# 浅谈压缩感知（二十三）：压缩感知重构算法之压缩采样匹配追踪（CoSaMP）

1. CoSaMP的算法流程
2. CoSaMP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

## 二、CS_CoSaMP的MATLAB实现(CS_CoSaMP.m)

function [ theta ] = CS_CoSaMP( y,A,K )
%   CS_CoSaOMP
%   Detailed explanation goes here
%   y = Phi * x
%   x = Psi * theta
%    y = Phi*Psi * theta
%   令 A = Phi*Psi, 则y=A*theta
%   K is the sparsity level
%   现在已知y和A，求theta
%   Reference:Needell D，Tropp J A．CoSaMP：Iterative signal recovery from
%   incomplete and inaccurate samples[J]．Applied and Computation Harmonic
%   Analysis，2009，26：301-321.
[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)
pos_num = []; %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 kk=1:K %最多迭代K次
%(1) Identification
product = A'*res; %传感矩阵A各列与残差的内积
[val,pos]=sort(abs(product),'descend');
Js = pos(1:2*K); %选出内积值最大的2K列
%(2) Support Merger
Is = union(pos_num,Js); %Pos_theta与Js并集
%(3) Estimation
%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
At = A(:,Is); %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 kk == 1
theta_ls = 0;
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
%(4) Pruning
[val,pos]=sort(abs(theta_ls),'descend');
%(5) Sample Update
pos_num = Is(pos(1:K));
theta_ls = theta_ls(pos(1:K));
%At(:,pos(1:K))*theta_ls是y在At(:,pos(1:K))列空间上的正交投影
res = y - At(:,pos(1:K))*theta_ls; %更新残差
if norm(res)<1e-6 %Repeat the steps until r=0
break; %跳出for循环
end
end
theta(pos_num)=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 = 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); %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_CoSaMP( 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

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 = 2*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
fprintf('K=%d,M=%d\n',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_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 = 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 CoSaMPMtoPercentage1000 %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 = 2*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、参考文章

http://blog.csdn.net/jbb0523/article/details/45441361

posted @ 2016-01-11 15:28  AndyJee  阅读(10926)  评论(3编辑  收藏  举报