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Low‐rank tensor completion with sparse regularization in a transformed domain
Author(s) -
Wang PingPing,
Li Liang,
Cheng GuangHui
Publication year - 2021
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2387
Subject(s) - tensor (intrinsic definition) , mathematics , matrix norm , rank (graph theory) , singular value decomposition , cartesian tensor , mathematical optimization , tucker decomposition , regularization (linguistics) , algorithm , artificial intelligence , tensor density , computer science , tensor field , exact solutions in general relativity , tensor decomposition , mathematical analysis , pure mathematics , combinatorics , eigenvalues and eigenvectors , physics , quantum mechanics
Tensor completion is a challenging problem with various applications, especially in recovering incomplete visual data. Considering a color image or gray video as a three‐dimensional tensor, many related models based on the low‐rank prior of the tensor have been proposed. However, the low‐rank prior may not be enough to recover the original tensor from the observed incomplete tensor. In this paper, we propose a tensor completion method to recover color images and gray videos by exploiting both the low‐rank and sparse prior of the observed tensor. Specifically, the tensor completion task can be formulated as a low‐rank minimization problem with a sparse regularizer. The low‐rank property is depicted by the tensor truncated nuclear norm based on tensor singular value decomposition which is a better approximation of tensor tubal rank than tensor nuclear norm. While the sparse regularizer is imposed by a ℓ 1 ‐norm in a discrete cosine transformation domain, which can better employ the local sparse property of the incomplete data. To solve the optimization problem, we employ an alternating direction method of multipliers in which we only need to solve several subproblems which have closed‐form solutions. Substantial experiments on real‐world images and videos show that the proposed method has better performances than the existing state‐of‐the‐art methods.