Abstract

Three-dimensional image distortion caused by mismatch between autostereoscopic displays and contents is analyzed. For a given three-dimensional object scene, the original light ray field in the contents and deformed one by the autostereoscopic displays are calculated. From the deformation of the light ray field, the distortion of the resultant three-dimensional image is finally deduced. The light ray field approach enables generalized distortion analysis across different autostereoscopic display techniques. The analysis result is verified experimentally when multi-view contents are applied to a multi-view display of non-matching parameters and a horizontal parallax only integral imaging display.

© 2012 OSA

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2012 (2)

2011 (4)

2010 (3)

V. Saveljev, “Image and observer regions in 3D displays,” J. Inform. Disp.11(2), 68–75 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

2009 (2)

2008 (1)

2007 (2)

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

J.-Y. Son, Y. N. Gruts, K.-D. Kwack, K.-H. Cha, and S.-K. Kim, “Stereoscopic image distortion in radial camera and projector configurations,” J. Opt. Soc. Am. A24(3), 643–650 (2007).
[CrossRef] [PubMed]

2006 (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE94(3), 591–607 (2006).
[CrossRef]

2005 (1)

2004 (3)

J.-H. Park, Y. Kim, J. Kim, S.-W. Min, and B. Lee, “Three-dimensional display scheme based on integral imaging with three-dimensional information processing,” Opt. Express12(24), 6020 (2004).
[CrossRef] [PubMed]

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
[CrossRef]

L. M. J. Meesters, W. A. IJsselsteijn, and P. J. H. Seuntiens, “A survey of perceptual evaluations and requirements of three-dimensional TV,” IEEE Trans. Circ. Syst. Video Tech.14(3), 381–391 (2004).
[CrossRef]

1998 (1)

1993 (1)

A. Woods, T. Docherty, and R. Koch, “Image distortions in stereoscopic video systems,” Proc. SPIE1915, 36–48 (1993).
[CrossRef]

1980 (1)

T. Okoshi, “Three-dimensional displays,” Proc. IEEE68(5), 548–564 (1980).
[CrossRef]

Arai, J.

Benzie, P.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Cha, K.-H.

Chen, N.

Choi, H.-J.

Docherty, T.

A. Woods, T. Docherty, and R. Koch, “Image distortions in stereoscopic video systems,” Proc. SPIE1915, 36–48 (1993).
[CrossRef]

Fukushima, R.

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
[CrossRef]

Gruts, Y. N.

Hahn, J.

Haino, Y.

Hamagishi, G.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Hirayama, Y.

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
[CrossRef]

Hisatake, Y.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Hong, J.

Hong, K.

Hopf, K.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Horikoshi, T.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Hoshino, H.

Hwang, J.-J.

IJsselsteijn, W. A.

L. M. J. Meesters, W. A. IJsselsteijn, and P. J. H. Seuntiens, “A survey of perceptual evaluations and requirements of three-dimensional TV,” IEEE Trans. Circ. Syst. Video Tech.14(3), 381–391 (2004).
[CrossRef]

Isono, H.

Javidi, B.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE94(3), 591–607 (2006).
[CrossRef]

A. Stern and B. Javidi, “Ray phase space approach for 3-D imaging and 3-D optical data representation,” J. Display Technol.1(1), 141–150 (2005).
[CrossRef]

Jeong, K.-M.

Jung, J.-H.

Kato, C.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Kawakita, M.

Kim, H.

Kim, J.

Kim, S.-K.

Kim, Y.

Koch, R.

A. Woods, T. Docherty, and R. Koch, “Image distortions in stereoscopic video systems,” Proc. SPIE1915, 36–48 (1993).
[CrossRef]

Koike, T.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Kwack, K.-D.

Lee, B.

Lee, B.-R.

Lee, K.-H.

Lee, M.-J.

Martinez-Berti, E.

Mashitani, K.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Meesters, L. M. J.

L. M. J. Meesters, W. A. IJsselsteijn, and P. J. H. Seuntiens, “A survey of perceptual evaluations and requirements of three-dimensional TV,” IEEE Trans. Circ. Syst. Video Tech.14(3), 381–391 (2004).
[CrossRef]

Min, S.-W.

Mishina, T.

Nomura, T.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Oi, R.

Okano, F.

Okoshi, T.

T. Okoshi, “Three-dimensional displays,” Proc. IEEE68(5), 548–564 (1980).
[CrossRef]

Okui, M.

Park, J.-H.

Park, S.-

Rakkolainen, I.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Ricolfe-Viala, C.

Sainov, V.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Saishu, T.

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
[CrossRef]

Sanchez-Salmeron, A.-J.

Sasaki, H.

Sato, M.

Saveljev, V.

V. Saveljev, “Image and observer regions in 3D displays,” J. Inform. Disp.11(2), 68–75 (2010).
[CrossRef]

Senoh, T.

Seuntiens, P. J. H.

L. M. J. Meesters, W. A. IJsselsteijn, and P. J. H. Seuntiens, “A survey of perceptual evaluations and requirements of three-dimensional TV,” IEEE Trans. Circ. Syst. Video Tech.14(3), 381–391 (2004).
[CrossRef]

Son, J.-Y.

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE94(3), 591–607 (2006).
[CrossRef]

A. Stern and B. Javidi, “Ray phase space approach for 3-D imaging and 3-D optical data representation,” J. Display Technol.1(1), 141–150 (2005).
[CrossRef]

Suehiro, K.

Surman, P.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Taira, K.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
[CrossRef]

Uehara, S.-I.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Ujike, H.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Umezu, N.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Urey, H.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

von Kopylow, C.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Watanabe, N.

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

Watson, J.

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

Woods, A.

A. Woods, T. Docherty, and R. Koch, “Image distortions in stereoscopic video systems,” Proc. SPIE1915, 36–48 (1993).
[CrossRef]

Yamamoto, K.

Yoon, Y.-S.

Yoshimura, M.

Yuuki, A.

T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

Yuyama, I.

Appl. Opt. (4)

IEEE Trans. Circ. Syst. Video Tech. (2)

L. M. J. Meesters, W. A. IJsselsteijn, and P. J. H. Seuntiens, “A survey of perceptual evaluations and requirements of three-dimensional TV,” IEEE Trans. Circ. Syst. Video Tech.14(3), 381–391 (2004).
[CrossRef]

P. Benzie, J. Watson, P. Surman, I. Rakkolainen, K. Hopf, H. Urey, V. Sainov, and C. von Kopylow, “A survey of 3DTV display: techniques and technologies,” IEEE Trans. Circ. Syst. Video Tech.17(11), 1647–1658 (2007).
[CrossRef]

J. Display Technol. (1)

J. Inform. Disp. (1)

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[CrossRef]

J. Opt. Soc. Am. A (3)

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[CrossRef]

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE94(3), 591–607 (2006).
[CrossRef]

Proc. SPIE (1)

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[CrossRef]

SID Tech. Dig. (3)

T. Saishu, K. Taira, R. Fukushima, and Y. Hirayama, “Distortion control in a one-dimensional integral imaging autostereoscopic display system with parallel optical beam groups,” SID Tech. Dig.35(1), 1438–1441 (2004).
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T. Horikoshi, S.-I. Uehara, T. Koike, C. Kato, K. Taira, G. Hamagishi, K. Mashitani, T. Nomura, A. Yuuki, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of 3D image quality on autostereoscopic displays: proposal of interocular 3D purity,” SID Tech. Dig.41(1), 331–334 (2010).
[CrossRef]

S.-I. Uehara, T. Horikoshi, C. Kato, T. Koike, G. Hamagishi, K. Mashitani, T. Nomura, K. Taira, A. Yuuki, N. Umezu, N. Watanabe, Y. Hisatake, and H. Ujike, “Characterization of motion parallax on multi-view/integral-imaging displays,” SID Tech. Dig.41(1), 661–664 (2010).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Concept of light ray field.

Fig. 2
Fig. 2

Light ray field of a single 3D point object.

Fig. 3
Fig. 3

Rays captured at several camera positions.

Fig. 4
Fig. 4

Light ray field in the MV contents.

Fig. 5
Fig. 5

Ray reconstruction characteristics.

Fig. 6
Fig. 6

Reconstructed light ray field.

Fig. 7
Fig. 7

Light ray field reconstruction by correct and incorrect display panels.

Fig. 8
Fig. 8

Light ray field and 3D object.

Fig. 9
Fig. 9

Image distortions when MV image contents are applied to non-matching MV display panel.

Fig. 10
Fig. 10

Image distortion when MV image contents are applied to HPO InIm display panel

Fig. 11
Fig. 11

Experimental setup

Fig. 12
Fig. 12

Experimental result: MV contents to MV display, real cylindrical object located at + 20mm.

Fig. 13
Fig. 13

Experimental result: MV contents to MV display, real cylindrical object located at + 10mm.

Fig. 14
Fig. 14

Experimental result: MV contents to HPO InIm display, real cylindrical object located at + 20mm.

Fig. 15
Fig. 15

Experimental result: MV contents to HPO InIm display, real cylindrical object located at + 9mm.

Equations (18)

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l( x, θ x )=f( x+ θ x z;z ).
l ( x, θ x )= f ( x+ θ x z ; z ).
f ( x+ θ x z ; z )=f( x+( θ x a z c )z;z ).
f ( u ; z )=f( u a z c z+ θ x ( z z );z ).
f ( x;z )=f( x a z c z;z ),
l ( x, θ x )= f ( x+ θ x z ; z )=f( x+ z c b z c θ x z;z ),
f ( u ; z )=f( u + θ x ( z c b z c z z );z ).
f ( x;z )=f( x; z c z c b z ),
l ( x, nkΔ x c x z c )=l( x, nΔ x c x z c ).
l ( x, θ x )=l( x, θ x (k1)nΔ x c z c )=l( x, 1 k ( θ x (k1)x z c ) ),
l ( x, θ x )= f ( x+ θ x z ; z )=f( x+ 1 k ( θ x ( k1 )x z c )z;z ).
f ( u ; z )=f( ( 1 (k1)z k z c ) u + θ x ( z k + ( k1 )z z k z c z );z ).
z= k z 1+ (k1) z / z c .
f ( x;z )=f( ( 1 (k1)z z c +(k1)z )x; kz 1+ (k1)z / z c ),
l ( x, θ xn )=l( x, x cn x z c ).
l ( x, θ x )=l( x, θ x Δ x c xΔ θ x z c Δ θ x ).
l ( x, θ x )= f ( x+ θ x z ; z )=f( x+ θ x Δ x c xΔ θ x z c Δ θ z;z ),
f ( x;z )=f( ( 1 Δ θ x z Δ x c +Δ θ x z )x;( z c Δ θ x Δ x c +Δ θ x z )z ).

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