Matlab implementation of compressed sensing reconstruction algorithm: ROMP regularization orthogonal matching pursuit algorithm

Orthogonal matching pursuit algorithm based on ROMP regularization

$ROMP$ algorithm is a greedy compressed sensing reconstruction algorithm, in$OMP$ added the regularization process based on. In addition, compared to$OMP$ algorithm,$ROMP$ selects themmmost relevant to the residual in each iteration$m$ column vectors. Below is$DeannaNeedell$和 R o m a n V e r s h y n i n Roman Vershynin ROMP ROMPgiven in$R$$o$$m$$a$$n$$V$$e$$r$$s$$h$$y$$n$$i$$n$$ROMP$ 's algorithm.

In the regularization process, you need to find all the subsets of the index set. Here you can useMatlab MatlabFf 2 n ff2nin$M$$a$$t$$l$$a$$b$$The\; ff2n$ function is realized. For regularization conditions, only satisfy$The$ absolute value of the largest element in$u$ is less than2 2 of theabsolute value of the smallest element$2$ times is enough.
In this paper, Gaussian random matrix is ​​used as the measurement matrix, and the length of the target signal to be observed is$N$ is$512$ , and the element values ​​are only$-1,0,1$ , the number of observations$M$ is$120$ , target signal sparsity$K$ is$10$。$The$ experimental results of$M$$a$$t$$l$$a$$b$ are as follows:

clear;
clc;
N=512;%待观测目标信号长度
K=10;%目标信号稀疏度
M=120;%观测值个数
maxnum=100;
x=zeros(N,1);
q=randperm(N);%随机打乱排序
t=1:N;
x(q(1:K))=sign(randn(K,1));%生成幅值为1，-1的K个非0元素的目标信号y
A=randn(M,N);%生成M*N的服从高斯分布的测量矩阵
A=orth(A')';%把测量矩阵按行正交化
y=A*x;%获得观测向量
x2=A'*y;%基于L2范数最小化估算的初值
S=ff2n(K);%20个元素所有可能的组合,(2^K)*K的矩阵
I=[];
A1=[];
r=y;%初始化时，残差r=y
for i=1:K
    u=A'*r;
    u1=abs(u);
    [temp,index]=sort(u1);%对向量u从小到大排序,index储存索引
    len1=length(u);
    J=index(len1-K+1:end);%储存K个的最大系数索引,此处存在问题，应该是非0
   if length(I)>=2*K
       break;
   else
       %正则化标准意思是选择各列向量与残差内积绝对值的最大值不能比最小值大两倍以上
       %(comparable coordinates)且能量最大的一组(with the maximal energy)
       [row,col]=size(S);%记录S的行数
       uu=[0;u];
       index1=S.*J';%存储所有子集在u中对应的索引
       temp2=uu(index1+1);%存储所有子集中的元素值，(2^K)个子集，一行看成一个向量
       abstemp=abs(temp2);
       max1=max(abstemp');
       abstemp1=abstemp';
       %abstemp1(find(abstemp1==0))=maxnum;
       min1=min(abstemp1(find(abstemp1~=0)));
       min1(1)=0;
       diff=abs(max1)-2*abs(min1);%最大值大于最小值两倍以上，则>0，
       %2倍关系可以换成其他的条件吗？
       index2=find(diff>0);%寻找满足最大值小于最小值两倍以上的索引
       if isempty(find(diff>0,1))
          disp('未发现满足正则条件的集合');
          break;
       end
       temp3=temp2(index2,:);%保留满足条件的子集
       temp4=temp3.*temp3;%中间变量，存储元素平方
       temp5=sqrt(sum(temp4,2));%按行求和，再开方，即求二范数
       [maxnorm,index3]=max(temp5);%会不会存在多个最大值？
       temp6=temp3(index3(1),:);%存储第一个最大范数的坐标向量
       temp7=temp2-temp6;%若temp6是temp2中某一行，则那一行全为0
       temp8=temp7;
       temp8(1,:)=[];
       J0=find(all(temp8== 0,2));%all(temp7== 0,2)寻找全是0的行 
       J0=index1(J0+1,:);
       if isempty(intersect(I,J0))~=0
           T=intersect(I,J0);%查找重复元素,并删除
           for jj=1:length(T)
               T1=find(J0==T(jj));
               J0(T1)=[];
           end
       end
       I=[I J0];%更新索引集合
       A1=A(:,I);
       y2=inv(A1'*A1)*A1'*y;%最小二乘法求y2
       r=y-A1*y2;
   end
end
y3=zeros(N,1);
y3(I)=y2;%储存恢复结果
plot(t,x,'r',t,y3,'g  *');
legend('red 代表原始信号','green 代表恢复的信号');
length(find(y3~=0));%在能正确恢复信号时，y3中非0系数是2K
% A1(:,1)'*A1(:,10)
% A(:,1)'*A(:,200)

Needell D , Vershynin R . Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit[J]. 2007.