Two Stage Continuous Domain Regularization for Piecewise Constant Image Restoration

The finite-rate-of-innovation (FRI) framework which corresponds a signal/image to a structured low-rank matrix is emerging as an alternative to the traditional sparse regularization. This is because such an off-the-grid approach is able to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we propose a two-stage off-the-grid regularization model for the image restoration. Given that the discontinuities/edges of the image lie in the zero level set of a band-limited periodic function, we can derive that the Fourier samples of the gradient of the image satisfy an annihilation relation, resulting in a low-rank two-fold Hankel matrix. In addition, since the singular value decomposition of a low-rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients, our approach consists of the following two stages. The first stage learns the tight wavelet frame system from a given measurement, and the second stage restores the image via the analysis approach based sparse regularization. The numerical results are presented to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low-rank matrix approaches.