Missing Image Data Reconstruction Based on Least-Squares Approach with Randomized SVD

Abstract

In this paper, we introduce an efficient algorithm for reconstructing incomplete images based on optimal least-squares (LS) approximation. Generally, LS method requires a low-rank basis set that can represent the overall characteristic of an image, which can be obtained optimally via the singular value decomposition (SVD). This basis is called proper orthogonal decomposition (POD) basis. To significantly decrease the computational cost of SVD, this work employs a randomized singular value decomposition (rSVD) to compute the basis from the available image pixels. In this work, to preserve the 2-dimensional structure of the image, the test image is first subdivided into many 2-dimensional small patches. The complete patches are used to compute the POD basis for reconstructing corrupted patches. For each incomplete patch, the known pixels in the neighborhood around the missing components are used in the LS approximation together with the POD basis in the reconstruction process. The numerical tests compare the execution time used in computing this optimal low-rank basis by using rSVD and SVD, as well as demonstrate the accuracy of the resulting image reconstructions.