Abstract
Various fundamental applications in computer vision and machine learning require finding the basis of a certain subspace. The obtained basis are then used to represent any observation in the subspace, often using only the major basis in order to reduce the dimensionality and suppress noise. Examples of such applications include face detection, motion estimation, activity recognition … etc. An increasing interest has been recently placed on this area as a result of significant advances in the mathematics of matrix rank optimization. Interestingly, as we will discuss in more detail in the coming sections, robust subspace estimation can be posed as a low rank-optimization problem, which can be solved efficiently using techniques such as the method of Augmented Lagrange Multiplier.In this book we discuss fundamental formulations and extensions for low-rank optimization-based subspace estimation and representation. By minimizing the rank of the matrix containing observations drawn from images, we demonstrate to the reader how to solve four fundamental computer vision problems, including video denosing, background subtraction, motion estimation, and activity recognition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Ali, M. Shah, A Lagrangian particle dynamics approach for crowd flow segmentation and stability analysis, in CVPR (Minneapolis, Minnesota, USA, 2007)
J.-F. Cai, E.J. Candes, Z. Shen, A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)
E. Candes, T. Tao, Decoding by linear programming. IEEE Trans. Inf. Theory 51, 4203 (2005)
E.J. Candes, X. Li, Y. Ma, J. Wright, Robust principal component analysis? (2009, arXiv:0912.3599v1)
D.L. Donoho, For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Commun. Pure Appl. Math. 59, 797–829 (2006)
M. Fazel, B. Recht, P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2010)
A. Gruber, Y. Weiss, Multibody factorization with uncertainty and missing data using the EM algorithm, in CVPR (Washington, DC, 2004)
H. Izadinia, M. Shah, Recognizing complex events using large margin joint low-level event model, in ECCV (Firenze, Italy, 2012)
I. Jolliffe, Principal Component Analysis (Springer, New York, 1986)
N. Joshiy, C. Lawrence Zitnicky, R. Szeliskiy, D.J. Kriegman, Image deblurring and denoising using color priors, in CVPR (Miami, FL, 2009)
I. Laptev, T. Lindeberg, Space-time interest points, in ICCV (Nice, France, 2003)
Z. Lin, M. Chen, L. Wu, Y. Ma, The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices, UIUC technical report, 2009
Z. Lin, G. Liu, Y. Yu, Robust subspace segmentation by low-rank representation, in ICML (Haifa, Israel, 2010)
X. Liu, O. Oreifej, M. Shah, Simultaneous video stabilization and moving object detection in turbulence, in PAMI (Augsburg, Germany, 2012)
O. Oreifej, S. Wu, M. Shah, Action recognition in videos acquired by a moving camera using motion decomposition of lagrangian particle trajectories, in ICCV, 2011
T. Pace, O. Oreifej, G. Shu, M. Shah, A two-stage reconstruction approach for seeing through water, in CVPR (Colorado Springs, CO, 2011)
B. Scholkopf, A. Smola, K.-R. Müller, Kernel principal component analysis, in Advances in Kernel Methods – Support Vector Learning, ed. by B. Schölkopf, C.J.C. Burges, A.J. Smola (MIT, Cambridge, 1999), pp. 327–352
M. Tipping, C. Bishop, Mixtures of probabilistic principal component analyzers. Neural Comput. 11, 443–482 (1999)
H. Wang, A. Klaser, C. Schmid, C. L. Liu, Action recognition by dense trajectories, in CVPR (Colorado Springs, CO, 2011)
L. Xu, J. Jia, Two-phase Kernel estimation for robust motion deblurring, in ECCV (Crete, Greece, 2010)
Y. Yang, M. Shah, Complex events detection using data-driven concepts, in ECCV (Firenze, Italy, 2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Oreifej, O., Shah, M. (2014). Introduction. In: Robust Subspace Estimation Using Low-Rank Optimization. The International Series in Video Computing, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-04184-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-04184-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04183-4
Online ISBN: 978-3-319-04184-1
eBook Packages: Computer ScienceComputer Science (R0)