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Generalized nonconvex regularization for tensor RPCA and its applications in visual inpainting

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Abstract

As a demonstrated and foremost approach of extracting the key features from corrupted observations, tensor robust principal component analysis has been considered in various fields of data processing related to tensors. It is usually modeled as a low-rank and sparse tensor decomposition problem and can be solved by minimizing a simple convex program. However, convex optimization methods often fail to deeply explore the rank and sparsity of tensors, which leads to suboptimality. Based on tensor singular value decomposition, in this work, we introduce generalized nonconvex regularizers accommodating most popular nonconvex (and possibly nonsmooth) surrogate functions to be used as effective approximations of the tensor rank function and \(\ell _{0}\)-norm. The established unified frame equips universality for a large group of nonconvex surrogate functions. Moreover, we consider tube-wise sparse noise in addition to entry-wise sparse noise, which provides a better way of handling structured corrupted observations arising from practical issues. We further develop an efficient algorithm with convergence guarantees to implement generalized nonconvex optimization based upon the alternating direction method of multipliers. The satisfactory performance results of the proposed method are verified by simulations and visual inpainting applications.

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Availability of data and materials

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Notes

  1. Originated from http://www.eecs.qmul.ac.uk/~phao/IP/Images/

  2. RelErrorL is usually much higher than that of the sparse component; thus, we only use the relative error of low-rank component recovery to measure the performance of algorithms from now on.

  3. https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/grouping/segbench/

  4. http://vision.ucsd.edu/~iskwak/ExtYaleDatabase/ExtYaleB.html

  5. http://trace.eas.asu.edu/yuv/

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12101512), the China Postdoctoral Science Foundation (Grant No. 2021M692681), the Natural Science Foundation of Chongqing, China (Grant No. cstc2021jcyj-bshX0155) and the Fundamental Research Funds for the Central Universities (Grant No. SWU120078).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Feng Zhang, Wenjin Qin & Jianjun Wang

  2. School of Mathematics and Statistics, Xi’an JiaoTong University, Xi’an, 710049, China

    Hailin Wang

  3. School of Mathematical Sciences/Research Center for Image and Vision Computing, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Xile Zhao

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  1. Feng Zhang
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Correspondence to Jianjun Wang.

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Zhang, F., Wang, H., Qin, W. et al. Generalized nonconvex regularization for tensor RPCA and its applications in visual inpainting. Appl Intell 53, 23124–23146 (2023). https://doi.org/10.1007/s10489-023-04744-9

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