Making sense of randomness: an approach for fast recovery of compressively sensed signals

In compressed sensing (CS) framework, a signal is sampled below Nyquist rate, and the acquired compressed samples are generally random in nature. However, for efficient estimation of the actual signal, the sensing matrix must preserve the relative distances among the acquired compressed samples. Provided this condition is fulfilled, we show that CS samples will preserve the envelope of the actual signal even at different compression ratios. Exploiting this envelope preserving property of CS samples, we propose a new fast dictionary learning (DL) algorithm which is able to extract prototype signals from compressive samples for efficient sparse representation and recovery of signals. These prototype signals are orthogonal intrinsic mode functions (IMFs) extracted using empirical mode decomposition (EMD), which is one of the popular methods to capture the envelope of a signal. The extracted IMFs are used to build the dictionary without even comprehending the original signal or the sensing matrix. Moreover, one can build the dictionary on-line as new CS samples are available. In particularly, to recover first $L$ signals ($\in\mathbb{R}^n$) at the decoder, one can build the dictionary in just $\mathcal{O}(nL\log n)$ operations, that is far less as compared to existing approaches. The efficiency of the proposed approach is demonstrated experimentally for recovery of speech signals.