Trust No One: Low Rank Matrix Factorization Using Hierarchical RANSAC

In this paper we present a system for performing low rank matrix factorization. Low-rank matrix factorization is an essential problem in many areas, including computer vision with applications in affine structure-from-motion, photometric stereo, and non-rigid structure from motion. We specifically target structured data patterns, with outliers and large amounts of missing data. Using recently developed characterizations of minimal solutions to matrix factorization problems with missing data, we show how these can be used as building blocks in a hierarchical system that performs bootstrapping on all levels. This gives a robust and fast system, with state-of-the-art performance.

[1]  M. Oskarsson,et al.  Classifying and solving minimal structure and motion problems with missing data , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[2]  Alexandre Bernardino,et al.  Unifying Nuclear Norm and Bilinear Factorization Approaches for Low-Rank Matrix Decomposition , 2013, 2013 IEEE International Conference on Computer Vision.

[3]  Takeo Kanade,et al.  An Iterative Image Registration Technique with an Application to Stereo Vision , 1981, IJCAI.

[4]  Andrew W. Fitzgibbon,et al.  Damped Newton algorithms for matrix factorization with missing data , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[5]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[6]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[7]  Lourdes Agapito,et al.  Dense Variational Reconstruction of Non-rigid Surfaces from Monocular Video , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[8]  Anders P. Eriksson,et al.  Efficient computation of robust low-rank matrix approximations in the presence of missing data using the L1 norm , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[9]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[10]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[11]  Takayuki Okatani,et al.  Efficient algorithm for low-rank matrix factorization with missing components and performance comparison of latest algorithms , 2011, 2011 International Conference on Computer Vision.

[12]  Long Quan,et al.  Minimal Projective Reconstruction Including Missing Data , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Magnus Oskarsson,et al.  Prime Rigid Graphs and Multidimensional Scaling with Missing Data , 2014, 2014 22nd International Conference on Pattern Recognition.

[14]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[15]  Yi Ma,et al.  The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices , 2010, Journal of structural biology.

[16]  Viktor Larsson,et al.  Rank Minimization with Structured Data Patterns , 2014, ECCV.

[17]  Henrik Aanæs,et al.  Robust Factorization , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Ruzena Bajcsy,et al.  Robust rank-4 affine factorization for structure from motion , 2013, 2013 IEEE Workshop on Applications of Computer Vision (WACV).

[19]  Magnus Oskarsson,et al.  On the minimal problems of low-rank matrix factorization , 2015, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[20]  Alessio Del Bue,et al.  Bilinear Modeling via Augmented Lagrange Multipliers (BALM) , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Fredrik Kahl,et al.  Structure from Motion with Missing Data is NP-Hard , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[22]  Carl Olsson,et al.  A Convex Approach to Low Rank Matrix Approximation with Missing Data , 2009, SCIA.

[23]  Ronen Basri,et al.  Photometric stereo with general, unknown lighting , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[24]  Takeo Kanade,et al.  Robust L/sub 1/ norm factorization in the presence of outliers and missing data by alternative convex programming , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[25]  Marc Pollefeys,et al.  A Factorization-Based Approach for Articulated Nonrigid Shape, Motion and Kinematic Chain Recovery From Video , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  Joan Serrat,et al.  An Iterative Multiresolution Scheme for SFM with Missing Data , 2009, Journal of Mathematical Imaging and Vision.

[27]  Adrien Bartoli,et al.  Batch Recovery of Multiple Views with Missing Data Using Direct Sparse Solvers , 2003, BMVC.

[28]  Viktor Larsson,et al.  Convex Envelopes for Low Rank Approximation , 2015, EMMCVPR.

[29]  Xuelong Li,et al.  Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[30]  Lebrecht Henneberg,et al.  Die graphische Statik der Starren Systeme , 1911 .

[31]  Dennis Strelow,et al.  General and nested Wiberg minimization , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[32]  Johan Karlsson,et al.  MDL patch correspondences on unlabeled images with occlusions , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[33]  Henning Biermann,et al.  Recovering non-rigid 3D shape from image streams , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[34]  Andrea Montanari,et al.  Matrix completion from a few entries , 2009, 2009 IEEE International Symposium on Information Theory.

[35]  Shuicheng Yan,et al.  Practical low-rank matrix approximation under robust L1-norm , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.