function [ theta ] = CS_OMP( y,A,iter )
%   CS_OMP
%   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
%   iter = 迭代次数 [m,n] = size(y);if m<ny = 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)At = zeros(M,iter); %用来迭代过程中存储A被选择的列pos_num = zeros(1,iter); %用来迭代过程中存储A被选择的列序号res = y; %初始化残差(residual)为yfor ii=1:iter %迭代t次，t为输入参数product = A'*res; %传感矩阵A各列与残差的内积[val,pos] = max(abs(product)); %找到最大内积绝对值，即与残差最相关的列At(:,ii) = A(:,pos); %存储这一列pos_num(ii) = pos; %存储这一列的序号A(:,pos) = zeros(M,1); %清零A的这一列，其实此行可以不要，因为它与残差正交% y=At(:,1:ii)*theta，以下求theta的最小二乘解(Least Square)theta_ls = (At(:,1:ii)'*At(:,1:ii))^(-1)*At(:,1:ii)'*y;%最小二乘解% At(:,1:ii)*theta_ls是y在At(:,1:ii)列空间上的正交投影res = y - At(:,1:ii)*theta_ls; %更新残差        endtheta(pos_num)=theta_ls;% 恢复出的theta
end

%压缩感知重构算法OMP测试
%以一维信号为例
clear all;close all;clc;
M = 64;%观测值个数
N = 256;%信号x的长度
K = 10;%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);%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_OMP(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

%   压缩感知重构算法测试CS_Reconstuction_MtoPercentage.m
%   绘制参考文献中的Fig.1
%   参考文献：Joel A. Tropp and Anna C. Gilbert
%   Signal Recovery From Random Measurements Via Orthogonal Matching
%   Pursuit，IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 12,
%   DECEMBER 2007.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
K_set = [4,12,20,28,36]; %信号x的稀疏度集合
Percentage = zeros(length(K_set),N); %存储恢复成功概率%% 主循环，遍历每组(K,M,N)
tic
for kk = 1:length(K_set)K = K_set(kk); %本次稀疏度M_set = K:5:N; %M没必要全部遍历，每隔5测试一个就可以了PercentageK = zeros(1,length(M_set)); %存储此稀疏度K下不同M的恢复成功概率for mm = 1:length(M_set)M = M_set(mm); %本次观测值个数P = 0;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); %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ytheta = CS_OMP(y,A,K); %恢复重构信号thetax_r = Psi * theta; % x=Psi * thetaif norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功P = P + 1;endendPercentageK(mm) = P/CNT*100; %计算恢复概率endPercentage(kk,1:length(M_set)) = PercentageK;
end
toc
save MtoPercentage1000 %运行一次不容易，把变量全部存储下来%% 绘图
S = ['-ks';'-ko';'-kd';'-kv';'-k*'];
figure;
for kk = 1:length(K_set)K = K_set(kk);M_set = K:5:N;L_Mset = length(M_set);plot(M_set,Percentage(kk,1:L_Mset),S(kk,:));%绘出x的恢复信号hold on;
end
hold off;
xlim([0 256]);
legend('K=4','K=12','K=20','K=28','K=36');
xlabel('Number of measurements(M)');
ylabel('Percentage recovered');
title('Percentage of input signals recovered correctly(N=256)(Gaussian)');

%   压缩感知重构算法测试CS_Reconstuction_KtoPercentage.m
%   绘制参考文献中的Fig.2
%   参考文献：Joel A. Tropp and Anna C. Gilbert
%   Signal Recovery From Random Measurements Via Orthogonal Matching
%   Pursuit，IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 12,
%   DECEMBER 2007.
%
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 = [52,100,148,196,244]; %测量值集合
Percentage = zeros(length(M_set),N); %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 = 1:5:ceil(M/2); %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PercentageM = zeros(1,length(K_set)); %存储此测量值M下不同K的恢复成功概率for kk = 1:length(K_set)K = K_set(kk); %本次信号x的稀疏度KP = 0;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); %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ytheta = CS_OMP(y,A,K); %恢复重构信号thetax_r = Psi * theta; % x=Psi * thetaif norm(x_r-x)<1e-6 %如果残差小于1e-6则认为恢复成功P = P + 1;endendPercentageM(kk) = P/CNT*100; %计算恢复概率endPercentage(mm,1:length(K_set)) = PercentageM;
end
toc
save KtoPercentage1000test %运行一次不容易，把变量全部存储下来%% 绘图
S = ['-ks';'-ko';'-kd';'-kv';'-k*'];
figure;
for mm = 1:length(M_set)M = M_set(mm);K_set = 1:5:ceil(M/2);L_Kset = length(K_set);plot(K_set,Percentage(mm,1:L_Kset),S(mm,:));%绘出x的恢复信号hold on;
end
hold off;
xlim([0 125]);
legend('M=52','M=100','M=148','M=196','M=244');
xlabel('Sparsity level(K)');
ylabel('Percentage recovered');
title('Percentage of input signals recovered correctly(N=256)(Gaussian)');