Adaptive structured recovery of compressive sensing via piecewise autoregressive modeling

In compressive sensing (CS) a challenge is to find a space in which the signal is sparse and hence recoverable faithfully and efficiently. Given the nonstationarity of many natural signals such as images, the sparse space varies in time/spatial domain. As such, CS recovery should be conducted in locally adaptive, signal-dependent spaces to counter the fact that the CS measurements are global and irrespective of signal structures. On the contrary most CS methods seek for a fixed set of bases (e.g., wavelets, DCT, and gradient spaces) for the entirety of a signal. To rectify this problem we propose a new framework for model-guided adaptive recovery of compressive sensing (MARX), and show how a piecewise autoregressive model can be integrated into the MARX framework to adapt to changing second order statistics of a signal in CS recovery. In addition, MARX offers a powerful mechanism of characterizing and exploiting structured sparsities of a signal, greatly restricting the CS solution space. A case study on CS-acquired images shows that the proposed MARX technique can increase the reconstruction quality by up to 8 dB over existing methods.