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CURE: Curvature Regularization for Missing Data Recovery
SIAM Journal on Imaging Sciences ( IF 2.1 ) Pub Date : 2020-12-10 , DOI: 10.1137/19m1261845
Bin Dong , Haocheng Ju , Yiping Lu , Zuoqiang Shi

SIAM Journal on Imaging Sciences, Volume 13, Issue 4, Page 2169-2188, January 2020.
Missing data recovery is an important and yet challenging problem in imaging and data science. Successful models often adopt certain carefully chosen regularization. Recently, the low dimensional manifold model (LDMM) was introduced by [S. Osher, Z. Shi, and W. Zhu, Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] and shown to be effective in image inpainting. The authors of [Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] observed that enforcing low dimensionality on the image patch manifold serves as a good image regularizer. In this paper, we observe that having only the low dimensional manifold regularization is not enough sometimes, and we need smoothness as well. For that, we introduce a new regularization by combining the low dimensional manifold regularization with a higher order \bf CUrvature \bf REgularization, which we call new regularization CURE for short. The key step of CURE is to solve a biharmonic equation on a manifold. We further introduce a weighted version of CURE, called WeCURE, in a similar manner as the weighted nonlocal Laplacian (WNLL) method [Z. Shi, S. Osher, and W. Zhu, Weighted nonlocal Laplacian on interpolation from sparse data, J. Sci. Comput., 73 (2017), pp. 1164--1177]. Numerical experiments for image inpainting and semisupervised learning show that the proposed CURE and WeCURE significantly outperform LDMM and WNLL, respectively.


中文翻译:

CURE:丢失数据恢复的曲率正则化

SIAM影像科学杂志,第13卷,第4期,第2169-2188页,2020年1月。
丢失数据恢复是成像和数据科学中的一个重要且具有挑战性的问题。成功的模型通常采用某些经过精心选择的正则化。最近,低维流形模型(LDMM)由[S. Osher,Z。Shi和W. Zhu,《图像处理的低维流形模型》,技术报告,凸轮报告16-04,加州大学洛杉矶分校,洛杉矶,加利福尼亚,2016年,被证明在图像修复中很有效。[用于图像处理的低维歧管模型,技术报告,凸轮报告16-04,加州大学洛杉矶分校,加利福尼亚,2016年]的作者观察到,在图像斑块歧管上强制使用低维可作为良好的图像正则化工具。在本文中,我们观察到有时仅具有低维流形正则化是不够的,并且还需要平滑度。为了那个原因,我们通过将低维流形正则化与高阶\ bf曲线\ bf规整化相结合来引入新的正则化,我们简称为新正则化CURE。CURE的关键步骤是在流形上求解双调和方程。我们以类似于加权非局部拉普拉斯算子(WNLL)方法的方式引入称为CURE的加权版本的CURE [Z. Shi,S. Osher和W. Zhu,《基于稀疏数据的插值的加权非局部拉普拉斯算子》,J。Sci。计算(73)(2017),1164--1177]。图像修复和半监督学习的数值实验表明,所提出的CURE和WeCURE分别明显优于LDMM和WNLL。我们简称为新的正则化CURE。CURE的关键步骤是在流形上求解双调和方程。我们以类似于加权非局部拉普拉斯算子(WNLL)方法的方式引入称为CURE的加权版本的CURE [Z. Shi,S. Osher和W. Zhu,《基于稀疏数据的插值的加权非局部拉普拉斯算子》,J。Sci。计算(73)(2017),1164--1177]。图像修复和半监督学习的数值实验表明,所提出的CURE和WeCURE分别明显优于LDMM和WNLL。我们简称为新的正则化CURE。CURE的关键步骤是在流形上求解双调和方程。我们以类似于加权非局部拉普拉斯算子(WNLL)方法的方式引入称为CURE的加权版本的CURE [Z. Shi,S. Osher和W. Zhu,《基于稀疏数据的插值的加权非局部拉普拉斯算子》,J。Sci。计算(73)(2017),1164--1177]。图像修复和半监督学习的数值实验表明,所提出的CURE和WeCURE分别明显优于LDMM和WNLL。科学 计算(73)(2017),1164--1177]。图像修复和半监督学习的数值实验表明,所提出的CURE和WeCURE分别明显优于LDMM和WNLL。科学 计算(73)(2017),1164--1177]。图像修复和半监督学习的数值实验表明,所提出的CURE和WeCURE分别明显优于LDMM和WNLL。
更新日期:2020-12-11
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