Common measurement matrix and matlab code of compressed sensing

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Preface

   The design of observation matrix is the second step of compressed sensing , Here is the commonly used measurement matrix matlab Code , For everyone to find and use . The code content is referenced from the following website , And modify some of the codes , If you want to explore its principle, you can enter this website to find ~

https://blog.csdn.net/jbb0523/article/details/44700735

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One 、 What is the measurement matrix ？

   The measurement matrix is used to enable people to see the observed values obtained by the instrument $y$. The concrete expression is ：$$y=\Phi x$$ among ,$x$ It's the original signal ,$y$ For observed signals . How many signals can be observed from the original signal , It is determined by the measurement matrix , Therefore, it will involve the concept of sampling rate . The usual concept of sampling rate is defined as the number of rows of the measurement matrix / Number of columns , namely $M/N$, If you are interested, you can consult the materials for learning ~

Two 、 Commonly used measurement matrix and its matlab Code

1、 Random Gaussian measurement matrix

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、 Random Bernoulli measurement matrix

function [ Phi ] = BernoulliMtx( M,N )
%BernoulliMtx Summary of this function goes here
%   Generate Bernoulli matrix 
%   M -- RowNumber
%   N -- ColumnNumber
%   Phi -- The Bernoulli matrix

    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);    % According to the formula, there should be this sentence 
end

3、 Partial Hadamard measurement matrix

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、 Partial Fourier measurement matrix

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、 Sparse random measurement matrix

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、 Toplitz measurement matrix

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

7、 Cyclic measurement matrix

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

summary

   The above is the code content to be shown in this section . I have consulted a lot of data in the process of learning compressed sensing , Also tested a lot of code , The code here does not lose money . If the code still deviates from its principle , You can also modify it according to the principle ~