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)

主要内容:

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

一、OMP的算法流程

二、OMP的MATLAB实现(CS_OMP.m)

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,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y
    for 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        
    end
    theta(pos_num)=theta_ls;% 恢复出的theta
end

三、一维信号的实验与结果(CS_Reconstuction_Test.m

%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

四、测量数M与重构成功概率关系的实验与结果(CS_Reconstuction_MtoPercentage.m)

%   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-6 %如果残差小于1e-6则认为恢复成功
                P = P + 1;
            end
       end
       PercentageK(mm) = P/CNT*100; %计算恢复概率
    end
    Percentage(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)');

五、稀疏度K与重构成功概率关系的实验与结果(CS_Reconstuction_KtoPercentage.m)

%   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K
       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-6 %如果残差小于1e-6则认为恢复成功
                P = P + 1;
            end
       end
       PercentageM(kk) = P/CNT*100; %计算恢复概率
    end
    Percentage(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)');

六、参考文章

http://blog.csdn.net/jbb0523/article/details/45268141

更多OMP请参考:浅谈压缩感知(九):正交匹配追踪算法OMP

posted @ 2016-01-08 15:07  AndyJee  阅读(8902)  评论(1编辑  收藏  举报