Abstract

The variation of pose, illumination, and expression continues to make face recognition a challenging problem. As a pre-processing step in holistic approaches, faces are usually aligned by eyes. The proposed method tries to perform a pixel alignment rather than eye alignment by mapping the geometry of faces to a reference face while keeping their own textures. The proposed geometry alignment not only creates a meaningful correspondence among every pixel of all faces, but also removes expression and pose variations effectively. The geometry alignment is performed pixel-wise, i.e., every pixel of the face is corresponded to a pixel of the reference face. In the proposed method, the information of intensity and geometry of faces is separated properly, trained by separate classifiers, and finally fused together to recognize human faces. Experimental results show a great improvement using the proposed method in comparison to eye-aligned recognition. For instance, at the false acceptance rate (FAR) of 0.001, the recognition rates are respectively improved by 24% and 33% in the Yale and AT&T datasets. In the labeled faces in the wild dataset, which is a challenging, big dataset, improvement is 20% at a FAR of 0.1.

© 2018 Optical Society of America

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References

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2017 (1)

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

2016 (2)

M. M. Kasar, D. Bhattacharyya, and T.-H. Kim, “Face recognition using neural network: a review,” Int. J. Security Appl. 10, 81–100 (2016).
[Crossref]

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

2015 (2)

C. Ding, C. Xu, and D. Tao, “Multi-task pose-invariant face recognition,” IEEE Trans. Image Process. 24, 980–993 (2015).
[Crossref]

F. Juefei-Xu, K. Luu, and M. Savvides, “Spartans: single-sample periocular-based alignment-robust recognition technique applied to non-frontal scenarios,” IEEE Trans. Image Process. 24, 4780–4795 (2015).
[Crossref]

2013 (1)

H. Mohammadzade and D. Hatzinakos, “Projection into expression subspaces for face recognition from single sample per person,” IEEE Trans. Affective Comput. 4, 69–82 (2013).
[Crossref]

2009 (2)

X. Zhang and Y. Gao, “Face recognition across pose: a review,” Pattern Recognit. 42, 2876–2896 (2009).
[Crossref]

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

2004 (1)

V. Perlibakas, “Distance measures for PCA-based face recognition,” Pattern Recognit. Lett. 25, 711–724 (2004).
[Crossref]

2003 (2)

J. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks 14, 117–126 (2003).
[Crossref]

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
[Crossref]

2001 (1)

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 681–685 (2001).
[Crossref]

2000 (1)

B. Moghaddam, T. Jebara, and A. Pentland, “Bayesian face recognition,” Pattern Recognit. 33, 1771–1782 (2000).
[Crossref]

1997 (2)

S.-H. Lin, S.-Y. Kung, and L.-J. Lin, “Face recognition/detection by probabilistic decision-based neural network,” IEEE Trans. Neural Networks 8, 114–132 (1997).
[Crossref]

P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[Crossref]

1996 (1)

P. S. Penev and J. J. Atick, “Local feature analysis: a general statistical theory for object representation,” Network 7, 477–500 (1996).
[Crossref]

1995 (2)

A. Lanitis, C. J. Taylor, and T. F. Cootes, “Automatic face identification system using flexible appearance models,” Image Vision Comput. 13, 393–401 (1995).
[Crossref]

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

1994 (2)

A. Pentland, B. Moghaddam, and T. Starner, “View-based and modular eigenspaces for face recognition,” Comput. Vision Pattern Recognit. 94, 84–91 (1994).

F. Samaria and S. Young, “Hmm-based architecture for face identification,” Image Vision Comput. 12, 537–543 (1994).
[Crossref]

1993 (1)

R. Brunelli and T. Poggio, “Face recognition: features versus templates,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1042–1052 (1993).
[Crossref]

1991 (1)

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).

AbdAlmageed, W.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Atick, J. J.

P. S. Penev and J. J. Atick, “Local feature analysis: a general statistical theory for object representation,” Network 7, 477–500 (1996).
[Crossref]

Baltrusaitis, T.

T. Baltrusaitis, P. Robinson, and L.-P. Morency, “Constrained local neural fields for robust facial landmark detection in the wild,” in Proceedings of the IEEE International Conference on Computer Vision Workshops (2013), pp. 354–361.

Belhumeur, P. N.

P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[Crossref]

Berg, T.

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

Bhattacharyya, D.

M. M. Kasar, D. Bhattacharyya, and T.-H. Kim, “Face recognition using neural network: a review,” Int. J. Security Appl. 10, 81–100 (2016).
[Crossref]

Bishop, C.

C. Bishop, Pattern Recognition and Machine Learning (Information Science and Statistics) (Springer, 2007).

Brunelli, R.

R. Brunelli and T. Poggio, “Face recognition: features versus templates,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1042–1052 (1993).
[Crossref]

Cameron, P.

I. Craw and P. Cameron, “Face recognition by computer,” in British Machine Vision Conference (1992), pp. 1–10.

Cao, X.

D. Chen, X. Cao, F. Wen, and J. Sun, “Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3025–3032.

Chellappa, R.

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
[Crossref]

Chen, D.

D. Chen, X. Cao, F. Wen, and J. Sun, “Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3025–3032.

Choi, J.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Cohn, J. F.

T. Kanade, J. F. Cohn, and Y. Tian, “Comprehensive database for facial expression analysis,” in Proceedings of the Fourth IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 2000), pp. 46–53.

Cooper, D. H.

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

Cootes, T. F.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 681–685 (2001).
[Crossref]

A. Lanitis, C. J. Taylor, and T. F. Cootes, “Automatic face identification system using flexible appearance models,” Image Vision Comput. 13, 393–401 (1995).
[Crossref]

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

G. J. Edwards, T. F. Cootes, and C. J. Taylor, “Face recognition using active appearance models,” in European Conference on Computer Vision (Springer, 1998), pp. 581–595.

A. Lanitis, C. J. Taylor, and T. F. Cootes, “A unified approach to coding and interpreting face images,” in Proceedings Fifth International Conference on Computer Vision (IEEE, 1995), pp. 368–373.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” in Proceedings of the European Conference on Computer Vision (Springer, 1998), Vol. 2, pp. 484–498.

D. Cristinacce and T. F. Cootes, “Feature detection and tracking with constrained local models,” in British Machine Vision Conference (2006), Vol. 1, pp. 3.

Corneanu, C.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Cox, D. D.

N. Pinto, J. J. DiCarlo, and D. D. Cox, “How far can you get with a modern face recognition test set using only simple features?” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2009), pp. 2591–2598.

Craw, I.

I. Craw and P. Cameron, “Face recognition by computer,” in British Machine Vision Conference (1992), pp. 1–10.

Cristinacce, D.

D. Cristinacce and T. F. Cootes, “Feature detection and tracking with constrained local models,” in British Machine Vision Conference (2006), Vol. 1, pp. 3.

DiCarlo, J. J.

N. Pinto, J. J. DiCarlo, and D. D. Cox, “How far can you get with a modern face recognition test set using only simple features?” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2009), pp. 2591–2598.

Ding, C.

C. Ding, C. Xu, and D. Tao, “Multi-task pose-invariant face recognition,” IEEE Trans. Image Process. 24, 980–993 (2015).
[Crossref]

Edwards, G. J.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 681–685 (2001).
[Crossref]

G. J. Edwards, T. F. Cootes, and C. J. Taylor, “Face recognition using active appearance models,” in European Conference on Computer Vision (Springer, 1998), pp. 581–595.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” in Proceedings of the European Conference on Computer Vision (Springer, 1998), Vol. 2, pp. 484–498.

Escalera, S.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Friedman, J.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2002).

Ganesh, A.

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

Gao, Y.

X. Zhang and Y. Gao, “Face recognition across pose: a review,” Pattern Recognit. 42, 2876–2896 (2009).
[Crossref]

Gärtner, B.

B. Gärtner and M. Hoffmann, Computational Geometry–Lecture Notes HS 2013 (ETH Zürich University, 2014), Chap. 6.

Graham, J.

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

Greitans, M.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Harel, S.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Harter, A. C.

F. S. Samaria and A. C. Harter, “Parameterisation of a stochastic model for human face identification,” in Proceedings of the Second IEEE Workshop on Applications of Computer Vision, 1994 (IEEE, 1994), pp. 138–142.

Hassner, T.

L. Wolf, T. Hassner, and Y. Taigman, “Descriptor based methods in the wild,” in Workshop on Faces in ‘Real-Life’ Images: Detection, Alignment, and Recognition (2008).

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Hastie, T.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2002).

Hatzinakos, D.

H. Mohammadzade and D. Hatzinakos, “Projection into expression subspaces for face recognition from single sample per person,” IEEE Trans. Affective Comput. 4, 69–82 (2013).
[Crossref]

Hayes, M. H.

A. V. Nefian and M. H. Hayes, “Hidden Markov models for face recognition,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1998), Vol. 5, pp. 2721–2724.

Hespanha, J. P.

P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[Crossref]

Hoffmann, M.

B. Gärtner and M. Hoffmann, Computational Geometry–Lecture Notes HS 2013 (ETH Zürich University, 2014), Chap. 6.

Hua, G.

H. Li and G. Hua, “Hierarchical-pep model for real-world face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 4055–4064.

Huang, G. B.

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

Jebara, T.

B. Moghaddam, T. Jebara, and A. Pentland, “Bayesian face recognition,” Pattern Recognit. 33, 1771–1782 (2000).
[Crossref]

Juefei-Xu, F.

F. Juefei-Xu, K. Luu, and M. Savvides, “Spartans: single-sample periocular-based alignment-robust recognition technique applied to non-frontal scenarios,” IEEE Trans. Image Process. 24, 4780–4795 (2015).
[Crossref]

Kalenichenko, D.

F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: a unified embedding for face recognition and clustering,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 815–823.

Kanade, T.

T. Kanade, J. F. Cohn, and Y. Tian, “Comprehensive database for facial expression analysis,” in Proceedings of the Fourth IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 2000), pp. 46–53.

T. Kanade, Computer Recognition of Human Faces (Birkhäuser, 1977), Vol. 47.

Kasar, M. M.

M. M. Kasar, D. Bhattacharyya, and T.-H. Kim, “Face recognition using neural network: a review,” Int. J. Security Appl. 10, 81–100 (2016).
[Crossref]

Kelly, M. D.

M. D. Kelly, Visual Identification of People by Computer (Stanford University of California, 1970).

Kim, J.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Kim, T.-H.

M. M. Kasar, D. Bhattacharyya, and T.-H. Kim, “Face recognition using neural network: a review,” Int. J. Security Appl. 10, 81–100 (2016).
[Crossref]

Kriegman, D. J.

P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[Crossref]

Kung, S.-Y.

S.-H. Lin, S.-Y. Kung, and L.-J. Lin, “Face recognition/detection by probabilistic decision-based neural network,” IEEE Trans. Neural Networks 8, 114–132 (1997).
[Crossref]

Lanitis, A.

A. Lanitis, C. J. Taylor, and T. F. Cootes, “Automatic face identification system using flexible appearance models,” Image Vision Comput. 13, 393–401 (1995).
[Crossref]

A. Lanitis, C. J. Taylor, and T. F. Cootes, “A unified approach to coding and interpreting face images,” in Proceedings Fifth International Conference on Computer Vision (IEEE, 1995), pp. 368–373.

Learned-Miller, E.

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

Lei, Z.

D. Yi, Z. Lei, and S. Z. Li, “Towards pose robust face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3539–3545.

Lekust, J.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Li, H.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

H. Li and G. Hua, “Hierarchical-pep model for real-world face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 4055–4064.

Li, S. Z.

D. Yi, Z. Lei, and S. Z. Li, “Towards pose robust face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3539–3545.

Lin, L.-J.

S.-H. Lin, S.-Y. Kung, and L.-J. Lin, “Face recognition/detection by probabilistic decision-based neural network,” IEEE Trans. Neural Networks 8, 114–132 (1997).
[Crossref]

Lin, S.-H.

S.-H. Lin, S.-Y. Kung, and L.-J. Lin, “Face recognition/detection by probabilistic decision-based neural network,” IEEE Trans. Neural Networks 8, 114–132 (1997).
[Crossref]

Lu, J.

J. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks 14, 117–126 (2003).
[Crossref]

J. Lu, K. Plataniotis, and A. Venetsanopoulos, “Kernel discriminant learning with application to face recognition,” in Support Vector Machines: Theory and Applications (Springer, 2005), pp. 275–296.

Luu, K.

F. Juefei-Xu, K. Luu, and M. Savvides, “Spartans: single-sample periocular-based alignment-robust recognition technique applied to non-frontal scenarios,” IEEE Trans. Image Process. 24, 4780–4795 (2015).
[Crossref]

Ma, Y.

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

Masi, I.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Medioni, G.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Moeslund, T. B.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Moghaddam, B.

B. Moghaddam, T. Jebara, and A. Pentland, “Bayesian face recognition,” Pattern Recognit. 33, 1771–1782 (2000).
[Crossref]

A. Pentland, B. Moghaddam, and T. Starner, “View-based and modular eigenspaces for face recognition,” Comput. Vision Pattern Recognit. 94, 84–91 (1994).

Mohammadzade, H.

H. Mohammadzade and D. Hatzinakos, “Projection into expression subspaces for face recognition from single sample per person,” IEEE Trans. Affective Comput. 4, 69–82 (2013).
[Crossref]

Morency, L.-P.

T. Baltrusaitis, P. Robinson, and L.-P. Morency, “Constrained local neural fields for robust facial landmark detection in the wild,” in Proceedings of the IEEE International Conference on Computer Vision Workshops (2013), pp. 354–361.

Nasrollahi, K.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Natarajan, P.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Nefian, A. V.

A. V. Nefian and M. H. Hayes, “Hidden Markov models for face recognition,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1998), Vol. 5, pp. 2721–2724.

Nevatia, R.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Nikisins, O.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Panagakis, Y.

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

Pantic, M.

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

Parkhi, O. M.

O. M. Parkhi, A. Vedaldi, and A. Zisserman, “Deep face recognition,” in British Machine Vision Conference (2015), Vol. 1, pp. 6.

Penev, P. S.

P. S. Penev and J. J. Atick, “Local feature analysis: a general statistical theory for object representation,” Network 7, 477–500 (1996).
[Crossref]

Pentland, A.

B. Moghaddam, T. Jebara, and A. Pentland, “Bayesian face recognition,” Pattern Recognit. 33, 1771–1782 (2000).
[Crossref]

A. Pentland, B. Moghaddam, and T. Starner, “View-based and modular eigenspaces for face recognition,” Comput. Vision Pattern Recognit. 94, 84–91 (1994).

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).

Pentland, A. P.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in Proceedings IEEE Computer Society Conference on Computer Vision and Pattern Recognition” (IEEE, 1991), pp. 586–591.

Perlibakas, V.

V. Perlibakas, “Distance measures for PCA-based face recognition,” Pattern Recognit. Lett. 25, 711–724 (2004).
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Philbin, J.

F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: a unified embedding for face recognition and clustering,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 815–823.

Phillips, P. J.

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
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P. J. Phillips, “Support vector machines applied to face recognition,” in Advances in Neural Information Processing Systems (1999), pp. 803–809.

Pinto, N.

N. Pinto, J. J. DiCarlo, and D. D. Cox, “How far can you get with a modern face recognition test set using only simple features?” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2009), pp. 2591–2598.

Plataniotis, K.

J. Lu, K. Plataniotis, and A. Venetsanopoulos, “Kernel discriminant learning with application to face recognition,” in Support Vector Machines: Theory and Applications (Springer, 2005), pp. 275–296.

Plataniotis, K. N.

J. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks 14, 117–126 (2003).
[Crossref]

Poggio, T.

R. Brunelli and T. Poggio, “Face recognition: features versus templates,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1042–1052 (1993).
[Crossref]

Ramesh, M.

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

Ranzato, M.

Y. Taigman, M. Yang, M. Ranzato, and L. Wolf, “Deepface: closing the gap to human-level performance in face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2014), pp. 1701–1708.

Rawls, S.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Robinson, P.

T. Baltrusaitis, P. Robinson, and L.-P. Morency, “Constrained local neural fields for robust facial landmark detection in the wild,” in Proceedings of the IEEE International Conference on Computer Vision Workshops (2013), pp. 354–361.

Rosenfeld, A.

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
[Crossref]

Sagonas, C.

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

Samaria, F.

F. Samaria and S. Young, “Hmm-based architecture for face identification,” Image Vision Comput. 12, 537–543 (1994).
[Crossref]

Samaria, F. S.

F. S. Samaria and A. C. Harter, “Parameterisation of a stochastic model for human face identification,” in Proceedings of the Second IEEE Workshop on Applications of Computer Vision, 1994 (IEEE, 1994), pp. 138–142.

Sastry, S. S.

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

Savvides, M.

F. Juefei-Xu, K. Luu, and M. Savvides, “Spartans: single-sample periocular-based alignment-robust recognition technique applied to non-frontal scenarios,” IEEE Trans. Image Process. 24, 4780–4795 (2015).
[Crossref]

Schroff, F.

F. Schroff, D. Kalenichenko, and J. Philbin, “Facenet: a unified embedding for face recognition and clustering,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 815–823.

Simón, M. O.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Starner, T.

A. Pentland, B. Moghaddam, and T. Starner, “View-based and modular eigenspaces for face recognition,” Comput. Vision Pattern Recognit. 94, 84–91 (1994).

Stegmann, M. B.

M. B. Stegmann, “Analysis and segmentation of face images using point annotations and linear subspace techniques,” (2002).

Sun, J.

D. Chen, X. Cao, F. Wen, and J. Sun, “Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3025–3032.

Sun, Y.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Sun, Z.

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Taigman, Y.

Y. Taigman, M. Yang, M. Ranzato, and L. Wolf, “Deepface: closing the gap to human-level performance in face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2014), pp. 1701–1708.

L. Wolf, T. Hassner, and Y. Taigman, “Descriptor based methods in the wild,” in Workshop on Faces in ‘Real-Life’ Images: Detection, Alignment, and Recognition (2008).

Tao, D.

C. Ding, C. Xu, and D. Tao, “Multi-task pose-invariant face recognition,” IEEE Trans. Image Process. 24, 980–993 (2015).
[Crossref]

Taylor, C. J.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 681–685 (2001).
[Crossref]

A. Lanitis, C. J. Taylor, and T. F. Cootes, “Automatic face identification system using flexible appearance models,” Image Vision Comput. 13, 393–401 (1995).
[Crossref]

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

A. Lanitis, C. J. Taylor, and T. F. Cootes, “A unified approach to coding and interpreting face images,” in Proceedings Fifth International Conference on Computer Vision (IEEE, 1995), pp. 368–373.

G. J. Edwards, T. F. Cootes, and C. J. Taylor, “Face recognition using active appearance models,” in European Conference on Computer Vision (Springer, 1998), pp. 581–595.

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” in Proceedings of the European Conference on Computer Vision (Springer, 1998), Vol. 2, pp. 484–498.

Tian, Y.

T. Kanade, J. F. Cohn, and Y. Tian, “Comprehensive database for facial expression analysis,” in Proceedings of the Fourth IEEE International Conference on Automatic Face and Gesture Recognition (IEEE, 2000), pp. 46–53.

Tibshirani, R.

T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2002).

Turk, M.

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).

Turk, M. A.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in Proceedings IEEE Computer Society Conference on Computer Vision and Pattern Recognition” (IEEE, 1991), pp. 586–591.

Vedaldi, A.

O. M. Parkhi, A. Vedaldi, and A. Zisserman, “Deep face recognition,” in British Machine Vision Conference (2015), Vol. 1, pp. 6.

Venetsanopoulos, A.

J. Lu, K. Plataniotis, and A. Venetsanopoulos, “Kernel discriminant learning with application to face recognition,” in Support Vector Machines: Theory and Applications (Springer, 2005), pp. 275–296.

Venetsanopoulos, A. N.

J. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks 14, 117–126 (2003).
[Crossref]

Wen, F.

D. Chen, X. Cao, F. Wen, and J. Sun, “Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3025–3032.

Wolf, L.

Y. Taigman, M. Yang, M. Ranzato, and L. Wolf, “Deepface: closing the gap to human-level performance in face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2014), pp. 1701–1708.

L. Wolf, T. Hassner, and Y. Taigman, “Descriptor based methods in the wild,” in Workshop on Faces in ‘Real-Life’ Images: Detection, Alignment, and Recognition (2008).

Wright, J.

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

Wu, Y.

W. AbdAlmageed, Y. Wu, S. Rawls, S. Harel, T. Hassner, I. Masi, J. Choi, J. Lekust, J. Kim, P. Natarajan, R. Nevatia, and G. Medioni, “Face recognition using deep multi-pose representations,” in IEEE Winter Conference on Applications of Computer Vision (WACV) (IEEE, 2016), pp. 1–9.

Xu, C.

C. Ding, C. Xu, and D. Tao, “Multi-task pose-invariant face recognition,” IEEE Trans. Image Process. 24, 980–993 (2015).
[Crossref]

Yang, A. Y.

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

Yang, M.

Y. Taigman, M. Yang, M. Ranzato, and L. Wolf, “Deepface: closing the gap to human-level performance in face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2014), pp. 1701–1708.

Yi, D.

D. Yi, Z. Lei, and S. Z. Li, “Towards pose robust face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3539–3545.

Young, S.

F. Samaria and S. Young, “Hmm-based architecture for face identification,” Image Vision Comput. 12, 537–543 (1994).
[Crossref]

Zafeiriou, S.

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

Zhang, X.

X. Zhang and Y. Gao, “Face recognition across pose: a review,” Pattern Recognit. 42, 2876–2896 (2009).
[Crossref]

Zhao, W.

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
[Crossref]

Zisserman, A.

O. M. Parkhi, A. Vedaldi, and A. Zisserman, “Deep face recognition,” in British Machine Vision Conference (2015), Vol. 1, pp. 6.

ACM Comput. Surv. (1)

W. Zhao, R. Chellappa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399–458 (2003).
[Crossref]

Comput. Vis. Image Underst. (1)

T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, “Active shape models-their training and application,” Comput. Vis. Image Underst. 61, 38–59 (1995).
[Crossref]

Comput. Vision Pattern Recognit. (1)

A. Pentland, B. Moghaddam, and T. Starner, “View-based and modular eigenspaces for face recognition,” Comput. Vision Pattern Recognit. 94, 84–91 (1994).

IEEE Trans. Affective Comput. (1)

H. Mohammadzade and D. Hatzinakos, “Projection into expression subspaces for face recognition from single sample per person,” IEEE Trans. Affective Comput. 4, 69–82 (2013).
[Crossref]

IEEE Trans. Image Process. (2)

F. Juefei-Xu, K. Luu, and M. Savvides, “Spartans: single-sample periocular-based alignment-robust recognition technique applied to non-frontal scenarios,” IEEE Trans. Image Process. 24, 4780–4795 (2015).
[Crossref]

C. Ding, C. Xu, and D. Tao, “Multi-task pose-invariant face recognition,” IEEE Trans. Image Process. 24, 980–993 (2015).
[Crossref]

IEEE Trans. Neural Networks (2)

J. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks 14, 117–126 (2003).
[Crossref]

S.-H. Lin, S.-Y. Kung, and L.-J. Lin, “Face recognition/detection by probabilistic decision-based neural network,” IEEE Trans. Neural Networks 8, 114–132 (1997).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (4)

P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[Crossref]

J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 210–227 (2009).
[Crossref]

R. Brunelli and T. Poggio, “Face recognition: features versus templates,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1042–1052 (1993).
[Crossref]

T. F. Cootes, G. J. Edwards, and C. J. Taylor, “Active appearance models,” IEEE Trans. Pattern Anal. Mach. Intell. 23, 681–685 (2001).
[Crossref]

IET Biometrics (1)

M. O. Simón, C. Corneanu, K. Nasrollahi, O. Nikisins, S. Escalera, Y. Sun, H. Li, Z. Sun, T. B. Moeslund, and M. Greitans, “Improved rgb-dt based face recognition,” IET Biometrics 5, 297–303 (2016).
[Crossref]

Image Vision Comput. (2)

F. Samaria and S. Young, “Hmm-based architecture for face identification,” Image Vision Comput. 12, 537–543 (1994).
[Crossref]

A. Lanitis, C. J. Taylor, and T. F. Cootes, “Automatic face identification system using flexible appearance models,” Image Vision Comput. 13, 393–401 (1995).
[Crossref]

Int. J. Comput. Vis. (1)

C. Sagonas, Y. Panagakis, S. Zafeiriou, and M. Pantic, “Robust statistical frontalization of human and animal faces,” Int. J. Comput. Vis. 122, 270–291 (2017).
[Crossref]

Int. J. Security Appl. (1)

M. M. Kasar, D. Bhattacharyya, and T.-H. Kim, “Face recognition using neural network: a review,” Int. J. Security Appl. 10, 81–100 (2016).
[Crossref]

J. Cogn. Neurosci. (1)

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71–86 (1991).

Network (1)

P. S. Penev and J. J. Atick, “Local feature analysis: a general statistical theory for object representation,” Network 7, 477–500 (1996).
[Crossref]

Pattern Recognit. (2)

B. Moghaddam, T. Jebara, and A. Pentland, “Bayesian face recognition,” Pattern Recognit. 33, 1771–1782 (2000).
[Crossref]

X. Zhang and Y. Gao, “Face recognition across pose: a review,” Pattern Recognit. 42, 2876–2896 (2009).
[Crossref]

Pattern Recognit. Lett. (1)

V. Perlibakas, “Distance measures for PCA-based face recognition,” Pattern Recognit. Lett. 25, 711–724 (2004).
[Crossref]

Other (32)

H. Li and G. Hua, “Hierarchical-pep model for real-world face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 4055–4064.

“AT&T Face Dataset,” http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html .

F. S. Samaria and A. C. Harter, “Parameterisation of a stochastic model for human face identification,” in Proceedings of the Second IEEE Workshop on Applications of Computer Vision, 1994 (IEEE, 1994), pp. 138–142.

“LFW Face Dataset,” http://vis-www.cs.umass.edu/lfw/ .

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

N. Pinto, J. J. DiCarlo, and D. D. Cox, “How far can you get with a modern face recognition test set using only simple features?” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2009), pp. 2591–2598.

L. Wolf, T. Hassner, and Y. Taigman, “Descriptor based methods in the wild,” in Workshop on Faces in ‘Real-Life’ Images: Detection, Alignment, and Recognition (2008).

D. Yi, Z. Lei, and S. Z. Li, “Towards pose robust face recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3539–3545.

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D. Chen, X. Cao, F. Wen, and J. Sun, “Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2013), pp. 3025–3032.

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Figures (15)

Fig. 1.
Fig. 1. Obtaining reference contour by averaging landmark contours of several neutral faces. The facial images are from the Yale dataset [36].
Fig. 2.
Fig. 2. Procedure of geometrical transformation and pixel-to-pixel face warping.
Fig. 3.
Fig. 3. Delaunay triangulation of face landmarks.
Fig. 4.
Fig. 4. Interpolation of x . The black points are the landmarks with known target coordinates and the yellow points are the interpolated target coordinates for the non-landmark pixels.
Fig. 5.
Fig. 5. Intensity interpolation.
Fig. 6.
Fig. 6. Example of geometrical transformation and pixel-to-pixel warping. The facial image is taken from the Cohn-Kanade dataset [38,39] (©Jeffrey Cohn).
Fig. 7.
Fig. 7. Illustration of Δ x and Δ y information for a sample warped face. (a)  Δ x information, (b)  Δ y information.
Fig. 8.
Fig. 8. Classification using the ensemble of patches.
Fig. 9.
Fig. 9. Overall structure of the proposed method.
Fig. 10.
Fig. 10. Several samples of pixel alignment in the Yale dataset [36].
Fig. 11.
Fig. 11. Several samples of pixel alignment in the AT&T dataset [44,45].
Fig. 12.
Fig. 12. Effect of size of patches in classification using the ensemble of patches.
Fig. 13.
Fig. 13. Comparison of classification using the whole face or the ensemble of patches.
Fig. 14.
Fig. 14. Comparison of the proposed method with eye-aligned face recognition on the (a) Yale dataset, (b) AT&T dataset, (c) Cohn-Kanade dataset, and (d) LFW dataset.
Fig. 15.
Fig. 15. ROC curves on the LFW dataset using the standard cross-validation protocol of the dataset.

Tables (2)

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Table 1. Results of the Proposed Method and Eye-Aligned Face Recognition in Specific False Alarm Rates

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Table 2. Comparison of the Proposed Method to the State-of-the-Art Results on the LFW Dataset

Equations (12)

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x = f ( x , y ) a 0 + a 1 x + a 2 y ,
I ( x , y ) = I ( x , y ) .
f p I = [ I ( 1 , 1 ) , I ( 1 , 2 ) , , I ( m , m ) ] T ,
f p Δ x = [ Δ x ( 1 , 1 ) , Δ x ( 1 , 2 ) , , Δ x ( m , m ) ] T ,
f p Δ y = [ Δ y ( 1 , 1 ) , Δ y ( 1 , 2 ) , , Δ y ( m , m ) ] T ,
W opt = arg max W | W T S b W | | W T S w W | ,
S w = i = 1 C x k X i ( x k μ i ) ( x k μ i ) T ,
S b = i = 1 C N i ( μ i μ ) ( μ i μ ) T ,
sim p I ( i , j ) = cos ( f ^ p , i I , f ^ p , j I ) = f ^ p , i I f ^ p , j I | f ^ p , i I | | f ^ p , j I | ,
sim p Δ x ( i , j ) = cos ( f ^ p , i Δ x , f ^ p , j Δ x ) = f ^ p , i Δ x f ^ p , j Δ x | f ^ p , i Δ x | | f ^ p , j Δ x | ,
sim p Δ y ( i , j ) = cos ( f ^ p , i Δ y , f ^ p , j Δ y ) = f ^ p , i Δ y f ^ p , j Δ y | f ^ p , i Δ y | | f ^ p , j Δ y | ,
sim ( i , j ) = p = 1 80 ( sim p I ( i , j ) + w sim p Δ x ( i , j ) + w sim p Δ y ( i , j ) ) ,

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