Recovery of piecewise finite-dimensional continuous signals by exploiting sparsity

Many inverse problems in science and engineering are formulated as recovery of piecewise finite-dimensional continuous (PFC) signals. Although the higher-order total variation (HTV) is known to be particularly effective for the sparsity-aware recovery of piecewise polynomials, it remains unclear so far whether the HTV can be extended to other signal models. In this paper, we present a convex regularizer which becomes a generalization of the HTV for the PFC signals. We first design a linear transformation which induces a certain group sparsity of samples of the PFC signals. This linear transformation is designed based on the fact that most of local samples can be interpolated by a fixed linear combination of known basis. Moreover, we provide theoretical evidence that the linear transformed samples have the group sparsity. Then, the proposed regularizer is designed to promote the group sparsity by using the ℓ_1,2 norm. A numerical experiment on recovery of piecewise sinusoidal signals shows the effectiveness of the proposed regularization.