Abstract

We describe a face recognition system based on two different three-dimensional (3D) sensors. We use 3D sensors to overcome the pose-variation problems that cannot be effectively solved in two-dimensional images. We acquire input data based on a structured-light system and compare it with 3D faces that are obtained from a 3D laser scanner. Owing to differences in structure between the input data and the 3D faces, we can generate the range images of the probe and stored images. For estimating the head pose of input data, we propose a novel error-compensated singular-value decomposition that geometrically estimates the rotation angle. Face recognition rates obtained with principal component analysis on various range images of 35 people in different poses show promising results.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2003

F. Tsalakanidou, D. Tzocaras, M. Strintzis, “Use of depth and colour eigenfaces for face recognition,” Pattern Recogn. Lett. 24, 1427–1435 (2003).
[CrossRef]

V. Blanz, T. Vetter, “Face recognition based on fitting a 3D morphable model,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1063–1074 (2003).
[CrossRef]

H. Song, U. Yang, K. Sohn, “3D face recognition under pose varying environments,” Lect. Notes Comput. Sci. 2908, 333–347 (2003).
[CrossRef]

2002

Y. Wang, C. Chua, Y. Ho, “Facial feature detection and face recognition from 2D and 3D images,” Pattern Recogn. Lett. 23, 1191–1202 (2002).
[CrossRef]

1995

R. Chellappa, C. L. Wilson, S. Sirohey, “Human and machine recognition of faces: a survey,” Proc. IEEE 83, 705–740 (1995).
[CrossRef]

1994

T. S. Huang, A. N. Netravali, “Motion and structure from feature correspondences: a review,” Proc. IEEE, 82, 252–268 (1994).
[CrossRef]

1991

M. A. Turk, A. P. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).
[CrossRef] [PubMed]

1989

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

1988

1987

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

B. K. P. Horn, “Closed-form solution of absolute orientation using unit quaternion,” J. Opt. Soc. Am. A 4, 629–642 (1987).
[CrossRef]

Achermann, B.

B. Achermann, X. Jiang, H. Bunke, “Face recognition using range images,” in Proceedings of International Conference on Virtual Systems and MultiMedia (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 129–136.

Arun, K. S.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Blanz, V.

V. Blanz, T. Vetter, “Face recognition based on fitting a 3D morphable model,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1063–1074 (2003).
[CrossRef]

Blostein, S. D.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Bowyer, K.

K. Chang, K. Bowyer, P. Flynn, “Face recognition using 2D and 3D facial data,” in Proceedings of the Multimodal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 25–32.

Bunke, H.

B. Achermann, X. Jiang, H. Bunke, “Face recognition using range images,” in Proceedings of International Conference on Virtual Systems and MultiMedia (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 129–136.

Cartoux, J. Y.

J. Y. Cartoux, J. T. LaPreste, M. Richetin, “Face authentication or recognition by profile extraction from range images,” in Proceedings of the Workshop on Interpretation of 3D Scenes (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1989), pp. 194–199.

Chang, K.

K. Chang, K. Bowyer, P. Flynn, “Face recognition using 2D and 3D facial data,” in Proceedings of the Multimodal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 25–32.

Chellappa, R.

R. Chellappa, C. L. Wilson, S. Sirohey, “Human and machine recognition of faces: a survey,” Proc. IEEE 83, 705–740 (1995).
[CrossRef]

W. Zhao, R. Chellappa, A. Rosenfeld, P. J. Phllips, “Face recognition: a survey,” (Center for Automation Research, University of Maryland at College Park, Md., 2000).

Chen, Q.

Q. Chen, H. Wu, T. Shioyama, T. Shimada, “Head pose estimation using both color and feature information,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2842–2847.

Q. Chen, H. Wu, T. Fukumoto, M. Yachida, “3D head pose estimation without feature tracking,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 88–93.
[CrossRef]

Chiaki, H.

H. T. Tanaka, M. Ikeda, H. Chiaki, “Curvature-based face surface recognition using spherical correlation,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 372–377.
[CrossRef]

Chua, C.

Y. Wang, C. Chua, Y. Ho, “Facial feature detection and face recognition from 2D and 3D images,” Pattern Recogn. Lett. 23, 1191–1202 (2002).
[CrossRef]

Chua, C. S.

C. S. Chua, F. Han, Y. K. Ho, “3D human face recognition using point signature,” in Proceedings of the Fourth International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 233–238.
[CrossRef]

Diaz, F. J.

A. B. Moreno, A. Sanchez, J. F. Velez, F. J. Diaz, “Face recognition using 3D surface-extracted descriptors,” in Proceedings of the Irish Machine Vision and Image Processing Conference (IMVIP2003), P. J. Morrow, B. W. Scotney, eds. (University of Ulster, Coleraine, Ireland).

Erlebacher, G.

C. Hesher, A. Srivastava, G. Erlebacher, “A novel technique for face recognition using range images,” in Proceedings of the Seventh International Symposium on Signal Processing and Its Applications (Institute of Electrical and Electronics Engineers, New York, 2003).
[CrossRef]

Flynn, P.

K. Chang, K. Bowyer, P. Flynn, “Face recognition using 2D and 3D facial data,” in Proceedings of the Multimodal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 25–32.

Fukumoto, T.

Q. Chen, H. Wu, T. Fukumoto, M. Yachida, “3D head pose estimation without feature tracking,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 88–93.
[CrossRef]

Gordon, G. G.

G. G. Gordon, “Face recognition based on depth maps and surface curvature,” in Geometric Methods in Computer Vision, B. C. Vemuri, ed., Proc. SPIE1570, 234–249 (1991).
[CrossRef]

G. G. Gordon, “Face recognition based on depth and curvature features,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 808–810.

Han, F.

C. S. Chua, F. Han, Y. K. Ho, “3D human face recognition using point signature,” in Proceedings of the Fourth International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 233–238.
[CrossRef]

Haralick, R. M.

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

Hattori, K.

K. Hattori, S. Matsumori, Y. Sato, “Estimating pose of human face based on symmetry plane using range and intensity images,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1183–1187.

Hesher, C.

C. Hesher, A. Srivastava, G. Erlebacher, “A novel technique for face recognition using range images,” in Proceedings of the Seventh International Symposium on Signal Processing and Its Applications (Institute of Electrical and Electronics Engineers, New York, 2003).
[CrossRef]

Hilden, H. M.

Ho, Y.

Y. Wang, C. Chua, Y. Ho, “Facial feature detection and face recognition from 2D and 3D images,” Pattern Recogn. Lett. 23, 1191–1202 (2002).
[CrossRef]

Ho, Y. K.

C. S. Chua, F. Han, Y. K. Ho, “3D human face recognition using point signature,” in Proceedings of the Fourth International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 233–238.
[CrossRef]

Horn, B. K. P.

Huang, T. S.

T. S. Huang, A. N. Netravali, “Motion and structure from feature correspondences: a review,” Proc. IEEE, 82, 252–268 (1994).
[CrossRef]

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Ikeda, M.

H. T. Tanaka, M. Ikeda, H. Chiaki, “Curvature-based face surface recognition using spherical correlation,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 372–377.
[CrossRef]

Jiang, X.

B. Achermann, X. Jiang, H. Bunke, “Face recognition using range images,” in Proceedings of International Conference on Virtual Systems and MultiMedia (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 129–136.

Joo, H. N.

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

Kawade, M.

S. Lao, Y. Sumi, M. Kawade, F. Tomita, “3D template matching for pose invariant face recognition using 3D facial model built with isoluminance line based stereo vision,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2911–2916.

Kim, M. B.

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

Lao, S.

S. Lao, Y. Sumi, M. Kawade, F. Tomita, “3D template matching for pose invariant face recognition using 3D facial model built with isoluminance line based stereo vision,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2911–2916.

LaPreste, J. T.

J. Y. Cartoux, J. T. LaPreste, M. Richetin, “Face authentication or recognition by profile extraction from range images,” in Proceedings of the Workshop on Interpretation of 3D Scenes (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1989), pp. 194–199.

Lee, C. N.

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

Lee, J. C.

J. C. Lee, E. Milios, “Matching range image of human faces,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 722–726.
[CrossRef]

Masuda, I.

T. Nagamine, T. Uemura, I. Masuda, “3D facial image analysis for human identification,” in Proceedings of the 11th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 324–327.

Matsumori, S.

K. Hattori, S. Matsumori, Y. Sato, “Estimating pose of human face based on symmetry plane using range and intensity images,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1183–1187.

Medioni, G.

G. Medioni, R. Waupotitsch, “Face recognition and modeling in 3D,” in Proceedings of the IEEE International Workshop on Analysis and Modeling of Faces and Gestures (AMFG 2003) (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 232–233.
[CrossRef]

Milios, E.

J. C. Lee, E. Milios, “Matching range image of human faces,” in Proceedings of the Third International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 722–726.
[CrossRef]

Moreno, A. B.

A. B. Moreno, A. Sanchez, J. F. Velez, F. J. Diaz, “Face recognition using 3D surface-extracted descriptors,” in Proceedings of the Irish Machine Vision and Image Processing Conference (IMVIP2003), P. J. Morrow, B. W. Scotney, eds. (University of Ulster, Coleraine, Ireland).

Nagamine, T.

T. Nagamine, T. Uemura, I. Masuda, “3D facial image analysis for human identification,” in Proceedings of the 11th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 324–327.

Negahdaripour, S.

Netravali, A. N.

T. S. Huang, A. N. Netravali, “Motion and structure from feature correspondences: a review,” Proc. IEEE, 82, 252–268 (1994).
[CrossRef]

Pentland, A. P.

M. A. Turk, A. P. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).
[CrossRef] [PubMed]

Phllips, P. J.

W. Zhao, R. Chellappa, A. Rosenfeld, P. J. Phllips, “Face recognition: a survey,” (Center for Automation Research, University of Maryland at College Park, Md., 2000).

Richetin, M.

J. Y. Cartoux, J. T. LaPreste, M. Richetin, “Face authentication or recognition by profile extraction from range images,” in Proceedings of the Workshop on Interpretation of 3D Scenes (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1989), pp. 194–199.

Rosenfeld, A.

W. Zhao, R. Chellappa, A. Rosenfeld, P. J. Phllips, “Face recognition: a survey,” (Center for Automation Research, University of Maryland at College Park, Md., 2000).

Sanchez, A.

A. B. Moreno, A. Sanchez, J. F. Velez, F. J. Diaz, “Face recognition using 3D surface-extracted descriptors,” in Proceedings of the Irish Machine Vision and Image Processing Conference (IMVIP2003), P. J. Morrow, B. W. Scotney, eds. (University of Ulster, Coleraine, Ireland).

Sato, Y.

K. Hattori, S. Matsumori, Y. Sato, “Estimating pose of human face based on symmetry plane using range and intensity images,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 1183–1187.

Shimada, T.

Q. Chen, H. Wu, T. Shioyama, T. Shimada, “Head pose estimation using both color and feature information,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2842–2847.

Shioyama, T.

Q. Chen, H. Wu, T. Shioyama, T. Shimada, “Head pose estimation using both color and feature information,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2842–2847.

Sirohey, S.

R. Chellappa, C. L. Wilson, S. Sirohey, “Human and machine recognition of faces: a survey,” Proc. IEEE 83, 705–740 (1995).
[CrossRef]

Sohn, K.

H. Song, U. Yang, K. Sohn, “3D face recognition under pose varying environments,” Lect. Notes Comput. Sci. 2908, 333–347 (2003).
[CrossRef]

Song, H.

H. Song, U. Yang, K. Sohn, “3D face recognition under pose varying environments,” Lect. Notes Comput. Sci. 2908, 333–347 (2003).
[CrossRef]

Srivastava, A.

C. Hesher, A. Srivastava, G. Erlebacher, “A novel technique for face recognition using range images,” in Proceedings of the Seventh International Symposium on Signal Processing and Its Applications (Institute of Electrical and Electronics Engineers, New York, 2003).
[CrossRef]

Strintzis, M.

F. Tsalakanidou, D. Tzocaras, M. Strintzis, “Use of depth and colour eigenfaces for face recognition,” Pattern Recogn. Lett. 24, 1427–1435 (2003).
[CrossRef]

Sumi, Y.

S. Lao, Y. Sumi, M. Kawade, F. Tomita, “3D template matching for pose invariant face recognition using 3D facial model built with isoluminance line based stereo vision,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2911–2916.

Tanaka, H. T.

H. T. Tanaka, M. Ikeda, H. Chiaki, “Curvature-based face surface recognition using spherical correlation,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 372–377.
[CrossRef]

Tomita, F.

S. Lao, Y. Sumi, M. Kawade, F. Tomita, “3D template matching for pose invariant face recognition using 3D facial model built with isoluminance line based stereo vision,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2911–2916.

Tsalakanidou, F.

F. Tsalakanidou, D. Tzocaras, M. Strintzis, “Use of depth and colour eigenfaces for face recognition,” Pattern Recogn. Lett. 24, 1427–1435 (2003).
[CrossRef]

Turk, M. A.

M. A. Turk, A. P. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).
[CrossRef] [PubMed]

Tzocaras, D.

F. Tsalakanidou, D. Tzocaras, M. Strintzis, “Use of depth and colour eigenfaces for face recognition,” Pattern Recogn. Lett. 24, 1427–1435 (2003).
[CrossRef]

Uemura, T.

T. Nagamine, T. Uemura, I. Masuda, “3D facial image analysis for human identification,” in Proceedings of the 11th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 324–327.

Vaidya, V. G.

R. M. Haralick, H. N. Joo, C. N. Lee, X. Zhuang, V. G. Vaidya, M. B. Kim, “Pose estimation from corresponding point data,” IEEE Trans. Syst. Man Cybern. 19, 1426–1446 (1989).
[CrossRef]

Velez, J. F.

A. B. Moreno, A. Sanchez, J. F. Velez, F. J. Diaz, “Face recognition using 3D surface-extracted descriptors,” in Proceedings of the Irish Machine Vision and Image Processing Conference (IMVIP2003), P. J. Morrow, B. W. Scotney, eds. (University of Ulster, Coleraine, Ireland).

Vetter, T.

V. Blanz, T. Vetter, “Face recognition based on fitting a 3D morphable model,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1063–1074 (2003).
[CrossRef]

Wang, Y.

Y. Wang, C. Chua, Y. Ho, “Facial feature detection and face recognition from 2D and 3D images,” Pattern Recogn. Lett. 23, 1191–1202 (2002).
[CrossRef]

Waupotitsch, R.

G. Medioni, R. Waupotitsch, “Face recognition and modeling in 3D,” in Proceedings of the IEEE International Workshop on Analysis and Modeling of Faces and Gestures (AMFG 2003) (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2003), pp. 232–233.
[CrossRef]

Wilson, C. L.

R. Chellappa, C. L. Wilson, S. Sirohey, “Human and machine recognition of faces: a survey,” Proc. IEEE 83, 705–740 (1995).
[CrossRef]

Wu, H.

Q. Chen, H. Wu, T. Fukumoto, M. Yachida, “3D head pose estimation without feature tracking,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 88–93.
[CrossRef]

Q. Chen, H. Wu, T. Shioyama, T. Shimada, “Head pose estimation using both color and feature information,” in Proceedings of the Fifteenth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2000), pp. 2842–2847.

Yachida, M.

Q. Chen, H. Wu, T. Fukumoto, M. Yachida, “3D head pose estimation without feature tracking,” in Proceedings of the Third International Conference on Automatic Face and Gesture Recognition (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 88–93.
[CrossRef]

Yang, U.

H. Song, U. Yang, K. Sohn, “3D face recognition under pose varying environments,” Lect. Notes Comput. Sci. 2908, 333–347 (2003).
[CrossRef]

Zhao, W.

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Figures (7)

Fig. 1
Fig. 1

Block diagram of the proposed algorithm.

Fig. 2
Fig. 2

Data images captured with the 3D FaceCam system and the 3D face model: (a) left pose, (b) frontal pose, (c) right pose.

Fig. 3
Fig. 3

Filling holes by use of bilinear interpolation.

Fig. 4
Fig. 4

ECSVD procedure.

Fig. 5
Fig. 5

Three-dimensional facial-feature extraction: top row, frontal pose; middle row, right pose; bottom row, left pose.

Fig. 6
Fig. 6

Cropped range images for face recognition: top row, range images obtained from the 3D FaceCam; bottom row, range images obtained from the Cyberware laser scanner.

Fig. 7
Fig. 7

Face recognition results based on correlation matching and PCA.

Tables (1)

Tables Icon

Table 1 Mean Absolute Rotation Error and Translational Error for Each Axis

Equations (21)

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I ( h , ϕ ) = [ r ( h , ϕ ) , R ( h , ϕ ) , G ( h , ϕ ) , B ( h , ϕ ) ] ,
F ( x i , y i , z i ) = ( F x - F min - x F max - x - F min - x σ , F y - F min - y F max - y - F min - y ɛ , F z - F min - z F max - z - F min - z z ) ,
R = R x R y R z = R SVD x R θ x R SVD y R θ y R SVD z R θ z ,
R x = R SVD x R θ x ,             R y = R SVD y R θ y ,             R z = R SVD z R θ z ,
f i = R f i + t = R x R y R z f i + t R SVD x R θ x R SVD y R θ y R SVD z R θ z f i + t ,
p i = R - 1 p i = R z - 1 R y - 1 R x - 1 p i = R θ z - 1 R SVD z - 1 R θ y - 1 R SVD y - 1 R θ x - 1 R SVD x - 1 p i ,
p = R x - 1 n = R θ x - 1 R SVD x - 1 n .
R x T = R x - 1 ,
R x = [ 1 0 0 0 cos θ x sin θ x 0 - sin θ x cos θ x ] ,
[ x 0 z ] = [ 1 0 0 0 cos θ x - sin θ x 0 sin θ x cos θ x ] × [ 1 0 0 0 cos θ SVD x - sin θ SVD x 0 sin θ SVD x cos θ SVD x ] [ x y z ] .
[ 1 0 0 0 cos θ SVD x - sin θ SVD x 0 sin θ SVD x cos θ SVD x ] [ x y z ] = [ 1 0 0 0 cos θ x sin θ x 0 - sin θ x cos θ x ] [ x 0 z ] [ x y cos θ SVD x - z sin θ SVD x y sin θ SVD x + z cos θ SVD x ] = [ x z sin θ x z cos θ x ] ,
y cos θ SVD x - z sin θ SVD x y sin θ SVD x + z cos θ SVD x = sin θ x cos θ x = tan θ x .
θ x = arctan ( y cos θ SVD x - z sin θ SVD x y sin θ SVD x + z cos θ SVD x ) .
p = R y - 1 p = R θ y - 1 R SVD y - 1 p ,
[ 0 0 z ] = [ cos θ SVD y 0 - sin θ SVD y 0 1 0 sin θ SVD y 0 cos θ SVD y ] × [ cos θ y 0 - sin θ y 0 1 0 sin θ y 0 cos θ y ] [ x 0 z ] , [ cos θ SVD y 0 - sin θ SVD y 0 1 0 sin θ SVD y 0 cos θ SVD y ] × [ x 0 z ] = [ cos θ y 0 sin θ y 0 1 0 - sin θ y 0 cos θ y ] [ 0 0 z ] ,
x cos θ SVD y - z sin θ SVD y x sin θ SVD y + z cos θ SVD y = sin θ y cos θ y = tan θ y .
θ y = arctan ( x cos θ SVD y - z sin θ SVD y x sin θ SVD y + z cos θ SVD y ) .
F = [ a ( a 2 + b 2 + c 2 ) 1 / 2 ,         b ( a 2 + b 2 + c 2 ) 1 / 2 ,             c ( a 2 + b 2 + c 2 ) 1 / 2 ] , [ a ( a 2 + b 2 + c 2 ) 1 / 2 ,             b ( a 2 + b 2 + c 2 ) 1 / 2 ,             0 ] ,
F = [ f x f y 0 ] = [ cos θ z - sin θ z 0 sin θ z cos θ z 0 0 0 1 ] [ 0 1 0 ] = [ - sin θ z cos θ z 0 ] ,
f x = a ( a 2 + b 2 + c 2 ) 1 / 2 = - sin θ z .
θ z = arcsin [ - a ( a 2 + b 2 + c 2 ) 1 / 2 ] .

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