浅谈压缩感知（二十一）：压缩感知重构算法之正交匹配追踪（OMP）

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<n
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,); %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(,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y
for ii=: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,); %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(:,:ii)*theta，以下求theta的最小二乘解(Least Square)
theta_ls = (At(:,:ii)'*At(:,1:ii))^(-1)*At(:,1:ii)'*y;%最小二乘解
% At(:,:ii)*theta_ls是y在At(:,:ii)列空间上的正交投影
res = y - At(:,:ii)*theta_ls; %更新残差
end
theta(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 = ;%观测值个数
N = ;%信号x的长度
K = ;%信号x的稀疏度
Index_K = randperm(N);
x = zeros(N,);
x(Index_K(:K)) = *randn(K,);%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.
% 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. , NO. ,
% DECEMBER .clear all;close all;clc;%% 参数配置初始化
CNT = ; %对于每组(K,M,N)，重复迭代次数
N = ; %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 = [,,,,]; %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 = :length(K_set)
K = K_set(kk); %本次稀疏度
M_set = K::N; %M没必要全部遍历，每隔5测试一个就可以了
PercentageK = zeros(,length(M_set)); %存储此稀疏度K下不同M的恢复成功概率
for mm = :length(M_set)
M = M_set(mm); %本次观测值个数
P = ;
for cnt = :CNT %每个观测值个数均运行CNT次
Index_K = randperm(N);
x = zeros(N,);
x(Index_K(:K)) = *randn(K,); %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y
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- %如果残差小于1e-6则认为恢复成功
P = P + ;
end
end
PercentageK(mm) = P/CNT*; %计算恢复概率
end
Percentage(kk,: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 = :length(K_set)
K = K_set(kk);
M_set = K::N;
L_Mset = length(M_set);
plot(M_set,Percentage(kk,:L_Mset),S(kk,:));%绘出x的恢复信号
hold on;
end
hold off;
xlim([ ]);
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.
% 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. , NO. ,
% DECEMBER .
%
clear all;close all;clc;%% 参数配置初始化
CNT = ; %对于每组(K,M,N)，重复迭代次数
N = ; %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 = [,,,,]; %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 = :length(M_set)
M = M_set(mm); %本次测量值个数
K_set = ::ceil(M/); %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(,length(K_set)); %存储此测量值M下不同K的恢复成功概率
for kk = :length(K_set)
K = K_set(kk); %本次信号x的稀疏度K
P = ;
for cnt = :CNT %每个观测值个数均运行CNT次
Index_K = randperm(N);
x = zeros(N,);
x(Index_K(:K)) = *randn(K,); %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y
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- %如果残差小于1e-6则认为恢复成功
P = P + ;
end
end
PercentageM(kk) = P/CNT*; %计算恢复概率
end
Percentage(mm,: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 = :length(M_set)
M = M_set(mm);
K_set = ::ceil(M/);
L_Kset = length(K_set);
plot(K_set,Percentage(mm,:L_Kset),S(mm,:));%绘出x的恢复信号
hold on;
end
hold off;
xlim([ ]);
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)');