快速NMF盲源分离算法

将秩一非负矩阵分解应用于盲源分离问题，把基于欧式距离的目标函数转化成二次函数的形式；施加稀疏性约束和正交性约束保证信号可分离性；利用二次函数的性质分别推得混合矩阵和源信号的迭代公式，从而得到一种基于秩一分解的快速NMF盲源分离算法（NMF-R1)。分析得到一次迭代更新NMF-R1算法比传统NMF盲源分离算法（NMF-BM)所需乘法次数少约30%，NMF-R1算法无矩阵求逆运算，NMF-BM算法还需2次矩阵求逆运算。图像信号的超定和欠定盲源分离仿真结果表明，NMF-R1算法都能分离出源信号， NMF-BM算法只能分离超定混合信号；NMF-R1算法与NMF-BM算法比，分离性能好、收敛速度快。 

Fast NMF Blind Source Separation Algorithm

By using non-negative matrix factorization with rank one into blind source separation, the objective function of the blind source separation based on the Euclidean distance is transformed into the form of a quadratic function. The constraints of sparisity and orthogonality are imposed on the blind source separation algorithm to guarantee its separability. The mixing matrix and source signals iterative formulas are derived by utilizing the property of the quadratic function, and then a fast NMF blind source separation algorithm based on rank one (NMF-R1) is obtained. The number of multiplications required for each update of NMF-R1 blind source separation algorithm is less about 30% than that of NMF-BM, besides NMF-R1 doesn't need computation of matrix inversion, but NMF-BM needs computation of two matrix inversion. The simulation results of blind source separation for image signals’ overdetermined and underdetermined mixing show that all of source signals can be separated by NMF-R1 algorithm, but only overdetermined mixed signals can be separated by NMF-BM algorithm, and NMF-R1 algorithm has the better separated performance and the faster convergence rate compared with NMF-BM algorithm.