Recovery in compressive sensing: a review

Abstract A very recent methodology of data sampling with a sub-Nyquist rate is compressive sensing (CS). CS theory is established based on the higher sparsity of the data containing information. It uses this fact for recovery of the data sampled with much fewer samples than required by the Nyquist sampling theorem. CS addresses two main questions: (i) how to construct a compressive sensing matrix which satisfies the uniqueness required for later recovery, (ii) how to recover the original much longer signal vector than the current available compressively sensed one. This chapter reviews mainly the related theory and solution for the latter question of the signal/image recovery from the compressively sensed signal vectors. It presents a review of the pre-requirements in compressive sensing for the recovery process and summarizes the related theorems. The recovery of the signals after being compressively sensed with and without noise is studied and the main methodologies used are explained.