Joint optimization of measurement matrix and sparse dictionary in compressive sensing

Compressive sensing, a novel signal acquisition method, is a joint sensing and compression which requires small number of measurement to reconstruct signal as compared to the conventional method, Shannon-Nyquist sampling theory, which requires at least twice the signal bandwidth. Compressive sensing exploits the sparse representations of signals. A signal is considered sparse if it can be represented by few non-zero coefficients using a suitable basis or dictionary. In order to achieve good reconstruction performance from a few number of measurement, compressive sensing requires a small mutual coherence between a measurement matrix and the basis or dictionary. The commonly used measurement matrix is random matrix because it has a small mutual coherence with many basis like Fourier and Wavelet. Note that, random matrix can further be optimized to achieve even smaller mutual coherence. This paper addresses the joint optimization between measurement matrix and sparse dictionary to minimize the average mutual coherence between them. Combination of KSVD and Equiangular Tight Frame (ETF) methods are used to perform this joint optimization. The joint optimized measurement matrix was used for image encoding to provide a compressive measurement. The simulation results showed that the joint optimization increases the PSNR of reconstructed image up to 77% and 15% compared to the random matrix and optimized measurement matrix only, respectively.