工程科学学报，第39卷.第8期：1254-1260.2017年8月 Chinese Journal of Engineering,Vol.39,No.8:1254-1260,August 2017 DOI:10.13374/j.issn2095-9389.2017.08.016;http://journals.ustb.edu.cn MMSE准则下基于玻尔兹曼机的快速重构算法 刘玲君2)四，谢中华)，冯久超”，杨萃12) 1)华南理工大学电子与信息学院，广州5106412)国家移动超声探测工程技术研究中心，广州510641 区通信作者，E-mail:liuO@gmail.com 摘要全连接的玻尔兹曼机模型可全面描述稀疏系数间统计依赖关系，但时间复杂度较高.为了提高基于玻尔兹曼机的 贝叶斯匹配追踪算法(BM-BMP)的重构速度和质量，本文提出一种改进算法.第一，将BM-BMP算法的最大后验概率(MAP) 估计评估值分解为上一次迭代的评估值与增量，使得每次迭代仅需计算增量，极大缩短了计算耗时.第二，利用显著最大后 验概率估计值平均的方式，有效近似最小均方误差(MMSE)估计，获得了更小的重构误差.实验结果表明，本文算法比 BM-BMP算法的运行时间平均缩短了73.66%，峰值信噪比(PSNR)值平均提高了0.57dB. 关键词稀疏信号重构；快速贝叶斯匹配追踪；玻尔兹曼机；最小均方误差 分类号TP391 Fast recovery algorithm based on Boltzmann machine and MMSE criterion LIU Ling-jun,XIE Zhong-hua),FENG Jiu-chao,YANG Cui) 1)School of Electronic and Information Engineering,South China University of Technology,Guangzhou 510641,China 2)National Engineering Technology Research Center for Mobile Ultrasonic Detection,Guangzhou 510641,China Corresponding author,E-mail:ljliu@gmail.com ABSTRACT Fully connected Boltzmann machine models can be used to provide a comprehensive description of statistical dependen- cies between sparse coefficients but with high time complexity.To improve the speed and quality of the Boltzmann machine-Bayesian matching pursuit (BM-BMP)method,an improved algorithm was proposed.First,the maximum a posteriori (MAP)estimation of the BM-BMP algorithm is decomposed into its value at the last iteration and an increment;thus,it only needs to calculate the increment in each iteration,which greatly reduces the computational time.Second,by calculating the mean of the significant MAP estimations,an effective approximation is obtained for the minimum mean square error(MMSE)estimation and a smaller reconstruction error is a- chieved.Compared with the BM-BMP,this method reduces the running time on average by 73.66%while improving the peak signal to noise ratio PSNR)by 0.57 dB. KEY WORDS sparse signal reconstruction;fast Bayesian matching pursuit;Boltzmann machine;minimum mean square error MMSE) 稀疏信号重构被广泛应用于信号和图像处理中，y的长度)，其中x是m×1的稀疏系数(m是矢量x的 例如：去噪、盲源分离、波达方向估计、压缩和采样、图长度)，A是n×m的冗余字典（因为m>n,所以A是 像复原等-).假设n×1维信号y=Ar+e(n是矢量冗余字典)，e是高斯白噪声，其均值为零方差为2. 收稿日期：2016-09-12 基金项目：国家自然科学基金资助项目(61327005,61302120)：广东省科技计划资助项目(2017A020214011):中央高校基本科研业务费资助 项目(2017M039)工程科学学报,第 39 卷,第 8 期:1254鄄鄄1260,2017 年 8 月 Chinese Journal of Engineering, Vol. 39, No. 8: 1254鄄鄄1260, August 2017 DOI: 10. 13374 / j. issn2095鄄鄄9389. 2017. 08. 016; http: / / journals. ustb. edu. cn MMSE 准则下基于玻尔兹曼机的快速重构算法 刘玲君1,2)苣 , 谢中华1) , 冯久超1) , 杨 萃1,2) 1) 华南理工大学电子与信息学院, 广州 510641 2) 国家移动超声探测工程技术研究中心, 广州 510641 苣 通信作者, E鄄mail: ljliu0@ gmail. com 摘 要 全连接的玻尔兹曼机模型可全面描述稀疏系数间统计依赖关系,但时间复杂度较高. 为了提高基于玻尔兹曼机的 贝叶斯匹配追踪算法(BM鄄BMP)的重构速度和质量,本文提出一种改进算法. 第一,将 BM鄄BMP 算法的最大后验概率(MAP) 估计评估值分解为上一次迭代的评估值与增量,使得每次迭代仅需计算增量,极大缩短了计算耗时. 第二,利用显著最大后 验概率估计值平均的方式,有效近似最小均方误差(MMSE) 估计,获得了更小的重构误差. 实验结果表明,本文算法比 BM鄄BMP算法的运行时间平均缩短了 73郾 66% ,峰值信噪比(PSNR)值平均提高了 0郾 57 dB. 关键词 稀疏信号重构; 快速贝叶斯匹配追踪; 玻尔兹曼机; 最小均方误差 分类号 TP391 Fast recovery algorithm based on Boltzmann machine and MMSE criterion LIU Ling鄄jun 1,2)苣 , XIE Zhong鄄hua 1) , FENG Jiu鄄chao 1) , YANG Cui 1,2) 1) School of Electronic and Information Engineering, South China University of Technology, Guangzhou 510641, China 2) National Engineering Technology Research Center for Mobile Ultrasonic Detection, Guangzhou 510641, China 苣 Corresponding author, E鄄mail: ljliu0@ gmail. com ABSTRACT Fully connected Boltzmann machine models can be used to provide a comprehensive description of statistical dependen鄄 cies between sparse coefficients but with high time complexity. To improve the speed and quality of the Boltzmann machine鄄Bayesian matching pursuit (BM鄄BMP) method, an improved algorithm was proposed. First, the maximum a posteriori (MAP) estimation of the BM鄄BMP algorithm is decomposed into its value at the last iteration and an increment; thus, it only needs to calculate the increment in each iteration, which greatly reduces the computational time. Second, by calculating the mean of the significant MAP estimations, an effective approximation is obtained for the minimum mean square error (MMSE) estimation and a smaller reconstruction error is a鄄 chieved. Compared with the BM鄄BMP, this method reduces the running time on average by 73郾 66% while improving the peak signal to noise ratio (PSNR) by 0郾 57 dB. KEY WORDS sparse signal reconstruction; fast Bayesian matching pursuit; Boltzmann machine; minimum mean square error (MMSE) 收稿日期: 2016鄄鄄09鄄鄄12 基金项目: 国家自然科学基金资助项目(61327005, 61302120); 广东省科技计划资助项目(2017A020214011); 中央高校基本科研业务费资助 项目(2017MS039) 稀疏信号重构被广泛应用于信号和图像处理中, 例如:去噪、盲源分离、波达方向估计、压缩和采样、图 像复原等[1鄄鄄2] . 假设 n 伊 1 维信号 y = Ax + e( n 是矢量 y 的长度),其中 x 是 m 伊 1 的稀疏系数(m 是矢量 x 的 长度),A 是 n 伊 m 的冗余字典(因为 m > n,所以 A 是 冗余字典),e 是高斯白噪声,其均值为零方差为 滓 2