L1 norm of high frequency components as a regularization term for compressed sensing reconstruction of image signals

This paper proposes a formulation of estimating a sparse image signal where the sparsity only occurs on the high frequency components. The formulation is for compressed sensing reconstruction when a compressed measurement signal is contaminated by impulsive noise. The approach is based on two prior well-known assumptions. First, adding an L₁-norm penalty on a compressible signal to the cost objective of the estimation problem will promote a sparsity in the signal. Secondly, an image signal should be sparse in high frequency domain, while it is dense in low frequency domain where most of its important information lies. Our approach is therefore to impose an L₁-norm penalty only on the signal components that need to be sparse while we allow the important low frequency part to be dense. The obtained formulation is a convex optimization which can be effectively solved by many available algorithms. The experiment demonstrated a superior impulsive noise tolerance of the proposed penalty function to a conventional scheme where all signal components are penalized by L₁-norm. At the noise probability of less than 0.1, the reconstruction was better both quantitatively and qualitatively.