  We propose in this paper a novel two-stage Projection Correction Modeling
(PCM) framework for image reconstruction from (non-uniform) Fourier
measurements. PCM consists of a projection stage (P-stage) motivated by the
multi-scale Galerkin method and a correction stage (C-stage) with an edge
guided regularity fusing together the advantages of total variation (TV) and
total fractional variation (TFV). 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 1D
signals and 2D images have also been presented to demonstrate the superior
performance of the proposed two-stage method to other classical one-stage
methods.
