Image Completion with Smooth Nonnegative Matrix Factorization

Nonnegative matrix factorization is an unsupervised learning method for part-based feature extraction and dimensionality reduction of nonnegative data with a variety of models, algorithms, structures, and applications. Smooth nonnegative matrix factorization assumes the estimated latent factors are locally smooth, and the smoothness is enforced by the underlying model or the algorithm. In this study, we extended one of the algorithms for this kind of factorization to an image completion problem. It is the B-splines ADMM-NMF (Nonnegative Matrix Factorization with Alternating Direction Method of Multipliers) that enforces smooth feature vectors by assuming they are represented by a linear combination of smooth basis functions, i.e. B-splines. The numerical experiments performed on several incomplete images show that the proposed method outperforms the other algorithms in terms of the quality of recovered images.

[1]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[2]  Yi Ma,et al.  Generalized Tensor Total Variation minimization for visual data recovery? , 2015, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[3]  Yin Zhang,et al.  Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm , 2012, Mathematical Programming Computation.

[4]  Xiaoming Yuan,et al.  Sparse and low-rank matrix decomposition via alternating direction method , 2013 .

[5]  Michael Möller,et al.  A Convex Model for Nonnegative Matrix Factorization and Dimensionality Reduction on Physical Space , 2011, IEEE Transactions on Image Processing.

[6]  Rafal Zdunek,et al.  Alternating direction method for approximating smooth feature vectors in Nonnegative Matrix Factorization , 2014, 2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP).

[7]  Andrzej Cichocki,et al.  B-Spline Smoothing of Feature Vectors in Nonnegative Matrix Factorization , 2014, ICAISC.

[8]  Liangpei Zhang,et al.  Hyperspectral Image Restoration Using Low-Rank Matrix Recovery , 2014, IEEE Transactions on Geoscience and Remote Sensing.

[9]  Yin Zhang,et al.  An alternating direction algorithm for matrix completion with nonnegative factors , 2011, Frontiers of Mathematics in China.

[10]  Rafal Zdunek,et al.  Modified HALS Algorithm for Image Completion and Recommendation System , 2017, ISAT.

[11]  Cédric Févotte,et al.  Alternating direction method of multipliers for non-negative matrix factorization with the beta-divergence , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[12]  Richard G. Baraniuk,et al.  Fast Alternating Direction Optimization Methods , 2014, SIAM J. Imaging Sci..

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

[14]  Xin Liu,et al.  Document clustering based on non-negative matrix factorization , 2003, SIGIR.

[15]  Takeo Kanade,et al.  Spatio-Temporal Frequency Analysis for Removing Rain and Snow from Videos , 2007 .

[16]  Andrzej Cichocki,et al.  Nonnegative Matrix and Tensor Factorization T , 2007 .

[17]  Yangyang Xu,et al.  Alternating proximal gradient method for nonnegative matrix factorization , 2011, ArXiv.

[18]  Christian Bauckhage,et al.  Hierarchical Convex NMF for Clustering Massive Data , 2010, ACML.

[19]  Sen Jia,et al.  Constrained Nonnegative Matrix Factorization for Hyperspectral Unmixing , 2009, IEEE Transactions on Geoscience and Remote Sensing.

[20]  Musa H. Asyali,et al.  Image Processing with MATLAB: Applications in Medicine and Biology , 2008 .

[21]  Andrzej Cichocki,et al.  Blind Image Separation Using Nonnegative Matrix Factorization with Gibbs Smoothing , 2007, ICONIP.

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

[23]  Rahul Mazumder,et al.  Non-negative matrix completion for bandwidth extension: A convex optimization approach , 2013, 2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP).

[24]  Michael W. Berry,et al.  Text Mining Using Non-Negative Matrix Factorizations , 2004, SDM.

[25]  Xiangfeng Wang,et al.  Nonnegative matrix factorization using ADMM: Algorithm and convergence analysis , 2016, 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[26]  Jiayu Zhou,et al.  Latent Fingerprint Value Prediction: Crowd-Based Learning , 2018, IEEE Transactions on Information Forensics and Security.

[27]  Soo-Chang Pei,et al.  Removing rain and snow in a single image using saturation and visibility features , 2014, 2014 IEEE International Conference on Multimedia and Expo Workshops (ICMEW).

[28]  Andrzej Cichocki,et al.  Smooth PARAFAC Decomposition for Tensor Completion , 2015, IEEE Transactions on Signal Processing.

[29]  Hairong Qi,et al.  Endmember Extraction From Highly Mixed Data Using Minimum Volume Constrained Nonnegative Matrix Factorization , 2007, IEEE Transactions on Geoscience and Remote Sensing.

[30]  Nancy Bertin,et al.  Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis , 2009, Neural Computation.

[31]  Rafal Zdunek,et al.  Approximation of Feature Vectors in Nonnegative Matrix Factorization with Gaussian Radial Basis Functions , 2012, ICONIP.

[32]  Tuomas Virtanen,et al.  Monaural Sound Source Separation by Nonnegative Matrix Factorization With Temporal Continuity and Sparseness Criteria , 2007, IEEE Transactions on Audio, Speech, and Language Processing.