Method Of Optimization And Construction On Measurement Matrix For Compressed Sensing

Compressed Sensing (CS) is a novel signal sampling theory for sparse or compressible signals. It implements signal sampling and data compression at the same time. So its sampling rate is far lower than Nyquist sampling rate. Measurement matrix plays an important role in signal sampling and signal reconstruction and it is the core part in compressed sensing. Therefore, the research on measurement matrix with good property both in theory and in practice is of great importance. In this paper, based on the research on compressed sensing and measurement matrices, we focus on the work as follows:First we compare the common measurement matrices and analyst their merits and demerits respectively. Combining the constraint conditions which measurement matrices are needed to satisfy, we deeply analyst the angle and method on improving the property of measurement matrices.(1) In order to improve reconstruction effect and reduce the number of measurements, we can optimize the common measurement matrices by reducing the mutual coherence between the measurement matrix and sparse transformed matrix. Gram matrix is constructed based on the product of the measurement matrix and sparse transformed matrix and we define the mutual coherent coefficient based on the off-diagonal elements of Gram matrix. So the method of optimizing a measurement matrix is equally to reduce the off-diagonal elements of the Gram matrix through a related algorithm. There are mainly two methods:iteration optimization method and efficient projection optimization method.(2) In order to design the measurement matrices which are easily for hardware implementation and with low computational complexity, we can construct sparse and structured matrices based on block. These matrices can be constructed by diagonally permuting sub-matrices with some structure. So these measurement matrices are sparse and structured.Aiming at the method of reducing the mutual coherence, iteration optimization methods have some disadvantages with many iteration times and high computation complexity. In this paper, we propose optimization method for measurement matrix based on eigenvalue decomposition. The main aim is reducing the new global mutual coherence. First the new global mutual coherent coefficient was defined based on all off-diagonal elements of the Gram matrix. After deriving the relationship between the global mutual coherent coefficient and the eigenvalues of the Gram matrix, we proposed an optimization model and the method of resolving the minimum of global mutual coherent coefficient. With this optimization method, the number of iteration times is much less and reconstruction effect is better.Aiming at the method of constructing sparse and deterministic measurement matrices by block and structure, the hardware performance of SBH needs to be improved further. In this paper, we propose the method of constructing block and sparse measurement matrices based on orthogonal vectors. These matrices can improve sensing efficiency and reduce computational complexity and can be easily applied to hardware implementation due to lower memory space and simple nonzero elements. The matrices constructed by this method have the advantage of diversified compression ratio. The reconstruction effect when we use the selected matrix is better than the effect when we use SBH matrix in image recovery. And the structure of this matrix is simpler and sparser than SBH matrix.