Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via Grassmannians

Affiliation(s)

Departmentof Mathematics and Statistics, California State University, Long Beach, USA.

Department of Mathematics, Colorado State University, Fort Collins, USA.

Departmentof Mathematics and Statistics, California State University, Long Beach, USA.

Department of Mathematics, Colorado State University, Fort Collins, USA.

ABSTRACT

Recent work has established that digital images of a human face, when collected with a fixed pose but under a variety of illumination conditions, possess discriminatory information that can be used in classification. In this paper we perform classification on Grassmannians to demonstrate that sufficient discriminatory information persists in feature patch (e.g., nose or eye patch) illumination spaces. We further employ the use of Karcher mean on the Grassmannians to demonstrate that this compressed representation can accelerate computations with relatively minor sacrifice on performance. The combination of these two ideas introduces a novel perspective in performing face recognition.

Recent work has established that digital images of a human face, when collected with a fixed pose but under a variety of illumination conditions, possess discriminatory information that can be used in classification. In this paper we perform classification on Grassmannians to demonstrate that sufficient discriminatory information persists in feature patch (e.g., nose or eye patch) illumination spaces. We further employ the use of Karcher mean on the Grassmannians to demonstrate that this compressed representation can accelerate computations with relatively minor sacrifice on performance. The combination of these two ideas introduces a novel perspective in performing face recognition.

Cite this paper

J. Chang, C. Peterson and M. Kirby, "Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via Grassmannians,"*Advances in Pure Mathematics*, Vol. 2 No. 4, 2012, pp. 226-242. doi: 10.4236/apm.2012.24033.

J. Chang, C. Peterson and M. Kirby, "Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via Grassmannians,"

References

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[4] J.-M. Chang, J. Beveridge, B. Draper, M. Kirby, H. Kley and C. Peterson, “Illumination Face Spaces are Idiosyncratic,” Proceedings of the International Conference on Image Processing & Computer Vision, Vol. 2, 2006, pp. 390-396.

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[6] K. Fukui and O. Yamaguchi, “Face Recognition Using Multi-Viewpoint Patterns for Robot Vision,” Proceedings of the International Symposium of Robotics Research, Vol. 15, 2005, pp. 192-201.

[7] R. Wang, S. Shan, X. Chen and W. Gao, “Manifold- Manifold Distance with Application to Face Recognition Based on Image Sets,” Proceedings of the Computer Vision and Pattern Recognition Conference, 23-28 June 2008, pp. 1-8.

[8] A. Fitzgibbon and A. Zisserman, “Joint Manifold Distance: A New Approach to Appearance Based Clustering,” Proceedings of the Computer Vision and Pattern Recognition Conference, Vol. 1, 2003, pp. 26-36.

[9] A. Srivastava, “A Bayesian Approach to Geometric Subspace Estimation,” IEEE Transactions on Signal Processing, Vol. 48, No. 5, 2000, pp. 1390-1400. doi:10.1109/78.839985

[10] X. Liu, A. Srivastava and K. Gallivan, “Optimal Linear Representations of Images for Object Recognition,” IEEE Transactions on Pattern Analysis and Machine Learning, Vol. 26, No. 5, 2004, pp. 662-666. doi:10.1109/TPAMI.2004.1273986

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[22] M. Nishiyama, M. Yuasa, T. Shibata, T. Wakasugi, T. Kawahara and O. Yamaguchi, “Recognizing Faces of Moving People by Hierarchical Image-Set Matching,” Proceedings of the Computer Vision and Pattern Recognition Conference, 2007, pp. 1-8.

[23] Q.-S. Sun, Z.-D. Liu, P.-A. Heng and D.-S. Xia, “A Theorem on the Generalized Canonical Projective Vectors,” Pattern Recognition, Vol. 38, No. 3, 2005, pp. 449- 452. doi:10.1016/j.patcog.2004.08.009

[24] Q.-S. Sun, P.-A. Heng, Z. Jin and D.-S. Xia, “Face Recognition Based on Generalized Canonical Correlation Analysis,” Advances in Intelligent Computing, Vol. 3645, 2005, pp. 958-967.

[25] J.-M. Chang, M. Kirby and C. Peterson, “Set-to-Set Face Recognition under Variations in Pose and Illumination,” Proceedings of the Biometrics Symposium, Baltimore, 11-13 September 2007, pp. 1-6. doi:10.1016/j.patrec.2009.06.002

[26] T. Wang and P. Shi, “Kernal Grassmannian Distances and Discriminate Analysis for Face Recognition from Image Sets,” Pattern Recognition Letters, Vol. 30, 2009, pp. 1161-1165. doi:10.1016/j.patrec.2009.06.002

[27] J. Hamm and D. Lee, “Grassmann Discriminate Analysis: a Unifying View on Subspace-Based Learning,” Proceedings of the 25th International Conference on Machine Learning, Vol. 307, 2008, pp. 376-383.

[28] M. Harandi, C. Sanderson, S. Shirazi and B. Lovell, “Graph Embedding Discriminant Analysis on Grassmannian Manifolds for Improved Image Set Matching,” Proceedings of the Compute Vision and Pattern Recognition Conference, 20-25 June 2011, pp. 2705-2712.

[29] H. Cevikalp and B. Triggs, “Face Recognition Based on Image Sets,” Proceedings of the Compute Vision and Pattern Recognition Conference, 13-18 June 2010, pp. 2567- 2573.

[30] A. Mansfield and J. Wayman, “Best Practices in Testing and Reporting of Biometric Devices: Version 2.01,” Tech. Rep. NPL Report CMSC 14/02, Centre for Mathematics and Scientific Computing, National Physical Laboratory, UK, 2002.

[31] H. Karcher, “Riemannian Center of Mass and Mollifier Smoothing,” Communications on Pure and Applied Mathematics, Vol. 30, 1977, pp. 509-541. doi:10.1002/cpa.3160300502

[32] W. Kendall, “Probability, Convexity and Harmonic Maps with Small Image I: Uniqueness and Fine Existence,” Proceedings of the London Mathematical Society, Vol. 61, 1990, pp. 371-406. doi:10.1112/plms/s3-61.2.371

[33] E. Begelfor and M. Werman, “Affine Invariance Revisited,” Proceedings of the Computer Vision and Pattern Recognition Conference, Vol. 2, 2006, pp. 2087-2094.

[34] J. Manton, “A Globally Convergent Numerical Algorithm for Computing the Center of Mass on Compact Lie groups,” Proceedings of the International Conference on Control, Automation, Robotics and Vision, Vol. 3, 2004, pp. 2211-2216.

[35] T. Sim, S. Baker and M. Bsat, “The CMU Pose, Illumination, and Expression Database,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 12, 2003, pp. 1615-1618. doi:10.1109/TPAMI.2003.1251154

[36] D. Bolme, B. Draper and J. Beveridge, “Average of Synthetic Exact Filters,” Proceedings of the Computer Vision and Pattern Recognition, 20-25 June 2009, pp. 2015- 2112.

[37] P. Viola and M. Jones, “Robust Real-Time Face Detection,” International Journal of Computer Vision, Vol. 57, No. 2, 2004, pp. 137-154. doi:10.1023/B:VISI.0000013087.49260.fb

[38] P. Belhumeur and D. Kriegman, “What Is the Set of Images of an Object under All Possible Lighting Conditions,” International Journal of Computer Vision, Vol. 28, No. 3, 1998, pp. 245-260. doi:10.1023/A:1008005721484

[1] A. Georghiades, P. Belhumeur and D. Kriegman, “From Few to Many: Illumination Cone Models for Face Recognition under Variable Lighting and Pose,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 6, 2001, pp. 643-660. doi:10.1109/34.927464

[2] R. Basri and D. Jacobs, “Lambertian Reflectance and Linear Subspaces,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 2, 2003, pp. 218-233. doi:10.1109/TPAMI.2003.1177153

[3] R. Gross, I. Matthews and S. Baker, “Fisher Light-Fields for Face Recognition across Pose and Illumination,” Proceedings of the German Symposium on Pattern Recognition (DAGM), Vol. 2449, 2002, pp. 481-489.

[4] J.-M. Chang, J. Beveridge, B. Draper, M. Kirby, H. Kley and C. Peterson, “Illumination Face Spaces are Idiosyncratic,” Proceedings of the International Conference on Image Processing & Computer Vision, Vol. 2, 2006, pp. 390-396.

[5] O. Yamaguchi, K. Fukui and K. Maeda, “Face Recognition Using Temporal Image Sequence,” Proceedings of the International Conference on Automatic Face and Gesture Recognition, 1998, pp. 318-323. doi:10.1109/AFGR.1998.670968

[6] K. Fukui and O. Yamaguchi, “Face Recognition Using Multi-Viewpoint Patterns for Robot Vision,” Proceedings of the International Symposium of Robotics Research, Vol. 15, 2005, pp. 192-201.

[7] R. Wang, S. Shan, X. Chen and W. Gao, “Manifold- Manifold Distance with Application to Face Recognition Based on Image Sets,” Proceedings of the Computer Vision and Pattern Recognition Conference, 23-28 June 2008, pp. 1-8.

[8] A. Fitzgibbon and A. Zisserman, “Joint Manifold Distance: A New Approach to Appearance Based Clustering,” Proceedings of the Computer Vision and Pattern Recognition Conference, Vol. 1, 2003, pp. 26-36.

[9] A. Srivastava, “A Bayesian Approach to Geometric Subspace Estimation,” IEEE Transactions on Signal Processing, Vol. 48, No. 5, 2000, pp. 1390-1400. doi:10.1109/78.839985

[10] X. Liu, A. Srivastava and K. Gallivan, “Optimal Linear Representations of Images for Object Recognition,” IEEE Transactions on Pattern Analysis and Machine Learning, Vol. 26, No. 5, 2004, pp. 662-666. doi:10.1109/TPAMI.2004.1273986

[11] G. Stewart and J.-G. Sun, “Matrix Perturbation Theory,” Academic Press, New York, 1990.

[12] T. Kanade, “Picture Processing System by Computer Complex and Recognition of Human Faces,” Ph.D. Thesis, Kyoto University, Kyoto, 1973.

[13] L. Wiskott, J.-M. Fellous, N. Krüger and C. Malsburg, “Face Recognition by Elastic Bunch Graph Matching,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 19, No. 7, 1997, pp. 775-779. doi:10.1109/34.598235

[14] R. Brunelli and T. Poggio, “Face Recognition: Feature Versus Templates,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 15, No. 10, 1993, pp. 1042-1052. doi:10.1109/34.254061

[15] Y. Kaya and K. Kobayasaki, “A Basic Study on Human Face Recognition,” Proceedings of the Frontiers of Pattern Recognition, 1972, pp. 265-290.

[16] J. Harris, “Algebraic Geometry: A First Course,” Springer, Berlin, 1992.

[17] A. Edelman, T. Arias and S. Smith, “The Geometry of Algorithms with Orthogonality Constraints,” SIAM Journal on Matrix Analysis and Applications, Vol. 20, No. 2, 1999, pp. 303-353. doi:10.1137/S0895479895290954

[18] A. Bj?rck and G. Golub, “Numerical Methods for Computing Angles between Linear Subspaces,” Mathematics of Computation, Vol. 27, No. 123, 1973, pp. 579-594.

[19] A. Knyazev and M. Argentati, “Principal Angles between Subspaces in a Based Scalar Product: Algorithms and Perturbation Estimates,” SIAM Journal of Scientific Computing, Vol. 23, No. 6, 2002, pp. 2008-2040. doi:10.1137/S1064827500377332

[20] J. Beveridge, B. Draper, J.-M. Chang, M. Kirby, H. Kley and C. Peterson, “Principal Angles Separate Subject Illumination Spaces in YDB and CMU-PIE,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 31, No. 2, 2009, pp. 351-356. doi:10.1109/TPAMI.2008.200

[21] M. Nishiyama, O. Yamaguchi and K. Fukui, “Face Recognition with the Multiple Constrained Mutual Subspace Method,” Proceedings of the International Conference on Audio- and Video-Based Biometric Person Authentication, 2005, pp. 71-80.

[22] M. Nishiyama, M. Yuasa, T. Shibata, T. Wakasugi, T. Kawahara and O. Yamaguchi, “Recognizing Faces of Moving People by Hierarchical Image-Set Matching,” Proceedings of the Computer Vision and Pattern Recognition Conference, 2007, pp. 1-8.

[23] Q.-S. Sun, Z.-D. Liu, P.-A. Heng and D.-S. Xia, “A Theorem on the Generalized Canonical Projective Vectors,” Pattern Recognition, Vol. 38, No. 3, 2005, pp. 449- 452. doi:10.1016/j.patcog.2004.08.009

[24] Q.-S. Sun, P.-A. Heng, Z. Jin and D.-S. Xia, “Face Recognition Based on Generalized Canonical Correlation Analysis,” Advances in Intelligent Computing, Vol. 3645, 2005, pp. 958-967.

[25] J.-M. Chang, M. Kirby and C. Peterson, “Set-to-Set Face Recognition under Variations in Pose and Illumination,” Proceedings of the Biometrics Symposium, Baltimore, 11-13 September 2007, pp. 1-6. doi:10.1016/j.patrec.2009.06.002

[26] T. Wang and P. Shi, “Kernal Grassmannian Distances and Discriminate Analysis for Face Recognition from Image Sets,” Pattern Recognition Letters, Vol. 30, 2009, pp. 1161-1165. doi:10.1016/j.patrec.2009.06.002

[27] J. Hamm and D. Lee, “Grassmann Discriminate Analysis: a Unifying View on Subspace-Based Learning,” Proceedings of the 25th International Conference on Machine Learning, Vol. 307, 2008, pp. 376-383.

[28] M. Harandi, C. Sanderson, S. Shirazi and B. Lovell, “Graph Embedding Discriminant Analysis on Grassmannian Manifolds for Improved Image Set Matching,” Proceedings of the Compute Vision and Pattern Recognition Conference, 20-25 June 2011, pp. 2705-2712.

[29] H. Cevikalp and B. Triggs, “Face Recognition Based on Image Sets,” Proceedings of the Compute Vision and Pattern Recognition Conference, 13-18 June 2010, pp. 2567- 2573.

[30] A. Mansfield and J. Wayman, “Best Practices in Testing and Reporting of Biometric Devices: Version 2.01,” Tech. Rep. NPL Report CMSC 14/02, Centre for Mathematics and Scientific Computing, National Physical Laboratory, UK, 2002.

[31] H. Karcher, “Riemannian Center of Mass and Mollifier Smoothing,” Communications on Pure and Applied Mathematics, Vol. 30, 1977, pp. 509-541. doi:10.1002/cpa.3160300502

[32] W. Kendall, “Probability, Convexity and Harmonic Maps with Small Image I: Uniqueness and Fine Existence,” Proceedings of the London Mathematical Society, Vol. 61, 1990, pp. 371-406. doi:10.1112/plms/s3-61.2.371

[33] E. Begelfor and M. Werman, “Affine Invariance Revisited,” Proceedings of the Computer Vision and Pattern Recognition Conference, Vol. 2, 2006, pp. 2087-2094.

[34] J. Manton, “A Globally Convergent Numerical Algorithm for Computing the Center of Mass on Compact Lie groups,” Proceedings of the International Conference on Control, Automation, Robotics and Vision, Vol. 3, 2004, pp. 2211-2216.

[35] T. Sim, S. Baker and M. Bsat, “The CMU Pose, Illumination, and Expression Database,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 25, No. 12, 2003, pp. 1615-1618. doi:10.1109/TPAMI.2003.1251154

[36] D. Bolme, B. Draper and J. Beveridge, “Average of Synthetic Exact Filters,” Proceedings of the Computer Vision and Pattern Recognition, 20-25 June 2009, pp. 2015- 2112.

[37] P. Viola and M. Jones, “Robust Real-Time Face Detection,” International Journal of Computer Vision, Vol. 57, No. 2, 2004, pp. 137-154. doi:10.1023/B:VISI.0000013087.49260.fb

[38] P. Belhumeur and D. Kriegman, “What Is the Set of Images of an Object under All Possible Lighting Conditions,” International Journal of Computer Vision, Vol. 28, No. 3, 1998, pp. 245-260. doi:10.1023/A:1008005721484