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A sparse optimization problem with hybrid \(L_2{\text {-}}L_p\) regularization for application of magnetic resonance brain images

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Abstract

Regularization techniques have been proved useful in an enormous variety of sparse optimization problem. In this paper, we introduce a new formulation of regularization with a hybrid \(L_2{\text {-}}L_p~(0< p <1)\) norm. Then we model a sparse optimization problem with hybrid \(L_2{\text {-}}L_p\) regularization. For solving the problem, we derive its local optimality conditions and develop a hybrid \(L_2{\text {-}}L_p\) algorithm. Moreover, we analyze the convergence of the algorithm. Finally, we apply our model and algorithm to the image recovery and deblurring for the magnetic resonance images of the brain. The numerical tests show that the effects of the recovery and deblurred images, respectively, obtained by the hybrid \(L_2{\text {-}}L_p\) algorithm with the suitable parameter p are better than those of other three algorithms, compared with the values of the signal-to-noise ratio.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Xuerui Gao & Yanqin Bai

  2. School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, China

    Qian Li

Authors
  1. Xuerui Gao
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  2. Yanqin Bai
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  3. Qian Li
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Correspondence to Yanqin Bai.

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This work was supported by the National Natural Science Foundations of China (Grant Number 11771275).

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Gao, X., Bai, Y. & Li, Q. A sparse optimization problem with hybrid \(L_2{\text {-}}L_p\) regularization for application of magnetic resonance brain images. J Comb Optim 42, 760–784 (2021). https://doi.org/10.1007/s10878-019-00479-x

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