Common sparse basis and matlab code for compressed sensing

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Preface

   The first step of compressed sensing is the sparse representation of the signal , So it is particularly important to find sparse bases , This blog will introduce the commonly used sparse basis and its matlab Code .

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One 、 What is a sparse base ？

   Sparse basis is a matrix that can sparse represent signals . Let one-dimensional discrete signal $x$, The length is $N$, Can be seen as $N$ Dimensional space $N\times 1$ The column vector , If this column vector contains $K$ It's not for 0 Elements , And $K<<N$, The signal $x$ yes $K$- Sparse signal , Have sparsity .
   If the signal is sparse , Then signal $x$ It can be expressed as ：$$x=\Psi s$$ among ,$\Psi$ Sparse basis （ sparse matrix ）.

Two 、 Some commonly used sparse bases

1、DFT（ Discrete Fourier ） Sparse basis

%%matlab Built-in function 
dftmtx(N)/sqrt(N);

%% Equivalent to the above 
fft(eye(N))/sqrt(N);

2、DCT（ Discrete cosine ） Sparse basis

%% matlab  Built-in function 
dctmtx(N);

%%  Expand and write as ：
[cc,rr] = meshgrid(0:n-1);
c = sqrt(2 / n) * cos(pi * (2*cc + 1) .* rr / (2 * n));
c(1,:) = c(1,:) / sqrt(2);

3、DWT（ wavelet ） Sparse basis

function ww=DWT(N)

[h,g]= wfilters('sym8','d');       %   Decompose low-pass and high pass filters   

% N=256;                           %   Matrix dimension ( The size is 2 Integer power of )
L=length(h);                       %   Filter length 
rank_max=log2(N);                  %   Maximum number of layers 
rank_min=double(int8(log2(L)))+1;  %   Minimum number of layers 
ww=1;   %   Preprocessing matrix 

%   Matrix construction 
for jj=rank_min:rank_max
    
    nn=2^jj;
    
    %   Construct vectors 
    p1_0=sparse([h,zeros(1,nn-L)]);
    p2_0=sparse([g,zeros(1,nn-L)]);
    
    %   Vector circumferential shift 
    for ii=1:nn/2
        p1(ii,:)=circshift(p1_0',2*(ii-1))';
        p2(ii,:)=circshift(p2_0',2*(ii-1))';
    end
    
    %   Construct an orthogonal matrix 
    w1=[p1;p2];
    mm=2^rank_max-length(w1);
    w=sparse([w1,zeros(length(w1),mm);zeros(mm,length(w1)),eye(mm,mm)]);
    ww=ww*w;
    
    clear p1;clear p2;
end

The author of the program is Javert , Department of electrical and electronic engineering, University of Hong Kong ,[email protected]

4、0-1 Random sparse basis （ It is also reasonable according to the sparse characteristics ）

%% Use rand Function generation 0-1 The number of intervals 
function A=get01(N)

B=rand(N,N);    % Will produce N That's ok N Column random number matrix 
A=round(B);     % Round the resulting matrix , You can get non  0  namely  1  Matrix 

end

summary

   The above is the content and code of the sparse base to be shown , If you have other commonly used sparse bases , Welcome to comment and share below ~