Multiscale geometric and spectral analysis of plane arrangements

Modeling data by multiple low-dimensional planes is an important problem in many applications such as computer vision and pattern recognition. In the most general setting where only coordinates of the data are given, the problem asks to determine the optimal model parameters (i.e., number of planes and their dimensions), estimate the model planes, and cluster the data accordingly. Though many algorithms have been proposed, most of them need to assume prior knowledge of the model parameters and thus address only the last two components of the problem. In this paper we propose an efficient algorithm based on multiscale SVD analysis and spectral methods to tackle the problem in full generality. We also demonstrate its state-of-the-art performance on both synthetic and real data.

[1]  Robert C. Bolles,et al.  Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography , 1981, CACM.

[2]  Allen Y. Yang,et al.  Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data , 2008, SIAM Rev..

[3]  Hans-Peter Kriegel,et al.  Subspace clustering , 2012, WIREs Data Mining Knowl. Discov..

[4]  David J. Kriegman,et al.  Clustering appearances of objects under varying illumination conditions , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[5]  Guangliang Chen,et al.  Spectral Curvature Clustering (SCC) , 2009, International Journal of Computer Vision.

[6]  David J. Kriegman,et al.  From Few to Many: Illumination Cone Models for Face Recognition under Variable Lighting and Pose , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  John Wright,et al.  Segmentation of Multivariate Mixed Data via Lossy Data Coding and Compression , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Guangliang Chen,et al.  Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling , 2008, Found. Comput. Math..

[9]  Lorenzo Rosasco,et al.  Some Recent Advances in Multiscale Geometric Analysis of Point Clouds , 2011 .

[10]  Christopher M. Bishop,et al.  Mixtures of Probabilistic Principal Component Analyzers , 1999, Neural Computation.

[11]  Guangliang Chen,et al.  Multiscale Geometric Methods for Data Sets II: Geometric Wavelets , 2011, ArXiv.

[12]  Ehsan Elhamifar,et al.  Sparse subspace clustering , 2009, 2009 IEEE Conference on Computer Vision and Pattern Recognition.

[13]  Gilad Lerman,et al.  Randomized hybrid linear modeling by local best-fit flats , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[14]  René Vidal,et al.  Segmenting Motions of Different Types by Unsupervised Manifold Clustering , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[15]  Kun Huang,et al.  A multiscale hybrid linear model for lossy image representation , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[16]  Yong Yu,et al.  Robust Subspace Segmentation by Low-Rank Representation , 2010, ICML.

[17]  René Vidal,et al.  A Benchmark for the Comparison of 3-D Motion Segmentation Algorithms , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[18]  Michael I. Jordan,et al.  Mixtures of Probabilistic Principal Component Analyzers , 2001 .

[19]  Mauro Maggioni,et al.  Multiscale Estimation of Intrinsic Dimensionality of Data Sets , 2009, AAAI Fall Symposium: Manifold Learning and Its Applications.

[20]  Kenichi Kanatani,et al.  Multi-Stage Unsupervised Learning for Multi-Body Motion Segmentation , 2004, IEICE Trans. Inf. Syst..

[21]  S. Shankar Sastry,et al.  Generalized principal component analysis (GPCA) , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[22]  Marc Pollefeys,et al.  A General Framework for Motion Segmentation: Independent, Articulated, Rigid, Non-rigid, Degenerate and Non-degenerate , 2006, ECCV.