{Application of the optimized 1-bit tensor completion method in the recovery of noisy digital images

Document Type : Original Paper


1 Faculty of Defense and Engineering, Imam Hossein University, Tehran, Iran

2 Department of Mathematics and Statistics, Imam Hossein Comprehensive University, Tehran, Iran,


Higher-order tensor structured data appear in many imaging scenarios, including hyperspectral imaging and colorful video. The recovery of a tensor from an incomplete set of its entries, known as tensor completion (TC), is significant in applications like compression. Moreover, in many illustrations, observations are not only incomplete but also highly quantized. Quantization is a critical step for high dimensional data transmission and storage in order to reduce storage requirements and power consumption, especially for energy-limited systems. In this paper, we propose a novel approach for the recovery of low-rank tensors from a small number of binary (1-bit) measurements. The proposed method called $1- bit$ Tensor Completion relies on the application of 1-bit matrix completion over different matricizations of the underlying tensor. Experimental results on hyperspectral images confirm that directly operating with the binary measurements, rather than treating them as real values, results in lower recovery error. Here a given third-order tensor with binary arrays is recovered. In practice, we open the tensor as a 3-matrix and apply the quantified tensor completion algorithm to all models of the matrix tensor. The data space here is distorted satellite spectral images for the purpose of image recovery.


Main Subjects

[1] Aidini A., Tsagkatakis G. and Tsagkatakis P. (2018). 1-Bit Tensor Completion, Image Processing: Algorithms and Systems, 6, 261-266.
[2] Bach F.R. (2008). Consistency of trace norm minimization, Journal of Machine Learning Research, 5, 1019-1048.
[3] Cande‘s E. J. and Recht B. (2009). Exact matrix completion via convex optimization, Foundations of Computational mathematics 9, 717-730.
[4] Gandy S., Recht B. and Yamada, I. (2011). Tensor completion and n-low-rank tensor recovery via convex optimizatio, Inverse Problems, 27, 1-19.
[5] Giannopoulos M., Savaki S., Tsagkatakis G. and Tsagkatakis P. (2009). Application of Tensor and Matrix Completion on Environmental Sensing Data, Springer.
[6] Li B., Zhang X., Li X. and Lu H. (2019). Tensor Completion From One-Bit Observations, IEEE Transactions on Image Processing, 28(1), 170-180.
[7] Liu J., Musialski P., Wonka P. and Ye J. (2013). Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(2), 208–220.
[8] Rovi A. and J. Thierselder (2010). Analysis of 2-Tensors, MAI mathematics: MS.C. thesis, Linkopings University.
[9] Song Q., Ge H., Hu J. and Hu X. (2019). Tensor Completion Algorithms in Big Data Analytics, ACM Transactions on Knowledge Discovery from Data (TKDD), 6(13), 1-48.
[10] Savvaki S., Tsagkatakis G., Panousopoulou A. and Tsagkatakis P. (2017). Matrix and Tensor Completion on a Human Activity Recognition Framework, IEEE journal of biomedical and health informatics, 21(6), 1554-1561.
[11] Zhao Q., Zhang Li. and Cichocki A. (2015). Bayesian CP-Factorization of incomplete Tensors with Automatic Rank Determination, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(37), 1751-1763.
[12] Xu Y., Hao R., Yin W. and Su Z. (2015). Parallel matrix factorization for low-rank tensor completion, American Institute of Mathematical Science, 9(2), 601-624.
[13] Wen Z., Yin W., and Zhang Y. (2012). Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Mathematical Programming Computation, Springer, 4, 333-361.
Volume 11, Issue 4 - Serial Number 4
December 2021
Pages 686-698
  • Receive Date: 25 July 1400
  • Revise Date: 14 August 1400
  • Accept Date: 28 August 1400
  • First Publish Date: 01 October 1400