

Fang, Q.

Frankowski, G.

G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing (Wiley, 1992), pp. 501-598.

Guo, H.

Guo, Y. F.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Halioua, M.

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

He, H.

He, X. Y.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Hu, Q. J.

Huang, P. S.

S. Zhang and P. S. Huang, "Phase error compensation for a 3-D shape measurement system based on the phase-shifting method," Proc. SPIE 6000, E1-E10 (2005).
[CrossRef]

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
[CrossRef]

J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
[CrossRef]

P. S. Huang, Q. J. Hu, and F. P. Chiang, "Double three-step phase-shifting algorithm," Appl. Opt. 41, 4503-4509 (2002).
[CrossRef] [PubMed]

C. Zhang, P. S. Huang, and F. P. Chiang, "Microscopic phase-shifting profilometry based on digital micromirror device technology," Appl. Opt. 41, 5896-5904 (2002).
[CrossRef] [PubMed]

Hummel, R.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.
[CrossRef]

Huth, T.

G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
[CrossRef]

Jia, P.

P. Jia, J. Kofman, and C. English, "Multiple-step triangular-pattern phase-shifting and the influence of number of steps and pitch on measurement accuracy," Appl. Opt. 46, 3253-3262 (2007).
[CrossRef] [PubMed]

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
[CrossRef]

Kim, S. W.

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. (Bellingham) 37, 1005-1010 (1998).
[CrossRef]

Kiryati, N.

G. Chazan and N. Kiryati, "Pyramidal intensity-ratio depth sensor," Tech. Rep. 121, Department of Electrical Engineering (Center for Communication and Information Technologies, Technion, Haifa, Israel, 1995).

Kofman, J.

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Multiple-step triangular-pattern phase-shifting and the influence of number of steps and pitch on measurement accuracy," Appl. Opt. 46, 3253-3262 (2007).
[CrossRef] [PubMed]

P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
[CrossRef]

Liu, H. C.

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

Liu, S.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Miyasaka, T.

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Photogrammetric Computer Vision (PCV02), Proceedings of the ISPRS Commission III (ISPRS, 2002), Vol. XXXIV, Part 3B, pp. 181-185.

Pan, J.

J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Zhang, C.

Zhang, S.

S. Zhang and P. S. Huang, "Phase error compensation for a 3-D shape measurement system based on the phase-shifting method," Proc. SPIE 6000, E1-E10 (2005).
[CrossRef]

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
[CrossRef]

Zheng, S.

Zou, D. Q.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Appl. Opt.

Opt. Eng. (Bellingham)

P. Jia, J. Kofman, and C. English, "Comparison of linear and non-linear calibration methods for phase-measuring profilometry," Opt. Eng. (Bellingham) 46, 043601 (2007).
[CrossRef]

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for the three-dimensional shape measurement," Opt. Eng. (Bellingham) 44, 123601 (2005).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (Bellingham) 46, 083201 (2007).
[CrossRef]

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. (Bellingham) 37, 1410-1419 (1998).
[CrossRef]

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. (Bellingham) 37, 1005-1010 (1998).
[CrossRef]

Opt. Lasers Eng.

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

Proc. SPIE

G. Frankowski, M. Chen, and T. Huth, "Real-time 3D shape measurement with digital strip projection by Texas Instruments Micromirror Devices DMDtrade," Proc. SPIE 3958, 90-105 (2000).
[CrossRef]

J. Pan, P. S. Huang, and F. Chiang, "Color-encoded digital fringe projection technique for high-speed 3D shape measurement--color coupling and imbalance compensation," Proc. SPIE 5265, 205-212 (2004).
[CrossRef]

S. Zhang and P. S. Huang, "Phase error compensation for a 3-D shape measurement system based on the phase-shifting method," Proc. SPIE 6000, E1-E10 (2005).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Repeated phase-offset measurement for error compensation in two-step triangular phase-shifting profilometry," Proc. SPIE 6375, 63750D (2006).
[CrossRef]

Other

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E.Wolf, ed. (Elsevier Science, 1988), pp. 349-393.
[CrossRef]

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing (Wiley, 1992), pp. 501-598.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.
[CrossRef]

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Photogrammetric Computer Vision (PCV02), Proceedings of the ISPRS Commission III (ISPRS, 2002), Vol. XXXIV, Part 3B, pp. 181-185.

G. Chazan and N. Kiryati, "Pyramidal intensity-ratio depth sensor," Tech. Rep. 121, Department of Electrical Engineering (Center for Communication and Information Technologies, Technion, Haifa, Israel, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Two-step triangular-pattern phase-shifting method: (a) two shifted triangular patterns; (b) intensity ratio with triangular wave shape; and (d) intensity-ratio ramp after removal of the triangular wave.

Fig. 2
Fig. 2

Defocus of the triangular pattern for different values of defocus coefficient S.

Fig. 3
Fig. 3

Defocused intensity ratio for different values of defocus coefficient S.

Fig. 4
Fig. 4

Defocused intensity-ratio ramp for different values of defocus coefficient S.

Fig. 5
Fig. 5

Average and maximum intensity-ratio errors for varying defocus coefficient S.

Fig. 6
Fig. 6

Nonlinear mapping of projector input grayscale values to camera-captured image intensity.

Fig. 7
Fig. 7

Triangular-pattern simulation with measured and ideal gamma curve functions. (a) Input triangular pattern and (b) simulated captured triangular pattern image.

Fig. 8
Fig. 8

Measured and ideal intensity ratio with periodic triangular shape.

Fig. 9
Fig. 9

Measured and ideal intensity-ratio ramp after conversion from the triangular shape.

Fig. 10
Fig. 10

Intensity-ratio error as a function of intensity ratio.

Fig. 11
Fig. 11

Schematic of the 3D shape measurement system based on triangular fringe-pattern projection.

Fig. 12
Fig. 12

Intensity-ratio compensation. (a) Wrapped intensity ratio before correction and (b) wrapped intensity ratio after correction.

Fig. 13
Fig. 13

Measurement errors of the two-step triangular-pattern phase-shifting method before and after applying intensity-ratio error compensation by measurement of a flat plate using a pitch of 16 pixels. Measurement errors, computed as the rms value, are based on errors in depth with respect to ground truth at all image pixels ( 648 × 494 ) of the measured plate.

Fig. 14
Fig. 14

Depth measurement of the measured flat plate at 20 mm from the reference position. (a) Measured profile of the flat plate before correction and (b) measured profile of the flat plate after correction.

Fig. 15
Fig. 15

Three-dimensional shape measurement of a human-head mask. (a)–(c) Before error compensation was applied and (d)–(f) after error compensation. The triangular pattern was generated with a pitch of 16 pixels.

Equations (12)

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

I 1 ( x , y ) = { 2 I m ( x , y ) T x + I min ( x , y ) + I m ( x , y ) 2 , x [ 0 , T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) + 3 I m ( x , y ) 2 , x [ T 4 , 3 T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) 3 I m ( x , y ) 2 , x [ 3 T 4 , T ) ] ,
I 2 ( x , y ) = { 2 I m ( x , y ) T x + I min ( x , y ) + I m ( x , y ) 2 , x [ 0 , T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) I m ( x , y ) 2 , x [ T 4 , 3 T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) + 5 I m ( x , y ) 2 , x [ 3 T 4 , T ) ] ,
I m ( x , y ) = I max ( x , y ) I min ( x , y ) ,
r 0 ( x , y ) = I 1 ( x , y ) I 2 ( x , y ) I m ( x , y ) .
r ( x , y ) = 2 × round ( R 1 2 ) + ( 1 ) R + 1 r 0 ( x , y ) , R = 1 , 2 , 3 , 4 ,
I 1 D ( x , y ) = { I 1 ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) > 0 I 1 ( x , y ) , when cos ( 2 π x T ) = 0 I 1 ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) < 0 ) ,
I 2 D ( x , y ) = { I 2 ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) > 0 I 2 ( x , y ) , when cos ( 2 π x T ) = 0 I 2 ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) < 0 ) ,
r 0 D ( x , y ) = I 1 D ( x , y ) I 2 D ( x , y ) Δ I D ( x , y ) ,
r 0 D ( x , y ) = { I 1 ( x , y ) I 2 ( x , y ) Δ I D ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) > 0 I 1 ( x , y ) I 2 ( x , y ) Δ I D ( x , y ) , when cos ( 2 π x T ) = 0 I 1 ( x , y ) I 2 ( x , y ) Δ I D ( x , y ) { 1 + 0.6 S [ cos ( 2 π x T ) ] S } , when cos ( 2 π x T ) < 0 ) .
Δ r 0 ( x , y ) = r 0 D ( x , y ) r 0 ( x , y ) .
Δ r a v e = 1 T x = 0 T Δ r 0 D ( x , y ) ,
Δ r max = max { Δ r 0 D ( x , y ) } .

Metrics