Skip to main content
Erschienen in: International Journal of Computer Vision 2-3/2015

01.09.2015

Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds

verfasst von: Mehrtash Harandi, Richard Hartley, Chunhua Shen, Brian Lovell, Conrad Sanderson

Erschienen in: International Journal of Computer Vision | Ausgabe 2-3/2015

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and dictionary learning in Grassmann manifolds, i.e., the space of linear subspaces. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping. This in turn enables us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we propose an algorithm for learning a Grassmann dictionary, atom by atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann sparse coding and dictionary learning algorithms through embedding into higher dimensional Hilbert spaces. Experiments on several classification tasks (gender recognition, gesture classification, scene analysis, face recognition, action recognition and dynamic texture classification) show that the proposed approaches achieve considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelized Affine Hull Method and graph-embedding Grassmann discriminant analysis.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
On an abstract Riemannian manifold \({\mathcal {M}}\), the gradient of a smooth real function f at a point \(x \in {\mathcal {M}}\), denoted by \(\mathrm {grad} f(x)\), is the element of \(T_x({\mathcal {M}})\) satisfying \(\langle \mathrm {grad}f(x), \zeta \rangle _x = Df_x[\zeta ]\) for all \(\zeta \in T_x({\mathcal {M}})\). Here, \(Df_x[\zeta ]\) denotes the directional derivative of f at x in the direction of \(\zeta \). The interested reader is referred to Absil et al. (2008) for more details on how the gradient of a function on Grassmann manifolds can be computed.
 
2
This is acknowledged by Ho et al. (2013).
 
Literatur
Zurück zum Zitat Absil, P.-A., Mahony, R., & Sepulchre, R. (2004). Riemannian geometry of grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica, 80(2), 199–220.CrossRefMathSciNetMATH Absil, P.-A., Mahony, R., & Sepulchre, R. (2004). Riemannian geometry of grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica, 80(2), 199–220.CrossRefMathSciNetMATH
Zurück zum Zitat Absil, P.-A., Mahony, R., & Sepulchre, R. (2008). Optimization algorithms on matrix manifolds. Princeton: Princeton University Press.CrossRefMATH Absil, P.-A., Mahony, R., & Sepulchre, R. (2008). Optimization algorithms on matrix manifolds. Princeton: Princeton University Press.CrossRefMATH
Zurück zum Zitat Aharon, M., Elad, M., & Bruckstein, A. (2006). K-svd: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11), 4311–4322.CrossRef Aharon, M., Elad, M., & Bruckstein, A. (2006). K-svd: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11), 4311–4322.CrossRef
Zurück zum Zitat Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006). Log-euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine, 56(2), 411–421.CrossRef Arsigny, V., Fillard, P., Pennec, X., & Ayache, N. (2006). Log-euclidean metrics for fast and simple calculus on diffusion tensors. Magnetic Resonance in Medicine, 56(2), 411–421.CrossRef
Zurück zum Zitat Basri, R., & Jacobs, D. W. (2003). Lambertian reflectance and linear subspaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(2), 218–233.CrossRef Basri, R., & Jacobs, D. W. (2003). Lambertian reflectance and linear subspaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(2), 218–233.CrossRef
Zurück zum Zitat Begelfor, E., & Werman, M. (2006). Affine invariance revisited. In Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2087–2094). Begelfor, E., & Werman, M. (2006). Affine invariance revisited. In Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2087–2094).
Zurück zum Zitat Candès, E. J., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.CrossRefMATH Candès, E. J., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.CrossRefMATH
Zurück zum Zitat Cetingul, H. E., & Vidal, R. (2009), Intrinsic mean shift for clustering on stiefel and grassmann manifolds. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1896–1902). Cetingul, H. E., & Vidal, R. (2009), Intrinsic mean shift for clustering on stiefel and grassmann manifolds. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1896–1902).
Zurück zum Zitat Cetingul, H.E., & Vidal, R. (2011). Sparse riemannian manifold clustering for HARDI segmentation. In IEEE International Symposium on Biomedical Imaging: From Nano to Macro (pp. 1750–1753). Cetingul, H.E., & Vidal, R. (2011). Sparse riemannian manifold clustering for HARDI segmentation. In IEEE International Symposium on Biomedical Imaging: From Nano to Macro (pp. 1750–1753).
Zurück zum Zitat Cetingul, H. E., Wright, M. J., Thompson, P. M., & Vidal, R. (2014). Segmentation of high angular resolution diffusion MRI using sparse riemannian manifold clustering. IEEE Transactions on Medical Imaging, 33(2), 301–317.CrossRef Cetingul, H. E., Wright, M. J., Thompson, P. M., & Vidal, R. (2014). Segmentation of high angular resolution diffusion MRI using sparse riemannian manifold clustering. IEEE Transactions on Medical Imaging, 33(2), 301–317.CrossRef
Zurück zum Zitat Cevikalp, H., & Triggs, B. (2010). Face recognition based on image sets. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2567–2573). Cevikalp, H., & Triggs, B. (2010). Face recognition based on image sets. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2567–2573).
Zurück zum Zitat Chan, A.B., & Vasconcelos, N. (2005). Probabilistic kernels for the classification of auto-regressive visual processes. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 846–851). Chan, A.B., & Vasconcelos, N. (2005). Probabilistic kernels for the classification of auto-regressive visual processes. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 846–851).
Zurück zum Zitat Chen, S., Sanderson, C., Harandi, M., & Lovell, B. C. (2013). Improved image set classification via joint sparse approximated nearest subspaces. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 452–459). Chen, S., Sanderson, C., Harandi, M., & Lovell, B. C. (2013). Improved image set classification via joint sparse approximated nearest subspaces. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 452–459).
Zurück zum Zitat Chikuse, Y. (2003). Statistics on special manifolds (Vol. 174). New York: Springer.MATH Chikuse, Y. (2003). Statistics on special manifolds (Vol. 174). New York: Springer.MATH
Zurück zum Zitat Dalal, N., & Triggs, B. (2005). Histograms of oriented gradients for human detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 886–893). Dalal, N., & Triggs, B. (2005). Histograms of oriented gradients for human detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 886–893).
Zurück zum Zitat Doretto, G., Chiuso, A., Wu, Y. N., & Soatto, S. (2003). Dynamic textures. International Journal of Computer Vision, 51, 91–109.CrossRefMATH Doretto, G., Chiuso, A., Wu, Y. N., & Soatto, S. (2003). Dynamic textures. International Journal of Computer Vision, 51, 91–109.CrossRefMATH
Zurück zum Zitat Elad, M. (2010). Sparse and redundant representations—From theory to applications in signal and image processing. New York: Springer.MATH Elad, M. (2010). Sparse and redundant representations—From theory to applications in signal and image processing. New York: Springer.MATH
Zurück zum Zitat Elhamifar, E., & Vidal, R. (2013). Sparse subspace clustering: Algorithm, theory, and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(11), 2765–2781.CrossRef Elhamifar, E., & Vidal, R. (2013). Sparse subspace clustering: Algorithm, theory, and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(11), 2765–2781.CrossRef
Zurück zum Zitat Gallivan, K. A., Srivastava, A., Liu, X., & Van Dooren, P. (2003). Efficient algorithms for inferences on Grassmann manifolds. In IEEE Workshop on Statistical Signal Processing (pp. 315–318). Gallivan, K. A., Srivastava, A., Liu, X., & Van Dooren, P. (2003). Efficient algorithms for inferences on Grassmann manifolds. In IEEE Workshop on Statistical Signal Processing (pp. 315–318).
Zurück zum Zitat Ghanem, B., & Ahuja, N. (2010). Maximum margin distance learning for dynamic texture recognition. Proceedings of the European Conference on Computer Vision (ECCV), 6312, 223–236. Ghanem, B., & Ahuja, N. (2010). Maximum margin distance learning for dynamic texture recognition. Proceedings of the European Conference on Computer Vision (ECCV), 6312, 223–236.
Zurück zum Zitat Goh, A., & Vidal, R. (2008). Clustering and dimensionality reduction on Riemannian manifolds. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–7). Goh, A., & Vidal, R. (2008). Clustering and dimensionality reduction on Riemannian manifolds. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–7).
Zurück zum Zitat Golub, G. H., & Van Loan, C. F. (1996). Matrix computations (3rd ed.). Baltimore: Johns Hopkins University Press.MATH Golub, G. H., & Van Loan, C. F. (1996). Matrix computations (3rd ed.). Baltimore: Johns Hopkins University Press.MATH
Zurück zum Zitat Gong, B., Shi, Y., Sha, F., & Grauman, K. (2012). Geodesic flow kernel for unsupervised domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2066–2073). Gong, B., Shi, Y., Sha, F., & Grauman, K. (2012). Geodesic flow kernel for unsupervised domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2066–2073).
Zurück zum Zitat Gopalan, R., Li, R., & Chellappa, R. (2014). Unsupervised adaptation across domain shifts by generating intermediate data representations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(11), 2288–2302.CrossRef Gopalan, R., Li, R., & Chellappa, R. (2014). Unsupervised adaptation across domain shifts by generating intermediate data representations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(11), 2288–2302.CrossRef
Zurück zum Zitat Guo, K., Ishwar, P., & Konrad, J. (2013). Action recognition from video using feature covariance matrices. IEEE Transactions on Image Processing (TIP), 22(6), 2479–2494.CrossRefMathSciNet Guo, K., Ishwar, P., & Konrad, J. (2013). Action recognition from video using feature covariance matrices. IEEE Transactions on Image Processing (TIP), 22(6), 2479–2494.CrossRefMathSciNet
Zurück zum Zitat Hamm, J., & Lee, D. D. (2008). Grassmann discriminant analysis: a unifying view on subspace-based learning. In Proceedings of the International Conference on Machine Learning (ICML) (pp. 376–383). Hamm, J., & Lee, D. D. (2008). Grassmann discriminant analysis: a unifying view on subspace-based learning. In Proceedings of the International Conference on Machine Learning (ICML) (pp. 376–383).
Zurück zum Zitat Harandi, M., Sanderson, C., Shen, C., & Lovell, B. C. (2013). Dictionary learning and sparse coding on Grassmann manifolds: An extrinsic solution. In: Proceedings of the International Conference on Computer Vision (ICCV). Harandi, M., Sanderson, C., Shen, C., & Lovell, B. C. (2013). Dictionary learning and sparse coding on Grassmann manifolds: An extrinsic solution. In: Proceedings of the International Conference on Computer Vision (ICCV).
Zurück zum Zitat Harandi, M.T., Hartley, R., Lovell, B. C., & Sanderson, C. (2015). Sparse coding on symmetric positive definite manifolds using bregman divergences. IEEE Transaction on Neural Networks and Learning Systems (TNNLS) PP(99):1–1. Harandi, M.T., Hartley, R., Lovell, B. C., & Sanderson, C. (2015). Sparse coding on symmetric positive definite manifolds using bregman divergences. IEEE Transaction on Neural Networks and Learning Systems (TNNLS) PP(99):1–1.
Zurück zum Zitat Harandi, M. T., Sanderson, C., Shirazi, S., & Lovell, B. C. (2011). Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2705–2712). Harandi, M. T., Sanderson, C., Shirazi, S., & Lovell, B. C. (2011). Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 2705–2712).
Zurück zum Zitat Hartley, R., Trumpf, J., Dai, Y., & Li, H. (2013). Rotation averaging. International Journal of Computer Vision, 103(3), 267–305.CrossRefMathSciNetMATH Hartley, R., Trumpf, J., Dai, Y., & Li, H. (2013). Rotation averaging. International Journal of Computer Vision, 103(3), 267–305.CrossRefMathSciNetMATH
Zurück zum Zitat Ho, J., Xie, Y., & Vemuri, B. (2013). On a nonlinear generalization of sparse coding and dictionary learning. In: Proceedings of the International Conference on Machine Learning (ICML) (pp. 1480–1488). Ho, J., Xie, Y., & Vemuri, B. (2013). On a nonlinear generalization of sparse coding and dictionary learning. In: Proceedings of the International Conference on Machine Learning (ICML) (pp. 1480–1488).
Zurück zum Zitat Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on pure and applied mathematics, 30(5), 509–541.CrossRefMathSciNetMATH Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on pure and applied mathematics, 30(5), 509–541.CrossRefMathSciNetMATH
Zurück zum Zitat Kim, M., Kumar, S., Pavlovic, V., & Rowley, H. (2008). Face tracking and recognition with visual constraints in real-world videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8). Kim, M., Kumar, S., Pavlovic, V., & Rowley, H. (2008). Face tracking and recognition with visual constraints in real-world videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8).
Zurück zum Zitat Kim, T.-K., & Cipolla, R. (2009). Canonical correlation analysis of video volume tensors for action categorization and detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(8), 1415–1428.CrossRef Kim, T.-K., & Cipolla, R. (2009). Canonical correlation analysis of video volume tensors for action categorization and detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(8), 1415–1428.CrossRef
Zurück zum Zitat Kim, T.-K., Kittler, J., & Cipolla, R. (2007). Discriminative learning and recognition of image set classes using canonical correlations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(6), 1005–1018.CrossRef Kim, T.-K., Kittler, J., & Cipolla, R. (2007). Discriminative learning and recognition of image set classes using canonical correlations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(6), 1005–1018.CrossRef
Zurück zum Zitat Kokiopoulou, E., Chen, J., & Saad, Y. (2011). Trace optimization and eigenproblems in dimension reduction methods. Numerical Linear Algebra with Applications, 18(3), 565–602.CrossRefMathSciNetMATH Kokiopoulou, E., Chen, J., & Saad, Y. (2011). Trace optimization and eigenproblems in dimension reduction methods. Numerical Linear Algebra with Applications, 18(3), 565–602.CrossRefMathSciNetMATH
Zurück zum Zitat Lee, J. M. (2012). Introduction to smooth manifolds (Vol. 218). New York: Springer.CrossRef Lee, J. M. (2012). Introduction to smooth manifolds (Vol. 218). New York: Springer.CrossRef
Zurück zum Zitat Li, B., Ayazoglu, M., Mao, T., Camps, O. I., & Sznaier, M. (2011). Activity recognition using dynamic subspace angles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 3193–3200). Li, B., Ayazoglu, M., Mao, T., Camps, O. I., & Sznaier, M. (2011). Activity recognition using dynamic subspace angles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 3193–3200).
Zurück zum Zitat Lui, Y. M. (2012). Human gesture recognition on product manifolds. Journal of Machine Learning Research, 13, 3297–3321.MathSciNetMATH Lui, Y. M. (2012). Human gesture recognition on product manifolds. Journal of Machine Learning Research, 13, 3297–3321.MathSciNetMATH
Zurück zum Zitat Mairal, J., Bach, F., & Ponce, J. (2012). Task-driven dictionary learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(4), 791–804.CrossRef Mairal, J., Bach, F., & Ponce, J. (2012). Task-driven dictionary learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(4), 791–804.CrossRef
Zurück zum Zitat Mairal, J., Bach, F., Ponce, J., & Sapiro, G. (2010). Online learning for matrix factorization and sparse coding. Journal of Machine Learning Research, 11, 19–60.MathSciNetMATH Mairal, J., Bach, F., Ponce, J., & Sapiro, G. (2010). Online learning for matrix factorization and sparse coding. Journal of Machine Learning Research, 11, 19–60.MathSciNetMATH
Zurück zum Zitat Mairal, J., Bach, F., Ponce, J., Sapiro, G., & Zisserman, A. (2008). Discriminative learned dictionaries for local image analysis. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8). IEEE. Mairal, J., Bach, F., Ponce, J., Sapiro, G., & Zisserman, A. (2008). Discriminative learned dictionaries for local image analysis. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8). IEEE.
Zurück zum Zitat Mairal, J., Elad, M., & Sapiro, G. (2008). Sparse representation for color image restoration. IEEE Transactions on Image Processing (TIP), 17(1), 53–69.CrossRefMathSciNet Mairal, J., Elad, M., & Sapiro, G. (2008). Sparse representation for color image restoration. IEEE Transactions on Image Processing (TIP), 17(1), 53–69.CrossRefMathSciNet
Zurück zum Zitat Manton, J. H. (2004). A globally convergent numerical algorithm for computing the centre of mass on compact lie groups. In International Conference on Control, Automation, Robotics and Vision 3 (pp. 2211–2216). Manton, J. H. (2004). A globally convergent numerical algorithm for computing the centre of mass on compact lie groups. In International Conference on Control, Automation, Robotics and Vision 3 (pp. 2211–2216).
Zurück zum Zitat Ojala, T., Pietikäinen, M., & Mäenpää, T. (2002). Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24, 971–987.CrossRef Ojala, T., Pietikäinen, M., & Mäenpää, T. (2002). Multiresolution gray-scale and rotation invariant texture classification with local binary patterns. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24, 971–987.CrossRef
Zurück zum Zitat Olshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583), 607–609.CrossRef Olshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583), 607–609.CrossRef
Zurück zum Zitat Ramamoorthi, R. (2002). Analytic PCA construction for theoretical analysis of lighting variability in images of a Lambertian object. IEEE Trans. Pattern Analysis and Machine Intelligence, 24(10), 1322–1333.CrossRef Ramamoorthi, R. (2002). Analytic PCA construction for theoretical analysis of lighting variability in images of a Lambertian object. IEEE Trans. Pattern Analysis and Machine Intelligence, 24(10), 1322–1333.CrossRef
Zurück zum Zitat Rao, S. R., Tron, R., Vidal, R., & Ma, Y. (2008). Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8). Rao, S. R., Tron, R., Vidal, R., & Ma, Y. (2008). Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1–8).
Zurück zum Zitat Ravichandran, A., Favaro, P., & Vidal, R. (2011). A unified approach to segmentation and categorization of dynamic textures. In Proceedings of the Asian Conference on Computer Vision (ACCV) (pp. 425–438). Springer. Ravichandran, A., Favaro, P., & Vidal, R. (2011). A unified approach to segmentation and categorization of dynamic textures. In Proceedings of the Asian Conference on Computer Vision (ACCV) (pp. 425–438). Springer.
Zurück zum Zitat Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326.CrossRef Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326.CrossRef
Zurück zum Zitat Sanderson, C., Harandi, M. T., Wong, Y., & Lovell, B. C. (2012). Combined learning of salient local descriptors and distance metrics for image set face verification. In Proceedings of the International Conference on Advanced Video and Signal-Based Surveillance (pp. 294–299). Sanderson, C., Harandi, M. T., Wong, Y., & Lovell, B. C. (2012). Combined learning of salient local descriptors and distance metrics for image set face verification. In Proceedings of the International Conference on Advanced Video and Signal-Based Surveillance (pp. 294–299).
Zurück zum Zitat Sankaranarayanan, A., Turaga, P., Baraniuk, R., & Chellappa, R. (2010). Compressive acquisition of dynamic scenes. Proceedings of the European Conference on Computer Vision (ECCV), 6311, 129–142. Sankaranarayanan, A., Turaga, P., Baraniuk, R., & Chellappa, R. (2010). Compressive acquisition of dynamic scenes. Proceedings of the European Conference on Computer Vision (ECCV), 6311, 129–142.
Zurück zum Zitat Shawe-Taylor, J., & Cristianini, N. (2004). Kernel methods for pattern analysis. Cambridge: Cambridge University Press.CrossRef Shawe-Taylor, J., & Cristianini, N. (2004). Kernel methods for pattern analysis. Cambridge: Cambridge University Press.CrossRef
Zurück zum Zitat Shirazi, S., Sanderson, C., McCool, C., & Harandi, M. T. (2015). Bags of affine subspaces for robust object tracking. Preprint: arXiv:1408.2313. Shirazi, S., Sanderson, C., McCool, C., & Harandi, M. T. (2015). Bags of affine subspaces for robust object tracking. Preprint: arXiv:​1408.​2313.
Zurück zum Zitat Srivastava, A., & Klassen, E. (2004). Bayesian and geometric subspace tracking. Advances in Applied Probability, 36(1), 43–56.CrossRefMathSciNetMATH Srivastava, A., & Klassen, E. (2004). Bayesian and geometric subspace tracking. Advances in Applied Probability, 36(1), 43–56.CrossRefMathSciNetMATH
Zurück zum Zitat Subbarao, R., & Meer, P. (2009). Nonlinear mean shift over Riemannian manifolds. International Journal of Computer Vision, 84(1), 1–20.CrossRef Subbarao, R., & Meer, P. (2009). Nonlinear mean shift over Riemannian manifolds. International Journal of Computer Vision, 84(1), 1–20.CrossRef
Zurück zum Zitat Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288.MathSciNetMATH Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288.MathSciNetMATH
Zurück zum Zitat Turaga, P., Veeraraghavan, A., Srivastava, A., & Chellappa, R. (2011). Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(11), 2273–2286.CrossRef Turaga, P., Veeraraghavan, A., Srivastava, A., & Chellappa, R. (2011). Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(11), 2273–2286.CrossRef
Zurück zum Zitat Turk, M., & Pentland, A. (1991). Eigenfaces for recognition. Journal of Cognitive Neuroscience, 3(1), 71–86.CrossRef Turk, M., & Pentland, A. (1991). Eigenfaces for recognition. Journal of Cognitive Neuroscience, 3(1), 71–86.CrossRef
Zurück zum Zitat Vemulapalli, R., Pillai, J. K., & Chellappa, R. (2013). Kernel learning for extrinsic classification of manifold features. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1782–1789). Vemulapalli, R., Pillai, J. K., & Chellappa, R. (2013). Kernel learning for extrinsic classification of manifold features. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1782–1789).
Zurück zum Zitat Viola, P., & Jones, M. J. (2004). Robust real-time face detection. International Journal of Computer Vision, 57(2), 137–154.CrossRef Viola, P., & Jones, M. J. (2004). Robust real-time face detection. International Journal of Computer Vision, 57(2), 137–154.CrossRef
Zurück zum Zitat Wang, J., Yang, J., Yu, K., Lv, F., Huang, T., & Gong, Y. (2010). Locality-constrained linear coding for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 3360–3367). Wang, J., Yang, J., Yu, K., Lv, F., Huang, T., & Gong, Y. (2010). Locality-constrained linear coding for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 3360–3367).
Zurück zum Zitat Wang, Y., & Mori, G. (2009). Human action recognition by semilatent topic models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(10), 1762–1774.CrossRef Wang, Y., & Mori, G. (2009). Human action recognition by semilatent topic models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(10), 1762–1774.CrossRef
Zurück zum Zitat Wikipedia. Min-max theorem – wikipedia, the free encyclopedia, 2015. [Online; accessed 27-May-2015]. Wikipedia. Min-max theorem – wikipedia, the free encyclopedia, 2015. [Online; accessed 27-May-2015].
Zurück zum Zitat Wright, J., Ma, Y., Mairal, J., Sapiro, G., Huang, T. S., & Yan, S. (2010). Sparse representation for computer vision and pattern recognition. Proceedings of the IEEE, 98(6), 1031–1044.CrossRef Wright, J., Ma, Y., Mairal, J., Sapiro, G., Huang, T. S., & Yan, S. (2010). Sparse representation for computer vision and pattern recognition. Proceedings of the IEEE, 98(6), 1031–1044.CrossRef
Zurück zum Zitat Wright, J., Yang, A. Y., Ganesh, A., Sastry, S. S., & Ma, Y. (2009). Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2), 210–227.CrossRef Wright, J., Yang, A. Y., Ganesh, A., Sastry, S. S., & Ma, Y. (2009). Robust face recognition via sparse representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(2), 210–227.CrossRef
Zurück zum Zitat Xu, Y., Quan, Y., Ling, H., & Ji, H. (2011). Dynamic texture classification using dynamic fractal analysis. In Proceedings of the International Conference on Computer Vision (ICCV). Xu, Y., Quan, Y., Ling, H., & Ji, H. (2011). Dynamic texture classification using dynamic fractal analysis. In Proceedings of the International Conference on Computer Vision (ICCV).
Zurück zum Zitat Yang, J., Yu, K., Gong, Y., & Huang, T. (2009). Linear spatial pyramid matching using sparse coding for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1794–1801). Yang, J., Yu, K., Gong, Y., & Huang, T. (2009). Linear spatial pyramid matching using sparse coding for image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 1794–1801).
Zurück zum Zitat Yu, K., & Zhang, T. (2010). Improved local coordinate coding using local tangents. In Proceedings of the International Conference on Machine Learning (ICML) (pp. 1215–1222). Yu, K., & Zhang, T. (2010). Improved local coordinate coding using local tangents. In Proceedings of the International Conference on Machine Learning (ICML) (pp. 1215–1222).
Zurück zum Zitat Yu, K., Zhang, T., & Gong, Y. (2009). Nonlinear learning using local coordinate coding. In Proceedings of the Advances in Neural Information Processing Systems (NIPS) 9 (p 1). Yu, K., Zhang, T., & Gong, Y. (2009). Nonlinear learning using local coordinate coding. In Proceedings of the Advances in Neural Information Processing Systems (NIPS) 9 (p 1).
Zurück zum Zitat Yu, S., Tan, T., Huang, K., Jia, K., & Wu, X. (2009). A study on gait-based gender classification. IEEE Transactions on Image Processing (TIP), 18(8), 1905–1910.CrossRefMathSciNet Yu, S., Tan, T., Huang, K., Jia, K., & Wu, X. (2009). A study on gait-based gender classification. IEEE Transactions on Image Processing (TIP), 18(8), 1905–1910.CrossRefMathSciNet
Zurück zum Zitat Yuan, C., Hu, W., Li, X., Maybank, S., & Luo, G. (2010). Human action recognition under log-euclidean Riemannian metric. In H. Zha, R.-I. Taniguchi, & S. Maybank editors, Proc. Asian Conference on Computer Vision (ACCV), volume 5994 of Lecture Notes in Computer Science, pages 343–353. Springer Berlin Heidelberg. Yuan, C., Hu, W., Li, X., Maybank, S., & Luo, G. (2010). Human action recognition under log-euclidean Riemannian metric. In H. Zha, R.-I. Taniguchi, & S. Maybank editors, Proc. Asian Conference on Computer Vision (ACCV), volume 5994 of Lecture Notes in Computer Science, pages 343–353. Springer Berlin Heidelberg.
Zurück zum Zitat Zhao, G., & Pietikäinen, M. (2007). Dynamic texture recognition using local binary patterns with an application to facial expressions. IEEE Trans. Pattern Analysis and Machine Intelligence, 29(6), 915–928.CrossRef Zhao, G., & Pietikäinen, M. (2007). Dynamic texture recognition using local binary patterns with an application to facial expressions. IEEE Trans. Pattern Analysis and Machine Intelligence, 29(6), 915–928.CrossRef
Zurück zum Zitat Zheng, S., Zhang, J., Huang, K., He, R., & Tan, T. (2011). Robust view transformation model for gait recognition. In International Conference on Image Processing (ICIP) (pp. 2073–2076). Zheng, S., Zhang, J., Huang, K., He, R., & Tan, T. (2011). Robust view transformation model for gait recognition. In International Conference on Image Processing (ICIP) (pp. 2073–2076).
Metadaten
Titel
Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds
verfasst von
Mehrtash Harandi
Richard Hartley
Chunhua Shen
Brian Lovell
Conrad Sanderson
Publikationsdatum
01.09.2015
Verlag
Springer US
Erschienen in
International Journal of Computer Vision / Ausgabe 2-3/2015
Print ISSN: 0920-5691
Elektronische ISSN: 1573-1405
DOI
https://doi.org/10.1007/s11263-015-0833-x

Weitere Artikel der Ausgabe 2-3/2015

International Journal of Computer Vision 2-3/2015 Zur Ausgabe