CURE: Curvature Regularization for Missing Data Recovery

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 CUrvature 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.