二维FIR滤波器约束最小二乘设计的最大分划松弛ADMM算法

约束二维有限脉冲响应（Finite Impulse Response，FIR）滤波器，现有设计算法计算复杂度高.针对二维FIR滤波器的约束最小二乘设计，本文应用交替方向乘子法（Alternating Direction Method of Multipliers，ADMM），研究其并行优化方法.通过模型的最大分划，并采用一种松弛技术，提出一个具有高度并行结构的最大分划松弛ADMM算法，分析了算法的计算复杂度，讨论了算法的收敛性，并给出了算法的参数设置方法.实验表明，最大分划松弛ADMM比非松弛的最大分划ADMM收敛快很多；与现有算法相比，提高了计算效率.GPU加速实验中获得的大加速比，表明了所提算法的高度并行性和可扩展性，在图像处理、计算机视觉、模式识别及机器学习等领域有广阔的应用前景.

A Maximally Split and Relaxed ADMM for Constrained Least-Squares Design of Two-Dimensional FIR Filters

For constrained two-dimensional (2-D) finite impulse response (FIR) filters,the computational complexity of the existing design algorithms is very high.Based on the alternating direction method of multipliers (ADMM),the parallel optimization of constrained least-squares (CLS) 2-D FIR filters was studied.By maximally splitting the problem into univariate subproblems and utilizing a relaxation technique,a maximally split and relaxed ADMM with a highly parallel computing architecture was proposed.The computational complexity and convergence of the algorithm were analyzed,and a practical scheme for selection of the algorithm parameters was provided.Experimental results show that the proposed maximally split and relaxed ADMM converges much faster than the maximally split unrelaxed ADMM.Compared with existing algorithms,the computational efficiency of the proposed algorithm is improved.The large acceleration ratios obtained by GPU demonstrate the high parallelism and scalability of the proposed algorithm,which is very valuable for applications of the algorithm in image processing,computer vision,pattern recognition and machine learning.