# 浅谈压缩感知（二十二）：压缩感知重构算法之正则化正交匹配追踪（ROMP）

1. ROMP的算法流程
2. ROMP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、ROMP的MATLAB实现(CS_ROMP.m)

1、正则化代码Regularize.m

function [val,pos] = Regularize(product,Kin)
%   Regularize
%   Detailed explanation goes here
%   product = A'*r_n;%传感矩阵A各列与残差的内积
%   K为稀疏度
%   pos为选出的各列序号
%   val为选出的各列与残差的内积值
%   Reference:Needell D，Vershynin R. Uniform uncertainty principle and
%   signal recovery via regularized orthogonal matching pursuit.
%   Foundations of Computational Mathematics, 2009,9(3): 317-334.
productabs = abs(product); %取绝对值
[productdes,indexproductdes] = sort(productabs,'descend'); %降序排列
for ii = length(productdes):-1:1
if productdes(ii)>1e-6 %判断productdes中非零值个数
break;
end
end
% Identify:Choose a set J of the K biggest coordinates
if ii>=Kin
J = indexproductdes(1:Kin); %集合J
Jval = productdes(1:Kin); %集合J对应的序列值
K = Kin;
else % or all of its nonzero coordinates,whichever is smaller
J = indexproductdes(1:ii);  %集合J
Jval = productdes(1:ii);  %集合J对应的序列值
K = ii;
end
% Regularize:Among all subsets J0∈J with comparable coordinates
MaxE = -1;  %循环过程中存储最大能量值
for kk = 1:K
J0_tmp = zeros(1,K);iJ0 = 1;
J0_tmp(iJ0) = J(kk);  %以J(kk)为本次寻找J0的基准(最大值)
Energy = Jval(kk)^2;  %本次寻找J0的能量
for mm = kk+1:K
if Jval(kk)<2*Jval(mm) %找到符合|u(i)|<=2|u(j)|的
iJ0 = iJ0 + 1; %J0自变量增1
J0_tmp(iJ0) = J(mm); %更新J0
Energy = Energy + Jval(mm)^2; %更新能量
else %不符合|u(i)|<=2|u(j)|的
break; %跳出本轮寻找，因为后面更小的值也不会符合要求
end
end
if Energy>MaxE %本次所得J0的能量大于前一组
J0 = J0_tmp(1:iJ0); %更新J0
MaxE = Energy; %更新MaxE，为下次循环做准备
end
end
pos = J0;
val = productabs(J0);
end

2、ROMP代码CS_ROMP.m

function [ theta ] = CS_ROMP( y,A,K )
%   CS_ROMP
%   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:Needell D，Vershynin R．Signal recovery from incomplete and
%   inaccurate measurements via regularized orthogonal matching pursuit[J]．
%   IEEE Journal on Selected Topics in Signal Processing，2010，4(2)：310—316.
[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,3*K); %用来迭代过程中存储A被选择的列
pos_num = zeros(1,2*K); %用来迭代过程中存储A被选择的列序号
Index = 0;
res = y; %初始化残差(residual)为y
%Repeat the following steps K times(or until |I|>=2K)
for ii=1:K %迭代K次
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
[val,pos] = Regularize(product,K); %按正则化规则选择原子
At(:,Index+1:Index+length(pos)) = A(:,pos); %存储这几列
pos_num(Index+1:Index+length(pos)) = pos; %存储这几列的序号
if Index+length(pos)<=M %At的行数大于列数，此为最小二乘的基础(列线性无关)
Index = Index+length(pos); %更新Index，为下次循环做准备
else %At的列数大于行数，列必为线性相关的,At(:,1:Index)'*At(:,1:Index)将不可逆
break; %跳出for循环
end
A(:,pos) = zeros(M,length(pos)); %清零A的这几列(其实此行可以不要,因为它们与残差正交)
%y=At(:,1:Index)*theta，以下求theta的最小二乘解(Least Square)
theta_ls = (At(:,1:Index)'*At(:,1:Index))^(-1)*At(:,1:Index)'*y; %最小二乘解
%At(:,1:Index)*theta_ls是y在At(:,1:Index)列空间上的正交投影
res = y - At(:,1:Index)*theta_ls; %更新残差
if norm(res)<1e-6 %Repeat the steps until r=0
break; %跳出for循环
end
if Index>=2*K %or until |I|>=2K
break; %跳出for循环
end
end
theta(pos_num(1:Index))=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 = 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_ROMP(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 = 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
kk
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_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 = 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 ROMPMtoPercentage %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、参考文章

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

posted @ 2016-01-11 15:10  AndyJee  阅读(5637)  评论(0编辑  收藏  举报