Image completion with approximate convex hull tensor decomposition

Abstract Many image completion methods are based on a low-rank approximation of the underlying image using matrix or tensor decomposition models. In this study, we assume that the image to be completed is represented by a multi-way array and can be approximated by a conical hull of subtensors in the observation space. If an observed tensor is near-separable along at least one mode, the extreme rays, represented by the selected subtensors, can be found by analyzing the corresponding convex hull. Following this assumption, we propose a geometric algorithm to address a low-rank image completion problem. The extreme rays are extracted with a segmented convex-hull algorithm that is suitable for performing noise-resistant non-negative tensor factorization. The coefficients of a conical combination of such rays are estimated using Douglas-Rachford splitting combined with the rank-two update least-squares algorithm. The proposed algorithm was applied to incomplete RGB images and a hyperspectral 3D array with a large number of randomly missing entries. Experiments confirm its good performance with respect to other well-known image completion methods.

[1]  Liqing Zhang,et al.  Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  David H Foster,et al.  Hyperspectral imaging in color vision research: tutorial. , 2019, Journal of the Optical Society of America. A, Optics, image science, and vision.

[3]  Rafal Zdunek,et al.  Image Completion with Nonnegative Matrix Factorization Under Separability Assumption , 2018, LVA/ICA.

[4]  Rafal Zdunek.  Initialization of Nonnegative Matrix Factorization with Vertices of Convex Polytope , 2012, ICAISC.

[5]  Yi Liu,et al.  Image completion using structure and texture GAN network , 2019, Neurocomputing.

[6]  Yu-Jin Zhang,et al.  Image inpainting via Weighted Sparse Non-negative Matrix Factorization , 2011, 2011 18th IEEE International Conference on Image Processing.

[7]  Jieping Ye,et al.  Tensor Completion for Estimating Missing Values in Visual Data , 2013, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Rafal Zdunek,et al.  Segmented convex-hull algorithms for near-separable NMF and NTF , 2019, Neurocomputing.

[9]  Yunyi Yan,et al.  Parallel Image Completion with Edge and Color Map , 2019, Applied Sciences.

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

[11]  J. Warga.  Minimizing Certain Convex Functions , 1963 .

[12]  Alexei A. Efros,et al.  Context Encoders: Feature Learning by Inpainting , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[13]  Christos Faloutsos,et al.  FastMap: a fast algorithm for indexing, data-mining and visualization of traditional and multimedia datasets , 1995, SIGMOD '95.

[14]  Antonio J. Plaza,et al.  Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches , 2012, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

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

[16]  Pan Zhou,et al.  Tensor Factorization for Low-Rank Tensor Completion , 2018, IEEE Transactions on Image Processing.

[17]  Joel A. Tropp,et al.  Factoring nonnegative matrices with linear programs , 2012, NIPS.

[18]  Sanjeev Arora,et al.  A Practical Algorithm for Topic Modeling with Provable Guarantees , 2012, ICML.

[19]  Suvrit Sra,et al.  Approximation Algorithms for Tensor Clustering , 2009, ALT.

[20]  Eli Shechtman,et al.  Image melding , 2012, ACM Trans. Graph..

[21]  Chong-Yung Chi,et al.  A Simplex Volume Maximization Framework for Hyperspectral Endmember Extraction , 2011, IEEE Transactions on Geoscience and Remote Sensing.

[22]  Patrick Pérez,et al.  Object removal by exemplar-based inpainting , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[23]  Venkatesh Saligrama,et al.  Topic Discovery through Data Dependent and Random Projections , 2013, ICML.

[24]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[25]  Jiaya Jia,et al.  Image completion with structure propagation , 2005, SIGGRAPH 2005.

[26]  Nicolas Gillis,et al.  Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation , 2013, SIAM J. Imaging Sci..

[27]  Vikas Sindhwani,et al.  Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization , 2012, ICML.

[28]  Dacheng Tao,et al.  Scalable completion of nonnegative matrices with the separable structure , 2016, AAAI 2016.

[29]  Wotao Yin,et al.  Parallel matrix factorization for low-rank tensor completion , 2013, ArXiv.

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

[31]  Ting-Zhu Huang,et al.  Laplace function based nonconvex surrogate for low-rank tensor completion , 2019, Signal Process. Image Commun..

[32]  Misha Elena Kilmer,et al.  Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[33]  Hongwei Liu,et al.  Rank-two residue iteration method for nonnegative matrix factorization , 2011, Neurocomputing.

[34]  P. Tseng.  Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization , 2001 .

[35]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[36]  Jianting Cao,et al.  High-order tensor completion via gradient-based optimization under tensor train format , 2019, Signal Process. Image Commun..

[37]  Byonghyo Shim,et al.  Low-Rank Matrix Completion: A Contemporary Survey , 2019, IEEE Access.

[38]  Hiroshi Ishikawa,et al.  Globally and locally consistent image completion , 2017, ACM Trans. Graph..