IEEE Trans 2009 Stagewise Weak Gradient Pursuits论文学习

function [ theta ] = CS_SWOMP( y,A,S,alpha ) 
%CS_SWOMP Summary of this function goes here 
%Version: 1.0 written by jbb0523 @2015-05-11 
%   Detailed explanation goes here 
%   y = Phi * x 
%   x = Psi * theta 
%   y = Phi*Psi * theta 
%   令 A = Phi*Psi, 则y=A*theta 
%   S is the maximum number of SWOMP iterations to perform 
%   alpha is the threshold parameter 
%   现在已知y和A，求theta 
%   Reference:Thomas Blumensath，Mike E. Davies．Stagewise weak gradient 
%   pursuits[J]．IEEE Transactions on Signal Processing，2009，57(11)：4333-4346． 
    if nargin < 4 
        alpha = 0.5;%alpha范围(0,1),默认值为0.5 
    end 
    if nargin < 3 
        S = 10;%S默认值为10 
    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 ss=1:S%最多迭代S次 
        product = A'*r_n;%传感矩阵A各列与残差的内积 
        sigma = max(abs(product)); 
        Js = find(abs(product)>=alpha*sigma);%选出大于阈值的列 
        Is = union(Pos_theta,Js);%Pos_theta与Js并集 
        if length(Pos_theta) == length(Is) 
            if ss==1 
                theta_ls = 0;%防止第1次就跳出导致theta_ls无定义 
            end 
            break;%如果没有新的列被选中则跳出循环 
        end 
        %At的行数要大于列数，此为最小二乘的基础(列线性无关) 
        if length(Is)<=M 
            Pos_theta = Is;%更新列序号集合 
            At = A(:,Pos_theta);%将A的这几列组成矩阵At 
        else%At的列数大于行数，列必为线性相关的,At'*At将不可逆 
            if ss==1 
                theta_ls = 0;%防止第1次就跳出导致theta_ls无定义 
            end 
            break;%跳出for循环 
        end 
        %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 
        if norm(r_n)<1e-6%Repeat the steps until r=0 
            break;%跳出for循环 
        end 
    end 
    theta(Pos_theta)=theta_ls;%恢复出的theta 
end 

%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_SWOMP( y,A); 
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 
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  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 
alpha_set = 0.1:0.1:1; 
K_set = [4,12,20,28,36];%信号x的稀疏度集合 
Percentage = zeros(N,length(K_set),length(alpha_set));%存储恢复成功概率 
%% 主循环，遍历每组(alpha,K,M,N) 
tic 
for tt = 1:length(alpha_set) 
    alpha = alpha_set(tt); 
    for kk = 1:length(K_set) 
        K = K_set(kk);%本次稀疏度 
        %M没必要全部遍历，每隔5测试一个就可以了 
        M_set=2*K:5:N; 
        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 
           fprintf('alpha=%f,K=%d,M=%dn',alpha,K,M); 
           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)/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 
                theta = CS_SWOMP(y,A,10,alpha);%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(1:length(M_set),kk,tt) = PercentageK; 
    end 
end 
toc 
save SWOMPMtoPercentage1000 %运行一次不容易，把变量全部存储下来 
%% 绘图 
for tt = 1:length(alpha_set) 
    S = ['-ks';'-ko';'-kd';'-kv';'-k*']; 
    figure; 
    for kk = 1:length(K_set) 
        K = K_set(kk); 
        M_set=2*K:5:N; 
        L_Mset = length(M_set); 
        plot(M_set,Percentage(1:L_Mset,kk,tt),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,alpha=',... 
        num2str(alpha_set(tt)),')(Gaussian)']); 
end 
for kk = 1:length(K_set) 
    K = K_set(kk); 
    M_set=2*K:5:N; 
    L_Mset = length(M_set); 
    S = ['-ks';'-ko';'-kd';'-k*';'-k+';'-kx';'-kv';'-k^';'-k<';'-k>']; 
    figure; 
    for tt = 1:length(alpha_set) 
        plot(M_set,Percentage(1:L_Mset,kk,tt),S(tt,:));%绘出x的恢复信号 
        hold on; 
    end 
    hold off; 
    xlim([0 256]); 
    legend('alpha=0.1','alpha=0.2','alpha=0.3','alpha=0.4','alpha=0.5',... 
        'alpha=0.6','alpha=0.7','alpha=0.8','alpha=0.9','alpha=1.0'); 
    xlabel('Number of measurements(M)'); 
    ylabel('Percentage recovered'); 
    title(['Percentage of input signals recovered correctly(N=256,K=',... 
        num2str(K),')(Gaussian)']);     
end 
y = Phi * x;%得到观测向量y 
%% 恢复重构信号x 
tic 
theta = CS_SWOMP( y,A); 
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