# 浅谈压缩感知（二十八）：压缩感知重构算法之广义正交匹配追踪(gOMP)

1. gOMP的算法流程
2. gOMP的MATLAB实现
3. 一维信号的实验与结果
4. 稀疏度K与重构成功概率关系的实验与结果

gOMP的算法流程：

## 二、gOMP的MATLAB实现(CS_gOMP.m)

function [ theta ] = CS_gOMP( y,A,K,S )
%   CS_gOMP
%   Detailed explanation goes here
%   y = Phi * x
%   x = Psi * theta
%    y = Phi*Psi * theta
%   令 A = Phi*Psi, 则y=A*theta
%   现在已知y和A，求theta
%   Reference: Jian Wang, Seokbeop Kwon, Byonghyo Shim.  Generalized
%   orthogonal matching pursuit, IEEE Transactions on Signal Processing,
%   vol. 60, no. 12, pp. 6202-6216, Dec. 2012.
%   Available at: http://islab.snu.ac.kr/paper/tsp_gOMP.pdf
if nargin < 4
S = round(max(K/4, 1));
end
[y_rows,y_columns] = size(y);
if y_rows<y_columns
y = y';%y should be a column vector
end
[M,N] = size(A);%传感矩阵A为M*N矩阵
theta = zeros(N,1);%用来存储恢复的theta(列向量)
Pos_theta = [];%用来迭代过程中存储A被选择的列序号
r_n = y;%初始化残差(residual)为y
for ii=1:K%迭代K次，K为稀疏度
product = A'*r_n;%传感矩阵A各列与残差的内积
[val,pos]=sort(abs(product),'descend');%降序排列
Sk = union(Pos_theta,pos(1:S));%选出最大的S个
if length(Sk)==length(Pos_theta)
if ii == 1
theta_ls = 0;
end
break;
end
if length(Sk)>M
if ii == 1
theta_ls = 0;
end
break;
end
At = A(:,Sk);%将A的这几列组成矩阵At
%y=At*theta，以下求theta的最小二乘解(Least Square)
theta_ls = (At'*At)^(-1)*At'*y;%最小二乘解
%At*theta_ls是y在At)列空间上的正交投影
r_n = y - At*theta_ls;%更新残差
Pos_theta = Sk;
if norm(r_n)<1e-6
break;%quit the iteration
end
end
theta(Pos_theta)=theta_ls;%恢复出的theta
end

## 三、一维信号的实验与结果

%压缩感知重构算法测试
clear all;close all;clc;
M = 128;%观测值个数
N = 256;%信号x的长度
K = 30;%信号x的稀疏度
Index_K = randperm(N);
x = zeros(N,1);
x(Index_K(1:K)) = 5*randn(K,1);%x为K稀疏的，且位置是随机的
Psi = eye(N);%x本身是稀疏的，定义稀疏矩阵为单位阵x=Psi*theta
Phi = randn(M,N)/sqrt(M);%测量矩阵为高斯矩阵
A = Phi * Psi;%传感矩阵
y = Phi * x;%得到观测向量y

%% 恢复重构信号x
tic
theta = CS_gOMP( y,A,K);
x_r = Psi * theta;% x=Psi * theta
toc

%% 绘图
figure;
plot(x_r,'k.-');%绘出x的恢复信号
hold on;
plot(x,'r');%绘出原信号x
hold off;
legend('Recovery','Original')
fprintf('\n恢复残差：');
norm(x_r-x)%恢复残差

## 四、稀疏数K与重构成功概率关系的实验与结果

%   压缩感知重构算法测试CS_Reconstuction_KtoPercentagegOMP.m
%   Reference: Jian Wang, Seokbeop Kwon, Byonghyo Shim.  Generalized
%   orthogonal matching pursuit, IEEE Transactions on Signal Processing,
%   vol. 60, no. 12, pp. 6202-6216, Dec. 2012.
%   Available at: http://islab.snu.ac.kr/paper/tsp_gOMP.pdf

clear all;close all;clc;

%% 参数配置初始化
CNT = 1000; %对于每组(K,M,N)，重复迭代次数
N = 256; %信号x的长度
Psi = eye(N); %x本身是稀疏的，定义稀疏矩阵为单位阵x=Psi*theta
M_set = [128]; %测量值集合
KIND = ['OMP      ';'ROMP     ';'StOMP    ';'SP       ';'CoSaMP   ';...
'gOMP(s=3)';'gOMP(s=6)';'gOMP(s=9)'];
Percentage = zeros(N,length(M_set),size(KIND,1)); %存储恢复成功概率

%% 主循环，遍历每组(K,M,N)
tic
for mm = 1:length(M_set)
M = M_set(mm); %本次测量值个数
K_set = 5:5:70; %信号x的稀疏度K没必要全部遍历，每隔5测试一个就可以了
%存储此测量值M下不同K的恢复成功概率
PercentageM = zeros(size(KIND,1),length(K_set));
for kk = 1:length(K_set)
K = K_set(kk); %本次信号x的稀疏度K
P = zeros(1,size(KIND,1));
fprintf('M=%d,K=%d\n',M,K);
for cnt = 1:CNT  %每个观测值个数均运行CNT次
Index_K = randperm(N);
x = zeros(N,1);
x(Index_K(1:K)) = 5*randn(K,1); %x为K稀疏的，且位置是随机的
Phi = randn(M,N)/sqrt(M); %测量矩阵为高斯矩阵
A = Phi * Psi; %传感矩阵
y = Phi * x; %得到观测向量y
%(1)OMP
theta = CS_OMP(y,A,K); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(1) = P(1) + 1;
end
%(2)ROMP
theta = CS_ROMP(y,A,K); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(2) = P(2) + 1;
end
%(3)StOMP
theta = CS_StOMP(y,A); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(3) = P(3) + 1;
end
%(4)SP
theta = CS_SP(y,A,K); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(4) = P(4) + 1;
end
%(5)CoSaMP
theta = CS_CoSaMP(y,A,K); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(5) = P(5) + 1;
end
%(6)gOMP,S=3
theta = CS_gOMP(y,A,K,3); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(6) = P(6) + 1;
end
%(7)gOMP,S=6
theta = CS_gOMP(y,A,K,6); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(7) = P(7) + 1;
end
%(8)gOMP,S=9
theta = CS_gOMP(y,A,K,9); %恢复重构信号theta
x_r = Psi * theta; % x=Psi * theta
if norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功
P(8) = P(8) + 1;
end
end
for iii = 1:size(KIND,1)
PercentageM(iii,kk) = P(iii)/CNT*100; %计算恢复概率
end
end
for jjj = 1:size(KIND,1)
Percentage(1:length(K_set),mm,jjj) = PercentageM(jjj,:);
end
end
toc
save KtoPercentage1000gOMP %运行一次不容易，把变量全部存储下来

%% 绘图
S = ['-ks';'-ko';'-yd';'-gv';'-b*';'-r.';'-rx';'-r+'];
figure;
for mm = 1:length(M_set)
M = M_set(mm);
K_set = 5:5:70;
L_Kset = length(K_set);
for ii = 1:size(KIND,1)
plot(K_set,Percentage(1:L_Kset,mm,ii),S(ii,:)); %绘出x的恢复信号
hold on;
end
end
hold off;
xlim([5 70]);
legend('OMP','ROMP','StOMP','SP','CoSaMP',...
'gOMP(s=3)','gOMP(s=6)','gOMP(s=9)');
xlabel('Sparsity level K');
ylabel('The Probability of Exact Reconstruction');
title('Prob. of exact recovery vs. the signal sparsity K(M=128,N=256)(Gaussian)');

## 六、参考文章

http://blog.csdn.net/jbb0523/article/details/45693027

posted @ 2016-01-18 10:02  AndyJee  阅读(4663)  评论(0编辑  收藏  举报