浅谈压缩感知（二十八）：压缩感知重构算法之广义正交匹配追踪(gOMP)

主要内容：

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

一、gOMP的算法流程

广义正交匹配追踪(Generalized OMP, gOMP)算法可以看作为OMP算法的一种推广。OMP每次只选择与残差相关最大的一个，而gOMP则是简单地选择最大的S个。之所以这里表述为"简单地选择"是相比于ROMP之类算法的，不进行任何其它处理，只是选择最大的S个而已。

gOMP的算法流程：

二、gOMP的MATLAB实现(CS_gOMP.m)

function [ theta ] = CS_gOMP( y,A,K,S )
%   CS_gOMP
%   Detailed explanation goes here
%   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
%   Reference: Jian Wang, Seokbeop Kwon, Byonghyo Shim.  Generalized 
%   orthogonal matching pursuit, IEEE Transactions on Signal Processing, 
%   vol. 60, no. 12, pp. 6202-6216, Dec. 2012. 
%   Available at: http://islab.snu.ac.kr/paper/tsp_gOMP.pdf
    if nargin < 4
        S = round(max(K/4, 1));
    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 ii=1:K%迭代K次，K为稀疏度
        product = A'*r_n;%传感矩阵A各列与残差的内积
        [val,pos]=sort(abs(product),'descend');%降序排列
        Sk = union(Pos_theta,pos(1:S));%选出最大的S个
        if length(Sk)==length(Pos_theta)
            if ii == 1
                theta_ls = 0;
            end
            break;
        end
        if length(Sk)>M
            if ii == 1
                theta_ls = 0;
            end
            break;
        end
        At = A(:,Sk);%将A的这几列组成矩阵At
        %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
        Pos_theta = Sk;
        if norm(r_n)<1e-6
            break;%quit the iteration
        end
    end
    theta(Pos_theta)=theta_ls;%恢复出的theta
end

三、一维信号的实验与结果

%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_gOMP( 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('
恢复残差：');
norm(x_r-x)%恢复残差

四、稀疏数K与重构成功概率关系的实验与结果

%   压缩感知重构算法测试CS_Reconstuction_KtoPercentagegOMP.m
%   Reference: Jian Wang, Seokbeop Kwon, Byonghyo Shim.  Generalized 
%   orthogonal matching pursuit, IEEE Transactions on Signal Processing, 
%   vol. 60, no. 12, pp. 6202-6216, Dec. 2012. 
%   Available at: http://islab.snu.ac.kr/paper/tsp_gOMP.pdf

clear all;close all;clc;
addpath(genpath('../../OMP/'))

%% 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 = [128]; %测量值集合
KIND = ['OMP      ';'ROMP     ';'StOMP    ';'SP       ';'CoSaMP   ';...
    'gOMP(s=3)';'gOMP(s=6)';'gOMP(s=9)'];
Percentage = zeros(N,length(M_set),size(KIND,1)); %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 = 5:5:70; %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
    %存储此测量值M下不同K的恢复成功概率
    PercentageM = zeros(size(KIND,1),length(K_set));
    for kk = 1:length(K_set)
       K = K_set(kk); %本次信号x的稀疏度K
       P = zeros(1,size(KIND,1));
       fprintf('M=%d,K=%d
',M,K);
       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
            %(1)OMP
            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(1) = P(1) + 1;
            end
            %(2)ROMP
            theta = CS_ROMP(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(2) = P(2) + 1;
            end
            %(3)StOMP
            theta = CS_StOMP(y,A); %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(3) = P(3) + 1;
            end
            %(4)SP
            theta = CS_SP(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(4) = P(4) + 1;
            end
            %(5)CoSaMP
            theta = CS_CoSaMP(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(5) = P(5) + 1;
            end
            %(6)gOMP,S=3
            theta = CS_gOMP(y,A,K,3); %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(6) = P(6) + 1;
            end
            %(7)gOMP,S=6
            theta = CS_gOMP(y,A,K,6); %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(7) = P(7) + 1;
            end
            %(8)gOMP,S=9
            theta = CS_gOMP(y,A,K,9); %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(8) = P(8) + 1;
            end
       end
       for iii = 1:size(KIND,1)
           PercentageM(iii,kk) = P(iii)/CNT*100; %计算恢复概率
       end
    end
    for jjj = 1:size(KIND,1)
        Percentage(1:length(K_set),mm,jjj) = PercentageM(jjj,:);
    end
end
toc
save KtoPercentage1000gOMP %运行一次不容易，把变量全部存储下来

%% 绘图
S = ['-ks';'-ko';'-yd';'-gv';'-b*';'-r.';'-rx';'-r+'];
figure;
for mm = 1:length(M_set)
    M = M_set(mm);
    K_set = 5:5:70;
    L_Kset = length(K_set);
    for ii = 1:size(KIND,1)
        plot(K_set,Percentage(1:L_Kset,mm,ii),S(ii,:)); %绘出x的恢复信号
        hold on;
    end
end
hold off;
xlim([5 70]);
legend('OMP','ROMP','StOMP','SP','CoSaMP',...
    'gOMP(s=3)','gOMP(s=6)','gOMP(s=9)');
xlabel('Sparsity level K');
ylabel('The Probability of Exact Reconstruction');
title('Prob. of exact recovery vs. the signal sparsity K(M=128,N=256)(Gaussian)');

结论：gOMP只是在OMP基础上修改了一下原子选择的个数，效果就好很多。

六、参考文章

http://blog.csdn.net/jbb0523/article/details/45693027