We propose in this paper a novel two-stage projection correction modeling (PCM) framework for image reconstruction from (nonuniform) Fourier measurements. PCM consists of a projection stage (P-stage) motivated by the multiscale Galerkin method and a correction stage (C-stage) with an edge guided regularity fusing together the advantages of total variation and total fractional variation. The P-stage allows for continuous modeling of the underlying image of interest. The given measurements are projected onto a space in which the image is well represented. We then enhance the reconstruction result at the C-stage that minimizes an energy functional consisting of a fidelity in the transformed domain and a novel edge guided regularity. We further develop efficient proximal algorithms to solve the corresponding optimization problem. Various numerical results in both one-dimensional signals and two-dimensional images have also been presented to demonstrate the superior performance of the proposed two-stage method to other classical one-stage methods.

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error, and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the ℓ1norm can be applied to improve the learning rate estimate of ℓ1-regularization in machine learning.