[转]压缩感知重构算法之分段正交匹配追踪(StOMP)

function [ theta ] = CS_StOMP( y,A,S,ts ) 
%CS_StOMP Summary of this function goes here 
%Version: 1.0 written by jbb0523 @2015-04-29 
%   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 StOMP iterations to perform 
%   ts is the threshold parameter 
%   现在已知y和A，求theta 
%   Reference:Donoho D L，Tsaig Y，Drori I，Starck J L．Sparse solution of 
%   underdetermined linear equations by stagewise orthogonal matching  
%   pursuit[J]．IEEE Transactions on Information Theory，2012，58(2)：1094—1121 
    if nargin < 4 
        ts = 2.5;%ts范围[2,3],默认值为2.5 
    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 = norm(r_n)/sqrt(M);%参见参考文献第3页Remarks(3) 
        Js = find(abs(product)>ts*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 = 64;%观测值个数 
N = 256;%信号x的长度 
K = 12;%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_StOMP(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