An Orthogonal Method for Measurement Matrix Optimization

Abstract

Compressive sensing theory states that signals can be sampled at a much smaller rate than that required by the Nyquist sampling theorem, because the sampling of a signal in the former is performed as a relatively small number of its linear measurements. Thus, the design of a measurement matrix is important in compressive sensing framework. A random measurement matrix optimization method is proposed in this study based on the incoherence principle of compressive sensing, which requires the mutual coherence of information operator to be small. The columns with mutual coherence are orthogonalized iteratively to decrease the mutual coherence of the information operator. The orthogonalization is realized by replacing the columns with the orthogonal matrix $$\mathbf {Q}$$Q of their QR factorization. An information operator with smaller mutual coherence is acquired after the optimization, leading to an improved measurement matrix in terms of its relationship with the information operator. Results of several experiments show that the improved measurement matrix can reduce its mutual coherence with dictionaries compared with the random measurement matrix. The signal reconstruction error also decreases when the optimized measurement matrix is utilized.