An Efficient Method for Robust Projection Matrix Design
This work addresses a bottleneck in Compressive Sensing for image processing by enabling efficient matrix design without training data, which is incremental but practical for broader applications.
The paper tackles the problem of designing robust projection matrices for Compressive Sensing systems when signals are not exactly sparse, by introducing a penalty function that eliminates the need for training data and sparse representation error matrices, achieving efficiency and effectiveness comparable to state-of-the-art methods.
Our objective is to efficiently design a robust projection matrix $Φ$ 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.