  Our objective is to efficiently design a robust projection matrix $\Phi$ for
the Compressive Sensing (CS) systems when applied to the signals that are not
exactly sparse. The optimal projection matrix is obtained by mainly minimizing
the average coherence of the equivalent dictionary. In order to drop the
requirement of the sparse representation error (SRE) for a set of training data
as in [15] [16], we introduce a novel penalty function independent of a
particular SRE matrix. Without requiring of training data, we can efficiently
design the robust projection matrix and apply it for most of CS systems, like a
CS system for image processing with a conventional wavelet dictionary in which
the SRE matrix is generally not available. Simulation results demonstrate the
efficiency and effectiveness of the proposed approach compared with the
state-of-the-art methods. In addition, we experimentally demonstrate with
natural images that under similar compression rate, a CS system with a learned
dictionary in high dimensions outperforms the one in low dimensions in terms of
reconstruction accuracy. This together with the fact that our proposed method
can efficiently work in high dimension suggests that a CS system can be
potentially implemented beyond the small patches in sparsity-based image
processing.
