Prior-Less Compressible Structure from Motion

Many non-rigid 3D structures are not modelled well through a low-rank subspace assumption. This is problematic when it comes to their reconstruction through Structure from Motion (SfM). We argue in this paper that a more expressive and general assumption can be made around compressible 3D structures. The vision community, however, has hitherto struggled to formulate effective strategies for recovering such structures after projection without the aid of additional priors (e.g. temporal ordering, rigid substructures, etc.). In this paper we present a "prior-less" approach to solve compressible SfM. Specifically, we demonstrate how the problem of SfM - assuming compressible 3D structures - can be theoretically characterized as a block sparse dictionary learning problem. We validate our approach experimentally by demonstrating reconstructions of 3D structures that are intractable using current state-of-theart low-rank SfM approaches.

[1]  Wotao Yin,et al.  Group sparse optimization by alternating direction method , 2013, Optics & Photonics - Optical Engineering + Applications.

[2]  Jing Xiao,et al.  A Closed-Form Solution to Non-Rigid Shape and Motion Recovery , 2004, International Journal of Computer Vision.

[3]  Anders P. Eriksson,et al.  Fast Convolutional Sparse Coding , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[4]  Aleix M. Martínez,et al.  Computing Smooth Time Trajectories for Camera and Deformable Shape in Structure from Motion with Occlusion , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[6]  K. Kreutz-Delgado,et al.  Basis selection in the presence of noise , 1998, Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284).

[7]  Hongdong Li,et al.  A Simple Prior-Free Method for Non-rigid Structure-from-Motion Factorization , 2012, International Journal of Computer Vision.

[8]  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).

[9]  F U GotardoPaulo,et al.  Computing Smooth Time Trajectories for Camera and Deformable Shape in Structure from Motion with Occlusion , 2011 .

[10]  Sridha Sridharan,et al.  Efficient Articulated Trajectory Reconstruction Using Dynamic Programming and Filters , 2012, ECCV.

[11]  Simon Lucey,et al.  Complex Non-rigid Motion 3D Reconstruction by Union of Subspaces , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[12]  Friedrich T. Sommer,et al.  When Can Dictionary Learning Uniquely Recover Sparse Data From Subsamples? , 2011, IEEE Transactions on Information Theory.

[13]  Kenneth P. Bogart,et al.  Introductory Combinatorics , 1977 .

[14]  Serge J. Belongie,et al.  Re-thinking non-rigid structure from motion , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[15]  Aaron Hertzmann,et al.  Nonrigid Structure-from-Motion: Estimating Shape and Motion with Hierarchical Priors , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Takeo Kanade,et al.  Shape and motion from image streams under orthography: a factorization method , 1992, International Journal of Computer Vision.

[17]  Simon Lucey,et al.  3D motion reconstruction for real-world camera motion , 2011, CVPR 2011.

[18]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[19]  Yaser Sheikh,et al.  3D reconstruction of a smooth articulated trajectory from a monocular image sequence , 2011, 2011 International Conference on Computer Vision.

[20]  Simon Lucey,et al.  Convolutional Sparse Coding for Trajectory Reconstruction , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Yaser Sheikh,et al.  In defense of orthonormality constraints for nonrigid structure from motion , 2009, 2009 IEEE Conference on Computer Vision and Pattern Recognition.

[22]  Aleix M. Martínez,et al.  Kernel non-rigid structure from motion , 2011, 2011 International Conference on Computer Vision.

[23]  L. Turner,et al.  Inverse of the Vandermonde matrix with applications , 1966 .

[24]  Takeo Kanade,et al.  Nonrigid Structure from Motion in Trajectory Space , 2008, NIPS.

[25]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[26]  Michael Elad,et al.  Analysis K-SVD: A Dictionary-Learning Algorithm for the Analysis Sparse Model , 2013, IEEE Transactions on Signal Processing.

[27]  Prateek Jain,et al.  Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization , 2013, SIAM J. Optim..

[28]  Bhaskar D. Rao,et al.  Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm , 1997, IEEE Trans. Signal Process..