Sparsity-Driven Impulsive Noise Removal: A Discrete Hermite Transform Case Study

This paper revisits a sparsity-assisted approach for the detection and removal of impulsive noise. Originally inspired by a sparse signal reconstruction methodology, which minimizes concentration measures through the variations of values of missing samples, the considered approach can perform a blind detection of impulsive disturbances, while the reconstruction is supported by the advances in the compressive sensing theory. The assumption is that the signal is sparse in a particular transform domain, which is the standard condition guaranteeing the reconstruction in the compressive sensing context. As a case study, an orthogonal form of the discrete Hermite transform is considered. Numerical results support the presented theory, indicating the efficiency of the method even in the challenging case of disturbances within the range of the signal.