The proximal alternating iterative hard thresholding method for minimization, with complexity

详细信息    查看全文

• 作者：Fan Yang^a ; ^{kiraofsin@yahoo.co.jp" class="auth_mail" title="E-mail the corresponding author} ; Yi Shen^a ; ^{yshen@zstu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Zhi Song Liu^b ; ^{liuzhisong@zjou.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Sparse approximation ; Alternating minimization ; Hard thresholding ; Tight wavelet frame ; Kurdyka&ndash ; Łojasiewicz property
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：311
• 期：Complete
• 页码：115-129
• 全文大小：964 K

文摘

Since digital images are usually sparse in the wavelet frame domain, some nonconvex minimization models based on wavelet frame have been proposed and sparse approximations have been widely used in image restoration in recent years. Among them, the proximal alternating iterative hard thresholding method is proposed in this paper to solve the nonconvex model based on wavelet frame. Through combining the proposed algorithm with the iterative hard thresholding algorithm which is well studied in compressed sensing theory, this paper proves that the complexity of the proposed method is class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303326&_mathId=si7.gif&_user=111111111&_pii=S0377042716303326&_rdoc=1&_issn=03770427&md5=f3d7fa815146e13f25f3b957a6de3aec">class="imgLazyJSB inlineImage" height="18" width="63" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716303326-si7.gif">class="mathContainer hidden">class="mathCode">$\mathcal{O}(1/\sqrt{k})$. On the other hand, a more general nonconvex–nonsmooth model is adopted and the pseudo proximal alternating linearized minimization method is developed to solve the above problem. With the Kurdyka–Łojasiewicz (KL) property, it is proved that the sequence generated by the proposed algorithm converges to some critical points of the corresponding model. Finally, the proposed method is applied to restore the blurred noisy gray images. As the numerical results reveal, the performance of the proposed method is comparable or better than some well-known convex image restoration methods.