Light field acquisition using a planar catadioptric system

Weitao Song; Yue Liu; Weiming Li; Yongtian Wang

doi:10.1364/OE.23.031126

Light field acquisition using a planar catadioptric system

Weitao Song, Yue Liu, Weiming Li, and Yongtian Wang

Optics Express
Vol. 23,
Issue 24,
pp. 31126-31135
(2015)
•https://doi.org/10.1364/OE.23.031126

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Weitao Song, Yue Liu, Weiming Li, and Yongtian Wang, "Light field acquisition using a planar catadioptric system," Opt. Express 23, 31126-31135 (2015)

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Related Topics
Table of Contents Category
- Geometric Optics and Optical Design
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Three-dimensional image processing (100.6890)
- Three-dimensional image acquisition (110.6880)
- Three-dimensional sensing (150.6910)

About this Article
History
- Original Manuscript: October 5, 2015
- Revised Manuscript: November 7, 2015
- Manuscript Accepted: November 15, 2015
- Published: November 20, 2015
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 11, Iss. 1

Accessible

Open Access

Abstract

We propose a novel light field acquisition method based on a planar catadioptric system. Only one camera with a large field-of-view (FOV) is required, and multiple virtual cameras can be created via an array of planar mirrors to acquire the light field from different views. The spatial distribution of the virtual cameras can be configured flexibly to acquire particular light rays, which can be controlled by simply changing the positions of the mirrors. Compared with previous systems, no aberration or reduction in light transmittance is introduced by the planar mirrors, which ensures image quality. In this study, the design method of the planar catadioptric system is provided and a calibration procedure of its computational model is analyzed in detail. The method is verified by a prototype system, with which the correct digital refocusing results are achieved using the acquired, calibrated light field.

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References

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Publication

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W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
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W. Song, Q. Zhu, Y. Liu, and Y. Wang, “Omnidirectional-view three-dimensional display based on rotating selective-diffusing screen and multiple mini-projectors,” Appl. Opt. 54(13), 4154–4160 (2015).
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T. E. Bishop and P. Favaro, “The light field camera: Extended depth of field, aliasing, and superresolution,” IEEE Trans. Pattern Anal. Mach. Intell. 34(5), 972–986 (2012).
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J. C. Yang, M. Everett, C. Buehler, and L. McMillan, “A Real-Time Distributed Light Field Camera,” in Proceedings of the 13th Eurographics workshop on Rendering (Eurographics Association, 2002), pp. 77–86.
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X. Cao, Z. Geng, and T. Li, “Dictionary-based light field acquisition using sparse camera array,” Opt. Express 22(20), 24081–24095 (2014).
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R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24(3), 735–744 (2005).
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[Crossref]
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A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26(3), 69 (2007).
[Crossref]
D. Lanman, “Mask-based Light Field Capture and Display,” Ph.D. Dissertation, Brown University, School of Engineering, 2010.
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D. Cho, M. Kim, and Y. Tai, “Modeling the calibration pipeline of the Lytro camera for high quality light-field image reconstruction,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2013), pp. 3280–3287.
[Crossref]
W. Li and Y. Li, “Generic camera model and its calibration for computational integral imaging and 3D reconstruction,” J. Opt. Soc. Am. A 28(3), 318–326 (2011).
[Crossref] [PubMed]
Y. Taguchi, A. Agrawal, A. Veeraraghavan, S. Ramalingam, and R. Raskar, “Axial-cones: modeling spherical catadioptric cameras for wide-angle light field rendering,” ACM Trans. Graph. 29(6), 172 (2010).
[Crossref]
D. Lanman, D. Crispell, M. Wachs, and G. Taubin, “Spherical catadioptric arrays: Construction, multi-view geometry, and calibration,” in International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE 2006) pp. 81–88.
[Crossref]
J. Gluckman and S. K. Nayar, “Planar catadioptric stereo: Geometry and calibration,” Int. J. Comput. Vis. 44(1), 65–79 (2001).
[Crossref]
K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
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[Crossref]
W. Li and Y. F. Li, “Single-camera panoramic stereo imaging system with a fisheye lens and a convex mirror,” Opt. Express 19(7), 5855–5867 (2011).
[Crossref] [PubMed]
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2015 (1)

W. Song, Q. Zhu, Y. Liu, and Y. Wang, “Omnidirectional-view three-dimensional display based on rotating selective-diffusing screen and multiple mini-projectors,” Appl. Opt. 54(13), 4154–4160 (2015).
[Crossref]

2014 (2)

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

X. Cao, Z. Geng, and T. Li, “Dictionary-based light field acquisition using sparse camera array,” Opt. Express 22(20), 24081–24095 (2014).
[Crossref] [PubMed]

2012 (2)

T. E. Bishop and P. Favaro, “The light field camera: Extended depth of field, aliasing, and superresolution,” IEEE Trans. Pattern Anal. Mach. Intell. 34(5), 972–986 (2012).
[Crossref] [PubMed]

T. Georgiev and A. Lumsdaine, “The multifocus plenoptic camera,” Proc. SPIE 8299, 829908 (2012).
[Crossref]

2011 (3)

W. Li and Y. Li, “Generic camera model and its calibration for computational integral imaging and 3D reconstruction,” J. Opt. Soc. Am. A 28(3), 318–326 (2011).
[Crossref] [PubMed]

W. Li and Y. F. Li, “Single-camera panoramic stereo imaging system with a fisheye lens and a convex mirror,” Opt. Express 19(7), 5855–5867 (2011).
[Crossref] [PubMed]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30(4), 95 (2011).
[Crossref]

2010 (2)

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Y. Taguchi, A. Agrawal, A. Veeraraghavan, S. Ramalingam, and R. Raskar, “Axial-cones: modeling spherical catadioptric cameras for wide-angle light field rendering,” ACM Trans. Graph. 29(6), 172 (2010).
[Crossref]

2009 (1)

J. H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009).
[Crossref] [PubMed]

2008 (1)

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

2007 (1)

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Trans. Graph. 26(3), 69 (2007).
[Crossref]

2006 (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

2005 (1)

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24(3), 735–744 (2005).
[Crossref]

2004 (1)

K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
[Crossref] [PubMed]

2002 (1)

C. Perwaß and L. Wietzke, “Single lens 3D-camera with extended depth-of-field,” Proc. SPIE 8291, 1–15 (2002).

2001 (2)

J. Gluckman and S. K. Nayar, “Planar catadioptric stereo: Geometry and calibration,” Int. J. Comput. Vis. 44(1), 65–79 (2001).
[Crossref]

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

1908 (1)

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. Theor. Appl. 7(1), 821–825 (1908).
[Crossref]

Agrawal, A.

Ahuja, N.

K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
[Crossref] [PubMed]

Bishop, T. E.

T. E. Bishop and P. Favaro, “The light field camera: Extended depth of field, aliasing, and superresolution,” IEEE Trans. Pattern Anal. Mach. Intell. 34(5), 972–986 (2012).
[Crossref] [PubMed]

Buehler, C.

J. C. Yang, M. Everett, C. Buehler, and L. McMillan, “A Real-Time Distributed Light Field Camera,” in Proceedings of the 13th Eurographics workshop on Rendering (Eurographics Association, 2002), pp. 77–86.

Cao, X.

X. Cao, Z. Geng, and T. Li, “Dictionary-based light field acquisition using sparse camera array,” Opt. Express 22(20), 24081–24095 (2014).
[Crossref] [PubMed]

Chen, H. H.

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

Cheng, D.

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

Cho, D.

D. Cho, M. Kim, and Y. Tai, “Modeling the calibration pipeline of the Lytro camera for high quality light-field image reconstruction,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2013), pp. 3280–3287.
[Crossref]

Curless, B.

T. Georgeiv, K. C. Zheng, B. Curless, D. Salesin, S. Nayar, and C. Intwala, “Spatio-angular resolution tradeoffs in integral photography,” in Proceedings of the 17th Eurographics conference on Rendering Techniques (Eurographics Association,2006), pp 263–272.

Dansereau, D.

D. Dansereau, O. Pizarro, and S. Williams, “Decoding, calibration and rectification for lenselet-based plenoptic cameras,” In IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2013), pp. 1027–1034.
[Crossref]

Everett, M.

Favaro, P.

T. E. Bishop and P. Favaro, “The light field camera: Extended depth of field, aliasing, and superresolution,” IEEE Trans. Pattern Anal. Mach. Intell. 34(5), 972–986 (2012).
[Crossref] [PubMed]

Fernandez, S.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Geng, Z.

X. Cao, Z. Geng, and T. Li, “Dictionary-based light field acquisition using sparse camera array,” Opt. Express 22(20), 24081–24095 (2014).
[Crossref] [PubMed]

Georgeiv, T.

Georgiev, T.

T. Georgiev and A. Lumsdaine, “The multifocus plenoptic camera,” Proc. SPIE 8299, 829908 (2012).
[Crossref]

Gluckman, J.

J. Gluckman and S. K. Nayar, “Planar catadioptric stereo: Geometry and calibration,” Int. J. Comput. Vis. 44(1), 65–79 (2001).
[Crossref]

Hadap, S.

M. W. Tao, S. Hadap, J. Malik, and R. Ramamoorthi, “Depth from combining defocus and correspondence using light-field cameras,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2013), pp. 673–680.
[Crossref]

Hanrahan, P.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques(ACM, 1996), pp. 31–42.

Heidrich, W.

Hong, K.

J. H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009).
[Crossref] [PubMed]

Horowitz, M. A.

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

Hua, H.

K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
[Crossref] [PubMed]

Intwala, C.

Javidi, B.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Kim, M.

Lanman, D.

Lee, B.

J. H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009).
[Crossref] [PubMed]

Lee, H. K.

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

Levoy, M.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques(ACM, 1996), pp. 31–42.

Liang, C. K.

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

Lin, T. H.

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

Lippmann, G.

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. Theor. Appl. 7(1), 821–825 (1908).
[Crossref]

Liu, C.

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

Liu, Y.

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

Llado, X.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Lumsdaine, A.

T. Georgiev and A. Lumsdaine, “The multifocus plenoptic camera,” Proc. SPIE 8299, 829908 (2012).
[Crossref]

Malik, J.

McMillan, L.

Mohan, A.

Nayar, S.

Nayar, S. K.

J. Gluckman and S. K. Nayar, “Planar catadioptric stereo: Geometry and calibration,” Int. J. Comput. Vis. 44(1), 65–79 (2001).
[Crossref]

Ng, R.

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24(3), 735–744 (2005).
[Crossref]

Park, J. H.

J. H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009).
[Crossref] [PubMed]

Perwaß, C.

C. Perwaß and L. Wietzke, “Single lens 3D-camera with extended depth-of-field,” Proc. SPIE 8291, 1–15 (2002).

Pizarro, O.

Pribanic, T.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Ramalingam, S.

Ramamoorthi, R.

Raskar, R.

Salesin, D.

Salvi, J.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Smulski, M.

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

Song, W.

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Taguchi, Y.

Tai, Y.

Tan, K. H.

K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
[Crossref] [PubMed]

Tao, M. W.

Tumblin, J.

Veeraraghavan, A.

Wang, Y.

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

Wetzstein, G.

Wietzke, L.

C. Perwaß and L. Wietzke, “Single lens 3D-camera with extended depth-of-field,” Proc. SPIE 8291, 1–15 (2002).

Wilburn, B. S.

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

Williams, S.

Wong, B. Y.

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

Yang, J. C.

Zheng, K. C.

Zhu, Q.

ACM Trans. Graph. (5)

C. K. Liang, T. H. Lin, B. Y. Wong, C. Liu, and H. H. Chen, “Programmable aperture photography: Multiplexed light field acquisition,” ACM Trans. Graph. 27(3), 55 (2008).
[Crossref]

R. Ng, “Fourier slice photography,” ACM Trans. Graph. 24(3), 735–744 (2005).
[Crossref]

Appl. Opt. (2)

J. H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009).
[Crossref] [PubMed]

Chin. Opt. Lett. (1)

W. Song, Y. Wang, D. Cheng, and Y. Liu, “Light f ield head-mounted display with correct focus cue using micro structure array,” Chin. Opt. Lett. 12(6), 060010 (2014).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

T. E. Bishop and P. Favaro, “The light field camera: Extended depth of field, aliasing, and superresolution,” IEEE Trans. Pattern Anal. Mach. Intell. 34(5), 972–986 (2012).
[Crossref] [PubMed]

K. H. Tan, H. Hua, and N. Ahuja, “Multiview panoramic cameras using mirror pyramids,” IEEE Trans. Pattern Anal. Mach. Intell. 26(7), 941–946 (2004).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

J. Gluckman and S. K. Nayar, “Planar catadioptric stereo: Geometry and calibration,” Int. J. Comput. Vis. 44(1), 65–79 (2001).
[Crossref]

J. Opt. Soc. Am. A (1)

W. Li and Y. Li, “Generic camera model and its calibration for computational integral imaging and 3D reconstruction,” J. Opt. Soc. Am. A 28(3), 318–326 (2011).
[Crossref] [PubMed]

J. Phys. Theor. Appl. (1)

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. Theor. Appl. 7(1), 821–825 (1908).
[Crossref]

Opt. Express (2)

X. Cao, Z. Geng, and T. Li, “Dictionary-based light field acquisition using sparse camera array,” Opt. Express 22(20), 24081–24095 (2014).
[Crossref] [PubMed]

W. Li and Y. F. Li, “Single-camera panoramic stereo imaging system with a fisheye lens and a convex mirror,” Opt. Express 19(7), 5855–5867 (2011).
[Crossref] [PubMed]

Pattern Recognit. (1)

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010).
[Crossref]

Proc. IEEE (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Proc. SPIE (3)

B. S. Wilburn, M. Smulski, H. K. Lee, and M. A. Horowitz, “Light field video camera,” Proc. SPIE 4674, 29–36 (2001).
[Crossref]

T. Georgiev and A. Lumsdaine, “The multifocus plenoptic camera,” Proc. SPIE 8299, 829908 (2012).
[Crossref]

C. Perwaß and L. Wietzke, “Single lens 3D-camera with extended depth-of-field,” Proc. SPIE 8291, 1–15 (2002).

Other (16)

D. Lanman, “Mask-based Light Field Capture and Display,” Ph.D. Dissertation, Brown University, School of Engineering, 2010.

D. Lanman, D. Crispell, M. Wachs, and G. Taubin, “Spherical catadioptric arrays: Construction, multi-view geometry, and calibration,” in International Symposium on 3D Data Processing, Visualization, and Transmission (IEEE 2006) pp. 81–88.
[Crossref]

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Supplementary Material (1)

Name	Description
» Visualization 1: AVI (13847 KB)	digital refocusing video using the calibrated light field

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Fig. 1 Schematic diagram of a light field acquisition method using a planar catadioptric system.

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Fig. 2 Schematic diagrams of (a) the position of virtual cameras & the placement of mirrors and (b) the constraints of mirror placements.

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Fig. 3 Multiple mirrors in a planar catadioptric system with virtual sensors on a plane (take 5 × 4 virtual cameras system as an example).

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Fig. 4 Module of the prototype of catadioptric light field capture system.

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Fig. 5 (a) One of the structured light images. (b) Structured light pattern decoding results in horizontal (X) direction. (c) Structured light pattern decoding results in vertical (Y) direction.

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Fig. 6 (a) Calibrated results of the virtual cameras. (b) Enlarged image of the calibrated results. (unit: mm)

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Fig. 7 (a) Experimental setup and (b) captured image by our developed system

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Fig. 8 Digital refocusing images using the captured image (Fig. 7(b)) by our developed system (see Visualization 1).

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Equations (13)

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\sum_{i = 0}^{m - 1} φ^{i} = Ψ, \sum_{j = 0}^{n - 1} π^{j} = Φ

r^{i j} = (\tan (\sum_{i = 0}^{i} φ^{i} - φ^{i} / 2 - Ψ / 2), \tan (\sum_{j = 0}^{j} π^{j} - π^{j} / 2 - Φ / 2), 1)

{\hat{n}}^{i j} = ({\hat{r}}^{i j} - {\hat{l}}^{i j}) / ‖ {\hat{r}}^{i j} - {\hat{l}}^{i j} ‖

M^{i j} = {({\hat{n}}_{x}^{i j}, {\hat{n}}_{y}^{i j}, {\hat{n}}_{z}^{i j}, - λ^{i j^{2}} / 2)}^{T}

λ^{i j} = \frac{- η}{{\hat{n}}_{x}^{i j} {\hat{l}}_{x}^{i j} + {\hat{n}}_{y}^{i j} {\hat{l}}_{y}^{i j} + {\hat{n}}_{z}^{i j} {\hat{l}}_{z}^{i j}}

[\begin{matrix} x_{p} \\ y_{p} \\ 1 \end{matrix}] = [\begin{matrix} f_{1} & α f_{1} & x_{0} \\ 0 & f_{2} & y_{0} \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} (1 + k_{1} r^{2} + k_{2} r^{4} + k_{5} r^{6}) x_{n} + 2 k_{3} x_{n} y_{n} + k_{4} (r^{2} + 2 x_{n}^{2}) \\ (1 + k_{1} r^{2} + k_{2} r^{4} + k_{5} r^{6}) y_{n} + 2 k_{4} x_{n} y_{n} + k_{3} (r^{2} + 2 y_{n}^{2}) \\ 1 \end{matrix}]

z_{c} {[\begin{matrix} x_{n} & y_{n} & 1 \end{matrix}]}^{T} = [\begin{matrix} R^{i j} & T^{i j} \end{matrix}] {[\begin{matrix} X & Y & Z & 1 \end{matrix}]}^{T}

f_{1} = \frac{U_{x}}{2 \tan (Ψ / 2)}, f_{2} = \frac{U_{y}}{2 \tan (Φ / 2)}

x_{0} = U_{x} / 2, y_{0} = U_{y} / 2

α = k_{1} = k_{2} = k_{3} = k_{4} = k_{5} = 0

{\begin{matrix} X r_{11}^{i j} + Y r_{12}^{i j} + Z r_{13}^{i j} - x_{n} X r_{31}^{i j} - x_{n} Y r_{32}^{i j} - x_{n} Z r_{33}^{i j} + t_{1}^{i j} - t_{3}^{i j} = 0 \\ X r_{21}^{i j} + Y r_{22}^{i j} + Z r_{23}^{i j} - y_{n} X r_{31}^{i j} - y_{n} Y r_{32}^{i j} - y_{n} Z r_{33}^{i j} + t_{2}^{i j} - t_{3}^{i j} = 0 \end{matrix}

l_{p} = - i n v (R^{i j}) T^{i j}

l_{v} = \frac{i n v (R^{i j}) {[\begin{matrix} x_{n} & y_{n} & 1 \end{matrix}]}^{T} - i n v (R^{i j}) T^{i j}}{‖ i n v (R^{i j}) {[\begin{matrix} x_{n} & y_{n} & 1 \end{matrix}]}^{T} - i n v (R^{i j}) T^{i j} ‖}

Abstract

References

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