Relationship between the robust statistics theory and sparse compressive sensed signals reconstruction

An analysis of robust estimation theory in the light of sparse signals reconstruction is considered. This approach is motivated by compressive sensing (CS) concept which aims to recover a complete signal from its randomly chosen, small set of samples. In order to recover missing samples, the authors define a new reconstruction algorithm. It is based on the property that the sum of generalised deviations of estimation errors, obtained from robust transform formulations, has different behaviour at signal and non-signal frequencies. Additionally, this algorithm establishes a connection between the robust estimation theory and CS. The effectiveness of the proposed approach is demonstrated on examples.