Smooth incomplete matrix factorization and its applications in image/video denoising

Low-rank matrix factorization with missing elements has many applications in computer vision. However, the original model without taking any prior information, which is to minimize the total reconstruction error of all the observed matrix elements, sometimes provides a physically meaningless solution in some applications. In this paper, we propose a regularized low-rank factorization model for a matrix with missing elements, called Smooth Incomplete Matrix Factorization (SIMF), and exploit a novel image/video denoising algorithm with the SIMF. Since data in many applications are usually of intrinsic spatial smoothness, the SIMF uses a 2D discretized Laplacian operator as a regularizer to constrain the matrix elements to be locally smoothly distributed. It is formulated as two optimization problems under the l"1 norm and the Frobenius norm, and two iterative algorithms are designed for solving them respectively. Then, the SIMF is extended to the tensor case (called Smooth Incomplete Tensor Factorization, SITF) by replacing the 2D Laplacian by a high-dimensional Laplacian. Finally, an image/video denoising algorithm is presented based on the proposed SIMF/SITF. Extensive experimental results show the effectiveness of our algorithm in comparison to other six algorithms.

[1]  Pedro M. Q. Aguiar,et al.  3D structure from video streams with partially overlapping images , 2002, Proceedings. International Conference on Image Processing.

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

[3]  João Paulo Costeira,et al.  Estimating 3D shape from degenerate sequences with missing data , 2009, Comput. Vis. Image Underst..

[4]  Franz Pernkopf,et al.  Sparse nonnegative matrix factorization with ℓ0-constraints , 2012, Neurocomputing.

[5]  S. Yun,et al.  An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems , 2009 .

[6]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

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

[8]  Yuxiao Hu,et al.  Learning a Spatially Smooth Subspace for Face Recognition , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[9]  F. O’Sullivan.  Discretized Laplacian Smoothing by Fourier Methods , 1991 .

[10]  René Vidal,et al.  Multiframe Motion Segmentation with Missing Data Using PowerFactorization and GPCA , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[11]  Shuicheng Yan,et al.  Active Subspace: Toward Scalable Low-Rank Learning , 2012, Neural Computation.

[12]  Takayuki Okatani,et al.  On the Wiberg Algorithm for Matrix Factorization in the Presence of Missing Components , 2007, International Journal of Computer Vision.

[13]  KanadeTakeo,et al.  Shape and motion from image streams under orthography , 1992 .

[14]  Jitendra Malik,et al.  A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[15]  Vincenzo Verardi.  Robust principal component analysis in Stata , 2009 .

[16]  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).

[17]  Junfeng Yang,et al.  Alternating Direction Algorithms for 1-Problems in Compressive Sensing , 2009, SIAM J. Sci. Comput..

[18]  Jian Yang,et al.  Low-rank representation based discriminative projection for robust feature extraction , 2013, Neurocomputing.

[19]  Robert Tibshirani,et al.  Spectral Regularization Algorithms for Learning Large Incomplete Matrices , 2010, J. Mach. Learn. Res..

[20]  G. Sapiro,et al.  A collaborative framework for 3D alignment and classification of heterogeneous subvolumes in cryo-electron tomography. , 2013, Journal of structural biology.

[21]  Pei Chen,et al.  Optimization Algorithms on Subspaces: Revisiting Missing Data Problem in Low-Rank Matrix , 2008, International Journal of Computer Vision.

[22]  João M. F. Xavier,et al.  Spectrally optimal factorization of incomplete matrices , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

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

[24]  Tamara G. Kolda,et al.  Categories and Subject Descriptors: G.4 [Mathematics of Computing]: Mathematical Software— , 2022 .

[25]  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).

[26]  S. Yun,et al.  An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems , 2009 .

[27]  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.

[28]  Zuowei Shen,et al.  Robust video denoising using low rank matrix completion , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[29]  Y. Zhang,et al.  Augmented Lagrangian alternating direction method for matrix separation based on low-rank factorization , 2014, Optim. Methods Softw..

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

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

[32]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[33]  Emmanuel J. Candès,et al.  A Singular Value Thresholding Algorithm for Matrix Completion , 2008, SIAM J. Optim..

[34]  Zhenyue Zhang,et al.  Successively alternate least square for low-rank matrix factorization with bounded missing data , 2010, Comput. Vis. Image Underst..

[35]  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.

[36]  Hongyuan Zha,et al.  Inducible regularization for low-rank matrix factorizations for collaborative filtering , 2012, Neurocomputing.

[37]  Harry Shum,et al.  Principal Component Analysis with Missing Data and Its Application to Polyhedral Object Modeling , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[38]  Hideki Hayakawa.  Photometric stereo under a light source with arbitrary motion , 1994 .