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csumushu

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虫号: 3531723
注册: 2014-11-11
性别: GG
专业: 信号理论与信号处理

[求助] 为什么ROMP的仿真性能比OMP还弱啊？

function [val,pos] = Regularize(product,Kin)
%Regularize Summary of this function goes here
% 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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ theta ] = CS_ROMP( y,A,K )
%CS_ROMP Summary of this function goes here
%Version: 1.0 written by jbb0523 @2015-04-24
% 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.
[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)
At = zeros(M,3*K);%用来迭代过程中存储A被选择的列
Pos_theta = zeros(1,2*K);%用来迭代过程中存储A被选择的列序号
Index = 0;
r_n = 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'*r_n;%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
      [~,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_theta(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
      r_n = y - At(:,1:Index)*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
      if Index>=2*K%or until |I|>=2K
         break;%跳出for循环
      end
end
theta(Pos_theta(1:Index))=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 = 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
sigma = 0.5;
e = sigma*randn(M,1);
y = Phi * x + e;%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

以上是ROMP的算法

function [ theta ] = CS_OMP( y,A,t )
%CS_OMP Summary of this function goes here
%Version: 1.0 written by jbb0523 @2015-04-18
% 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
[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)
At = zeros(M,t);%用来迭代过程中存储A被选择的列
Pos_theta = zeros(1,t);%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:t%迭代t次，t为输入参数
      product = A'*r_n;%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
      At(:,ii) = A(:,pos);%存储这一列
      Pos_theta(ii) = pos;%存储这一列的序号
      A(:,pos) = zeros(M,1);%清零A的这一列，其实此行可以不要，因为它与残差正交
      %y=At(:,1:ii)*theta，以下求theta的最小二乘解(Least Square)
      theta_ls = (At(:,1:ii)'*At(:,1:ii))^(-1)*At(:,1:ii)'*y;%最小二乘解
      %At(:,1:ii)*theta_ls是y在At(:,1:ii)列空间上的正交投影
      r_n = y - At(:,1:ii)*theta_ls;%更新残差
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 = 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
sigma = 0.5;
e = sigma*randn(M,1);
y = Phi * x + e;%得到观测向量y
%% 恢复重构信号x
tic
theta = CS_OMP(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
mse(x_r-x)

以上是OMP的算法

算法来自OMP和ROMP
唯一的改变是在观测值上加了噪声

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