Abstract

This paper describes a depth from defocus (DFD) method to recover the depth of scene from two images taken by a liquid crystal (LC) lens imaging system. The system is composed of a camera module and an LC lens. The system’s focal length is electrically adjusted by the voltages that are applied on the LC lens. We use two images that are taken at maximum positive and negative, respectively, and the LC lens’s optical power, to obtain the depth information via DFD. The principle is described and the experimental results are successfully obtained. The method is simple in that it needs not to involve any mechanical lens movements in the imaging system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

2015 (1)

M. Kawamura and S. Ishikuro, “Feature extraction from multiply focal images by using a liquid crystal lens,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 613(1), 51–58 (2015).
[Crossref]

2012 (1)

A. N. Joseph Raj and R. C. Staunton, “Rational filters design for depth from defocus,” Pattern Recognit. 45(1), 198–207 (2012).
[Crossref]

2011 (1)

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

2010 (2)

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

2008 (1)

A. S. Malik and T. S. Choi, “A novel algorithm for estimation of depth map using image focus for 3D shape recovery in the presence of noise,” Pattern Recognit. 41(7), 2200–2225 (2008).
[Crossref]

2007 (1)

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

2004 (1)

1998 (1)

M. Watanabe and S. K. Nayar, “Rational filters for passive depth from defocus,” Int. J. Comput. Vis. 27(3), 203–225 (1998).
[Crossref]

1987 (1)

P. Grossmann, “Depth from focus,” Pattern Recognit. Lett. 5(1), 63–69 (1987).
[Crossref]

Alenya, G.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: A survey,” in Proceeding of IEEE Conference on Sensors Journal (IEEE, 2011), pp. 1917–1926.

Bekaert, P.

M. Gerrits and P. Bekaert, “Local stereo matching with segmentation-based outlier rejection,” in Proceeding of IEEE Canadian Conference on Computer and Robot Vision (IEEE, 2006), pp. 66–72.
[Crossref]

Bodini, I.

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “A depth from defocus measurement system using a liquid lens objective for extended depth range,” in Proceeding of IEEE Transactions on Instrumentation and Measurement (IEEE, 2017), pp. 441–450.
[Crossref]

Chang, C. C.

C. H. Wu, Y. N. Sun, and C. C. Chang, “Three-dimensional modeling from endoscopic video using geometric constraints via feature positioning,” in Proceeding of IEEE Transactions on Bio-medical Engineering (IEEE, 2007), pp. 1199–1211.

Chen, S. J.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Choi, T. S.

A. S. Malik and T. S. Choi, “A novel algorithm for estimation of depth map using image focus for 3D shape recovery in the presence of noise,” Pattern Recognit. 41(7), 2200–2225 (2008).
[Crossref]

Dillmann, R.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Docchio, F.

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “A depth from defocus measurement system using a liquid lens objective for extended depth range,” in Proceeding of IEEE Transactions on Instrumentation and Measurement (IEEE, 2017), pp. 441–450.
[Crossref]

Duan, K.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Foix, S.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: A survey,” in Proceeding of IEEE Conference on Sensors Journal (IEEE, 2011), pp. 1917–1926.

Gerrits, M.

M. Gerrits and P. Bekaert, “Local stereo matching with segmentation-based outlier rejection,” in Proceeding of IEEE Canadian Conference on Computer and Robot Vision (IEEE, 2006), pp. 66–72.
[Crossref]

Gokstorp, M.

M. Gokstorp, “Computing depth from out-of-focus blur using a local frequency representation,” in Proceeding of 12th International Conference on Pattern Recognition (IEEE, 1994), pp. 153–158.
[Crossref]

Grossmann, P.

P. Grossmann, “Depth from focus,” Pattern Recognit. Lett. 5(1), 63–69 (1987).
[Crossref]

Gumpp, T.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Guo, Y. K.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Horn, E.

E. Horn and N. Kiryati, “Toward optimal structured light patterns,” in Proceeding of IEEE International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE, 1997), pp. 28–35.

Ishikuro, S.

M. Kawamura and S. Ishikuro, “Feature extraction from multiply focal images by using a liquid crystal lens,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 613(1), 51–58 (2015).
[Crossref]

Jiang, L.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Joseph Raj, A. N.

A. N. Joseph Raj and R. C. Staunton, “Rational filters design for depth from defocus,” Pattern Recognit. 45(1), 198–207 (2012).
[Crossref]

Kang, B.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Kawamura, M.

M. Kawamura and S. Ishikuro, “Feature extraction from multiply focal images by using a liquid crystal lens,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 613(1), 51–58 (2015).
[Crossref]

Kim, C.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Kim, J.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Kim, S.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Kiryati, N.

E. Horn and N. Kiryati, “Toward optimal structured light patterns,” in Proceeding of IEEE International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE, 1997), pp. 28–35.

Kohlhaas, R.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Lancini, M.

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “A depth from defocus measurement system using a liquid lens objective for extended depth range,” in Proceeding of IEEE Transactions on Instrumentation and Measurement (IEEE, 2017), pp. 441–450.
[Crossref]

Lee, K.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Lee, S.

B. Kang, S. Kim, S. Lee, K. Lee, J. Kim, and C. Kim, “Harmonic distortion free distance estimation in ToF camera,” Proc. SPIE 7864, 786403 (2011).
[Crossref]

Li, J. F.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Lin, C. H.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Lin, X.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Malik, A. S.

A. S. Malik and T. S. Choi, “A novel algorithm for estimation of depth map using image focus for 3D shape recovery in the presence of noise,” Pattern Recognit. 41(7), 2200–2225 (2008).
[Crossref]

May, S.

S. May, B. Werner, H. Surmann, and K. Pervolz, “3D time-of-flight cameras for mobile robotics,” in Proceeding of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2007), pp. 790–795.

Nayar, S. K.

M. Watanabe and S. K. Nayar, “Rational filters for passive depth from defocus,” Int. J. Comput. Vis. 27(3), 203–225 (1998).
[Crossref]

Pasinetti, S.

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “A depth from defocus measurement system using a liquid lens objective for extended depth range,” in Proceeding of IEEE Transactions on Instrumentation and Measurement (IEEE, 2017), pp. 441–450.
[Crossref]

Pentland, A. P.

A. P. Pentland, “A new sense for depth of field,” in Proceeding of IEEE Transactions on Pattern Analysis and Machine Intelligence (IEEE, 1987), pp. 523–531.
[Crossref]

Pervolz, K.

S. May, B. Werner, H. Surmann, and K. Pervolz, “3D time-of-flight cameras for mobile robotics,” in Proceeding of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2007), pp. 790–795.

Powell, R. A.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Sansoni, G.

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
[Crossref]

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “A depth from defocus measurement system using a liquid lens objective for extended depth range,” in Proceeding of IEEE Transactions on Instrumentation and Measurement (IEEE, 2017), pp. 441–450.
[Crossref]

Sato, S.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. 43(35), 6407–6412 (2004).
[Crossref] [PubMed]

Schamm, T.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Shafer, S. A.

Y. Xiong and S. A. Shafer, “Depth from focusing and defocusing,” in Proceeding of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 68–73.
[Crossref]

Staunton, R. C.

A. N. Joseph Raj and R. C. Staunton, “Rational filters design for depth from defocus,” Pattern Recognit. 45(1), 198–207 (2012).
[Crossref]

Strand, M.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Sun, Y. N.

C. H. Wu, Y. N. Sun, and C. C. Chang, “Three-dimensional modeling from endoscopic video using geometric constraints via feature positioning,” in Proceeding of IEEE Transactions on Bio-medical Engineering (IEEE, 2007), pp. 1199–1211.

Surmann, H.

S. May, B. Werner, H. Surmann, and K. Pervolz, “3D time-of-flight cameras for mobile robotics,” in Proceeding of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2007), pp. 790–795.

Takahashi, S.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

Tang, Q.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Torras, C.

S. Foix, G. Alenya, and C. Torras, “Lock-in time-of-flight (ToF) cameras: A survey,” in Proceeding of IEEE Conference on Sensors Journal (IEEE, 2011), pp. 1917–1926.

Tsai, H. L.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Uchida, M.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

Wang, B.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. 43(35), 6407–6412 (2004).
[Crossref] [PubMed]

Watanabe, M.

M. Watanabe and S. K. Nayar, “Rational filters for passive depth from defocus,” Int. J. Comput. Vis. 27(3), 203–225 (1998).
[Crossref]

Werner, B.

S. May, B. Werner, H. Surmann, and K. Pervolz, “3D time-of-flight cameras for mobile robotics,” in Proceeding of IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2007), pp. 790–795.

Wu, C. H.

C. H. Wu, Y. N. Sun, and C. C. Chang, “Three-dimensional modeling from endoscopic video using geometric constraints via feature positioning,” in Proceeding of IEEE Transactions on Bio-medical Engineering (IEEE, 2007), pp. 1199–1211.

Xiao, H.

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
[Crossref]

Xin, Y.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Xiong, Y.

Y. Xiong and S. A. Shafer, “Depth from focusing and defocusing,” in Proceeding of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1993), pp. 68–73.
[Crossref]

Yamaguchi, M.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

Yanase, S.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

Ye, M.

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
[Crossref]

M. Ye, B. Wang, and S. Sato, “Liquid-crystal lens with a focal length that is variable in a wide range,” Appl. Opt. 43(35), 6407–6412 (2004).
[Crossref] [PubMed]

Zhu, J. H.

J. F. Li, Y. K. Guo, J. H. Zhu, X. Lin, Y. Xin, K. Duan, and Q. Tang, “Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision,” Opt. Lasers Eng. 45(11), 1077–1087 (2007).
[Crossref]

Zollner, J.

T. Schamm, M. Strand, T. Gumpp, R. Kohlhaas, J. Zollner, and R. Dillmann, “ Vision and ToF-based driving assistance for a personal transporter,” in Proceeding of IEEE Conference on Advanced Robotics (IEEE, 2009), pp. 1–6.

Appl. Opt. (1)

Int. J. Comput. Vis. (1)

M. Watanabe and S. K. Nayar, “Rational filters for passive depth from defocus,” Int. J. Comput. Vis. 27(3), 203–225 (1998).
[Crossref]

Jpn. J. Appl. Phys. (1)

M. Ye, B. Wang, M. Uchida, S. Yanase, S. Takahashi, M. Yamaguchi, and S. Sato, “Low-voltage-driving liquid crystal lens,” Jpn. J. Appl. Phys. 49(10), 100204 (2010).
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Meas. Sci. Technol. (1)

C. H. Lin, R. A. Powell, L. Jiang, H. Xiao, S. J. Chen, and H. L. Tsai, “Real-time depth measurement for micro-holes drilled by lasers,” Meas. Sci. Technol. 21(2), 025307 (2010).
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Mol. Cryst. Liq. Cryst. (Phila. Pa.) (1)

M. Kawamura and S. Ishikuro, “Feature extraction from multiply focal images by using a liquid crystal lens,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 613(1), 51–58 (2015).
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Opt. Lasers Eng. (2)

S. Pasinetti, I. Bodini, M. Lancini, F. Docchio, and G. Sansoni, “Automatic selection of focal lengths in a depth from defocus measurement system based on liquid lenses,” Opt. Lasers Eng. 96, 68–74 (2017).
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Proc. SPIE (1)

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Figures (5)

Fig. 1
Fig. 1 Structure of LC lens.
Fig. 2
Fig. 2 DFD optic model of imaging system with LC lens.
Fig. 3
Fig. 3 Images with maximum positive (a) and negative (b) optical power.
Fig. 4
Fig. 4 Depth map.
Fig. 5
Fig. 5 Relationship between the depth value and the reciprocal of the object distance.

Tables (2)

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Table 1 Optical power and rms aberration.

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Table 2 Object distance and depth value.

Equations (12)

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  n e ( θ )= n e n o / n e 2 sin 2 θ+ n o 2 cos 2 θ .
P g + P max + d P g P max + .
P= P max ( 1 )P= P min +( 1+ )P.
1 u + 1 v 0 =P 1 u + 1 v 1 = P max 1 u + 1 v 2 = P min
R 1 =a v 0 ( 1 )P= a f g u 0 u 0 f g (1)P.
R 2 =a v 0 ( 1+ )P= a f g u 0 u 0 f g (1+)P.
h( x,y )= 1 π R 2 1 2R x 2 + y 2
H( p,q )= 1 πR p 2 + q 2 j 1 (2πR p 2 + q 2 ),
h 1 ( x,y )= h 1 (x,y;(1)P,a f g )= 1 π [a f g ( 1 )P] 2 ( 1 2a f g(1)P x 2 + y 2 )
h 2 ( x,y )= h 2 (x,y;(1+)P,a f g )= 1 π [a f g ( 1+ )P] 2 ( 1 2a f g(1+)P x 2 + y 2 .
M P = M(p,q) P(p,q) = H 2 (u,v,( 1+ )P,a f g ) H 1 (u,v,( 1 )P,a f g ) H 2 (u,v,( 1+ )P,a f g )+ H 1 (u,v,( 1 )P,a f g ) ,
f r f 2 0 = 1.22( u 0 f g ) 2a f g u 0 ( 1+1 )P .

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