浅谈压缩感知（十八）：常见测量矩阵及其实现

1. 随机高斯矩阵

   

   MATLAB实现：

   function [ Phi ] = GaussMtx( M,N )
   %GaussMtx Summary of this function goes here
   %   Generate Bernoulli matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The Gauss matrix
   
   %% Generate Gauss matrix   
       Phi = randn(M,N);
       %Phi = Phi/sqrt(M);
   end

2. 随机伯努利矩阵

   

   MATLAB实现：

   function [ Phi ] = BernoulliMtx( M,N )
   %BernoulliMtx Summary of this function goes here
   %   Generate Bernoulli matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The Bernoulli matrix
   
   %% (1)Generate Bernoulli matrix(The first kind)
   % 1--P=0.5   -1--P=0.5
       Phi = randi([0,1],M,N);%If your MATLAB version is too low,please use randint instead
       Phi(Phi==0) = -1;
       %Phi = Phi/sqrt(M);
   % %% (2)Generate Bernoulli matrix(The second kind)
   % % 1--P=1/6   -1--P=1/6  0--2/3
   %     Phi = randi([-1,4],M,N);%If your MATLAB version is too low,please use randint instead
   %     Phi(Phi==2) = 0;%P=1/6
   %     Phi(Phi==3) = 0;%P=1/6
   %     Phi(Phi==4) = 0;%P=1/6
   %     %Phi = Phi*sqrt(3/M);
   end

3. 部分哈达玛矩阵

   

   MATLAB实现：

   function [ Phi ] = PartHadamardMtx( M,N )
   %PartHadamardMtx Summary of this function goes here
   %   Generate part Hadamard matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The part Hadamard matrix
   
   %% parameter initialization
   %Because the MATLAB function hadamard handles only the cases where n, n/12,
   %or n/20 is a power of 2
       L_t = max(M,N);%Maybe L_t does not meet requirement of function hadamard
       L_t1 = (12 - mod(L_t,12)) + L_t;
       L_t2 = (20 - mod(L_t,20)) + L_t; 
       L_t3 = 2^ceil(log2(L_t));
       L = min([L_t1,L_t2,L_t3]);%Get the minimum L
   %% Generate part Hadamard matrix   
       Phi = [];
       Phi_t = hadamard(L);
       RowIndex = randperm(L);
       Phi_t_r = Phi_t(RowIndex(1:M),:);
       ColIndex = randperm(L);
       Phi = Phi_t_r(:,ColIndex(1:N));
   end

4. 部分傅里叶矩阵

   

   MATLAB实现：

   function [ Phi ] = PartFourierMtx( M,N )
   %PartFourierMtx Summary of this function goes here
   %   Generate part Fourier matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The part Fourier matrix
   
   %% Generate part Fourier matrix   
       Phi_t = fft(eye(N,N))/sqrt(N);%Fourier matrix
       RowIndex = randperm(N);
       Phi = Phi_t(RowIndex(1:M),:);%Select M rows randomly
       %normalization
       for ii = 1:N
           Phi(:,ii) = Phi(:,ii)/norm(Phi(:,ii));
       end
   end

5. 稀疏随机矩阵

   

   MATLAB实现：

   function [ Phi ] = SparseRandomMtx( M,N,d )
   %SparseRandomMtx Summary of this function goes here
   %   Generate SparseRandom matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   d -- The number of '1' in every column,d<M 
   %   Phi -- The SparseRandom matrix
   
   %% Generate SparseRandom matrix   
       Phi = zeros(M,N);
       for ii = 1:N
           ColIdx = randperm(M);
           Phi(ColIdx(1:d),ii) = 1;
       end
   end

6. 托普利兹矩阵和循环矩阵

   

   MATLAB实现：

   function [ Phi ] = ToeplitzMtx( M,N )
   %ToeplitzMtx Summary of this function goes here
   %   Generate Toeplitz matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The Toeplitz matrix
   
   %% Generate a random vector
   %     %(1)Gauss
   %     u = randn(1,2*N-1);
       %(2)Bernoulli
       u = randi([0,1],1,2*N-1);
       u(u==0) = -1;
   %% Generate Toeplitz matrix   
       Phi_t = toeplitz(u(N:end),fliplr(u(1:N)));
       Phi = Phi_t(1:M,:);
   end

   function [ Phi ] = CirculantMtx( M,N )
   %CirculantMtx Summary of this function goes here
   %   Generate Circulant matrix 
   %   M -- RowNumber
   %   N -- ColumnNumber
   %   Phi -- The Circulant matrix
   
   %% Generate a random vector
   %     %(1)Gauss
   %     u = randn(1,N);
       %(2)Bernoulli
       u = randi([0,1],1,N);
       u(u==0) = -1;
   %% Generate Circulant matrix   
       Phi_t = toeplitz(circshift(u,[1,1]),fliplr(u(1:N)));
       Phi = Phi_t(1:M,:);
   end