├── Measurement matrixs
    ├── BernoulliMtx.m
    ├── CirculantMtx.m
    ├── PartFourierMtx.m
    ├── PartHadamardMtx.m
    ├── SparseRandomMtx.m
    ├── ToeplitzMtx.m
    ├── other simple matrixs.txt
    └── 本文件夹中均为测量矩阵的实现.txt
├── README.md
├── Reconstruction algorithms
    ├── BP_linprog.m
    ├── CS_CoSaMP.m
    ├── CS_GBP.m
    ├── CS_IHT.m
    ├── CS_IRLS.m
    ├── CS_OMP.m
    ├── CS_ROMP.m
    ├── CS_SAMP.m
    ├── CS_SP.m
    └── 本文件夹中均为重构算法的实现.txt
├── Refactoring articles
    ├── CS Recovery Algorithms.pdf
    ├── CoSaMP  Iterative signal recovery from incomplete and inaccurate samples.pdf
    ├── Compressed sensing.pdf
    ├── IHT Iterative hard thresholding for compressed sensing.pdf
    ├── OMP Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit.pdf
    ├── RIHT Robust iterative hard thresholding for compressed sensing.pdf
    ├── RIP The restricted isometry property and its implications for compressed sensing.pdf
    ├── ROMP Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit.pdf
    ├── SAMP  Sparsity Adaptive Matching Pursuit Algotithm For Practical Compressive Sensing.pdf
    ├── SP Subspace Pursuit for Compressive Sensing Signal Reconstruction.pdf
    ├── Signal Space CoSaMP for Sparse Recovery With Redundant Dictionaries With Redundant  Dictionary.pdf
    ├── StOMP.pdf
    └── 本文件夹中大多为重构相关文章.txt
├── Some examples
    ├── Demo_CS_BP.m
    ├── Demo_CS_CoSaMP.m
    ├── Demo_CS_CoSaMP_2.m
    ├── Demo_CS_GBP.m
    ├── Demo_CS_IHT.m
    ├── Demo_CS_IHT_2.m
    ├── Demo_CS_IRLS.m
    ├── Demo_CS_OMP.m
    ├── Demo_CS_OMP_2.m
    ├── Demo_CS_ROMP.m
    ├── Demo_CS_SAMP.m
    ├── Demo_CS_SP.m
    ├── Demo_CS_SP_2.m
    ├── lena.bmp
    └── 一些一维和二维信号CS的Demo.txt
└── Sparse basis
    ├── DHartleyTmtx.m
    ├── DWT.m
    ├── DWangTmtx.m
    ├── other simple basis.txt
    └── 本文件夹中是为稀疏基的实现.txt


/Measurement matrixs/BernoulliMtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/BernoulliMtx.m


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/Measurement matrixs/CirculantMtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/CirculantMtx.m


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/Measurement matrixs/PartFourierMtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/PartFourierMtx.m


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/Measurement matrixs/PartHadamardMtx.m:
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/Measurement matrixs/SparseRandomMtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/SparseRandomMtx.m


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/Measurement matrixs/ToeplitzMtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/ToeplitzMtx.m


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/Measurement matrixs/other simple matrixs.txt:
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1 | 
2 | %% 注：以下均是 matlab 程序，并产生的为压缩感知计算中的 测量矩阵 
3 | %% 其中， N 、M 为测量矩阵的尺寸，N 的值为原信号长度
4 | 
5 | 高斯矩阵
6 |  Phi = randn(M,N);


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/Measurement matrixs/本文件夹中均为测量矩阵的实现.txt:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Measurement matrixs/本文件夹中均为测量矩阵的实现.txt


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/README.md:
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 1 | # Compressed-sensing-code
 2 | 
 3 | 此库中保存些许与压缩感知相关参数、重构的方法等代码，具体说明如下 <br>
 4 | 
 5 | - Measurement matrixs ：存有多种测量矩阵的实现代码
 6 | - Reconstruction algorithms ：存有多种重构算法的实现，包括最经典的 OMP、SAMP 等方法
 7 | - Refactoring articles ：保存了 CS 中多种重构算法的解释文件
 8 | - Some examples ：CS的实验案例
 9 | - Sparse basis ：保存了典型稀疏基的实现代码
10 | 
11 | 注意：此中代码多为 `.m` 文件，且每个路径中都有相关的txt文件说明各自所实现的内容！
12 | 


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/Reconstruction algorithms/BP_linprog.m:
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 1 | function [ alpha ] = BP_linprog( s,Phi )
 2 | %BP_linprog(Basis Pursuit with linprog) Summary of this function goes here
 3 | %Version: 1.0 written by jbb0523 @2016-07-21 
 4 | %Reference:Chen S S, Donoho D L, Saunders M A. Atomic decomposition by
 5 | %basis pursuit[J]. SIAM review, 2001, 43(1): 129-159.(Available at: 
 6 | %http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.37.4272&rep=rep1&type=pdf)
 7 | %   Detailed explanation goes here
 8 | %   s = Phi * alpha (alpha is a sparse vector)  
 9 | %   Given s & Phi, try to derive alpha
10 |     [s_rows,s_columns] = size(s);  
11 |     if s_rows<s_columns  
12 |         s = s';%s should be a column vector  
13 |     end 
14 |     p = size(Phi,2);
15 |     %according to section 3.1 of the reference
16 |     c = ones(2*p,1);
17 |     A = [Phi,-Phi];
18 |     b = s;
19 |     lb = zeros(2*p,1);
20 |     x0 = linprog(c,[],[],A,b,lb);
21 |     alpha = x0(1:p) - x0(p+1:2*p);
22 | end
23 | 


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/Reconstruction algorithms/CS_CoSaMP.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Reconstruction algorithms/CS_CoSaMP.m


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/Reconstruction algorithms/CS_GBP.m:
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  1 | function hat_x=CS_GBP(y,T_Mat,m)
  2 | % y=T_Mat*x, T_Mat is n-by-m
  3 | % y - measurements
  4 | % T_Mat - combination of random matrix and sparse representation basis,A
  5 | % m - size of the original signal
  6 | % the sparsity is length(y)/4
  7 | 
  8 | cnt=length(y); 
  9 | 
 10 | x=y';                         % x is row vector
 11 | D=[T_Mat,-T_Mat]';            % each row of D is a atom
 12 | 
 13 | 
 14 | OPT_VERBOSE_ON = 0; % commentary level
 15 | epsilon = 1.0e-6; % error tolerance
 16 | [n, d] = size(D); % problem size
 17 | pinv_epsilon = 1.0e-12; %% epsilon for PINV_ITER
 18 | exit_flag = 0; % default exit_flag
 19 | 
 20 | % Compute the initial atom
 21 | t = 1; % iteration index
 22 | [alpha_0, k] = max(D*x');
 23 | i_list = [k];
 24 | alpha_list = [alpha_0];
 25 | n_atoms = length(i_list); % number of atoms in the current representation
 26 | normal = x / sqrt(sum(x.^2)); % the normal to the hyperplane
 27 | xApprox = alpha_0*D(k,:); % the current approximation to x
 28 | d_H = dot(D(k,:), normal); % the distance of the hyperplane to the origin
 29 | xApprox_H = (d_H / dot(xApprox, normal))*xApprox; % xApprox projected onto the hyperplane
 30 | err_vec = x - xApprox; %
 31 | v = err_vec - dot(err_vec, normal)*normal;
 32 | v = v / sqrt(sum(v.^2));
 33 | %
 34 | Psi = [D(k,:)];
 35 | Psi_biortho = Psi;
 36 | 
 37 | % Iterate
 38 | for times=1:cnt/4
 39 |     % Exit when error is acceptable
 40 |     error = sqrt(sum(err_vec.^2));
 41 |     if OPT_VERBOSE_ON,
 42 |         fprintf('t = %i; i_t = %i; nAtoms = %i; error = %f\n', t, k, n_atoms, error);
 43 |     end
 44 | 
 45 |     if error < epsilon,
 46 |         fprintf('... GBP complete.\n');
 47 |         break;
 48 |     elseif isnan(error),
 49 |         fprintf('GBP -- Error: Exiting.\n');
 50 |         exit_flag = -2;
 51 |     end
 52 | 
 53 |     % Exit when number of atoms equals dimension
 54 |     if n_atoms==d,
 55 |         fprintf('Basis found.\n');
 56 |         break;
 57 |     end
 58 |     % Otherwise, next iteration
 59 |     t = t + 1;
 60 | 
 61 |     % Project atoms onto n-v plane
 62 |     indices = setdiff(1:n, i_list);  %delete the elements of i_list from 1:n
 63 |     D_n0 = D(indices,:) * normal';  % select the next atom from the residuals
 64 |     D_v0 = D(indices,:) * v';
 65 |     D_n = D_n0 - repmat(dot(xApprox_H, normal), length(indices), 1);
 66 |     D_v = D_v0 - repmat(dot(xApprox_H, v), length(indices), 1);
 67 |     % Compute angle
 68 |     D_theta = atan2(-D_n, D_v);
 69 |     D_theta = D_theta + (D_theta < 0)*2*pi; % re-center for min()
 70 | 
 71 |     % Order by angle and select the next intersected atom
 72 |     [junk, i_Dsorted] = sort(D_theta);
 73 |     k = indices(i_Dsorted(1));
 74 |     psi_k = D(k,:);
 75 | 
 76 |     % Compute auxiliary variables and coefficients
 77 |     % Compute orthogonal component of psi_k
 78 |     psi_k_ortho = psi_k;
 79 |     for i = 1:n_atoms,
 80 |         beta(i) = dot(psi_k, Psi_biortho(i,:));
 81 |         psi_k_ortho = psi_k_ortho - beta(i)*Psi(i,:);
 82 |     end
 83 |     % Compute new biorthogonal vector and adjust existing biorthogonal vectors
 84 |     psi_k_biortho = psi_k_ortho / sum(psi_k_ortho.^2);
 85 |     for i = 1:n_atoms,
 86 |         Psi_biortho(i,:) = Psi_biortho(i,:) - beta(i)*psi_k_biortho;
 87 |     end
 88 |     % Compute new coefficient and adjust existing coefficients
 89 |     alpha_k = dot(x, psi_k_biortho);
 90 |     for i = 1:n_atoms,
 91 |         alpha_list(i) = alpha_list(i) - beta(i)*alpha_k;
 92 |     end
 93 | 
 94 |     % Check if the new representation is convex (with respect to the new coefficient)
 95 |     % If not, there is a numerical error.
 96 |     if alpha_k < 0,
 97 |         fprintf('GBP -- Error: Hyperplane rotation failed. Exiting. \n');
 98 |         exit_flag = -1;
 99 |         return;
100 |     end
101 | 
102 |     % Update lists
103 |     Psi = [Psi; psi_k];
104 |    
105 |     Psi_biortho = [Psi_biortho; psi_k_biortho];
106 |     alpha_list = [alpha_list, alpha_k];
107 |     i_list = [i_list, k];
108 |     n_atoms = n_atoms + 1;
109 | 
110 |     %Test biorthogonality
111 |     AAplus = Psi*Psi_biortho';
112 |     Ierr = max(max(abs(AAplus - eye(size(AAplus)))));
113 |     if Ierr > epsilon,
114 |         % run pinv_iter with epsilon=epsilon and c=1
115 |         X = Psi_biortho';
116 |         while Ierr > epsilon,
117 |             X = X * (2*eye(size(AAplus)) - AAplus);
118 |             AAplus = Psi*X;
119 |             Ierr = max(max(abs(AAplus - eye(size(AAplus)))));
120 |             %fprintf('... pinv: I error: %f\n', Ierr);
121 |         end
122 |         % pinv_iter complete
123 |         Psi_biortho = X';
124 |         alpha_list = (Psi_biortho*x')';
125 |     end
126 | 
127 |     % Test if conv(Psi) contains extraneous atoms,
128 |     % otherwise, continue wrapping
129 |     while sum(alpha_list<=0)~=0,
130 |         if OPT_VERBOSE_ON,
131 |             fprintf('... Deleting atom. \n');
132 |         end
133 |         % Find an extraneous atom
134 |         [alpha_j, j] = min(alpha_list);
135 |         % Delete it
136 |         psi_j_biortho = Psi_biortho(j,:);
137 |         gamma = Psi_biortho*psi_j_biortho' / sum(psi_j_biortho.^2);
138 |         Psi = Psi([1:j-1,j+1:n_atoms],:);
139 |         
140 |         Psi_biortho = Psi_biortho - gamma*psi_j_biortho;
141 |         Psi_biortho = Psi_biortho([1:j-1,j+1:n_atoms],:);
142 | 
143 |         % Test biorthogonality
144 |         AAplus = Psi*Psi_biortho';
145 |         Ierr = max(max(abs(AAplus - eye(size(AAplus)))));
146 |         if Ierr > epsilon,
147 |             % run pinv_iter with epsilon=epsilon and c=1
148 |             X = Psi_biortho';
149 |             while Ierr > epsilon,
150 |                 X = X * (2*eye(size(AAplus)) - AAplus);
151 |                 AAplus = Psi*X;
152 |                 Ierr = max(max(abs(AAplus - eye(size(AAplus)))));
153 |                 %fprintf('... pinv: I error: %f\n', Ierr);
154 |             end
155 |             % pinv_iter complete
156 |             Psi_biortho = X';
157 |             alpha_list = (Psi_biortho*x')';
158 |         else
159 |             alpha_list = alpha_list - alpha_j*gamma';
160 |             alpha_list = alpha_list([1:j-1,j+1:n_atoms]);
161 |         end
162 | 
163 |         % Update the representation
164 |         i_list = i_list([1:j-1,j+1:n_atoms]);
165 |         n_atoms = length(i_list);
166 |     end
167 | 
168 |     % Compute the new approximation
169 |     xApprox = alpha_list * Psi;
170 |     % Next hyperplane
171 |     normal = (-D_n(i_Dsorted(1)))*v + (D_v(i_Dsorted(1)))*normal;
172 |     normal = normal / sqrt(sum(normal.^2));
173 |     d_H = dot(D(i_list(1),:), normal);
174 |     xApprox_H = (d_H / dot(xApprox, normal))*xApprox;
175 |     err_vec = x - xApprox;
176 |     v = err_vec - dot(err_vec, normal)*normal; 
177 |     v = v / sqrt(sum(v.^2));
178 | end
179 | 
180 | hat_x=zeros(1,size(D,1));
181 | hat_x(i_list)=alpha_list;         % recovered sparse representation
182 | end


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/Reconstruction algorithms/CS_IHT.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Reconstruction algorithms/CS_IHT.m


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/Reconstruction algorithms/CS_IRLS.m:
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 1 | function hat_x=CS_IRLS(y,T_Mat,m)
 2 | % y=T_Mat*x, T_Mat is n-by-m
 3 | % y - measurements
 4 | % T_Mat - combination of random matrix and sparse representation basis
 5 | % m - size of the original signal
 6 | % the sparsity is length(y)/4
 7 | 
 8 | %hat_x_tp=(T_Mat'*T_Mat)^(-1)*T_Mat'*y; % initialization, leading to warning
 9 | hat_x_tp=T_Mat'*y; 
10 | 
11 | epsilong=1;
12 | 
13 | p=1;                                   % solution for l-norm p
14 | times=1;
15 | while (epsilong>10e-9) && (times<length(y)/4)
16 |     
17 |     weight=(hat_x_tp.^2+epsilong).^(p/2-1); 
18 |     Q_Mat=diag(1./weight,0);
19 |     
20 |     hat_x=Q_Mat*T_Mat'*inv(T_Mat*Q_Mat*T_Mat')*y;
21 |     
22 |     if(norm(hat_x-hat_x_tp,2) < sqrt(epsilong)/100)
23 |         epsilong=epsilong/10;
24 |     end
25 |     
26 |     hat_x_tp=hat_x;
27 |     times=times+1;
28 | end


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/Reconstruction algorithms/CS_OMP.m:
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/Reconstruction algorithms/本文件夹中均为重构算法的实现.txt:
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/Refactoring articles/CS Recovery Algorithms.pdf:
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/Refactoring articles/CoSaMP  Iterative signal recovery from incomplete and inaccurate samples.pdf:
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/Refactoring articles/Compressed sensing.pdf:
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/Refactoring articles/IHT Iterative hard thresholding for compressed sensing.pdf:
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/Refactoring articles/OMP Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit.pdf:
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/Refactoring articles/RIHT Robust iterative hard thresholding for compressed sensing.pdf:
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/Refactoring articles/RIP The restricted isometry property and its implications for compressed sensing.pdf:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Refactoring articles/RIP The restricted isometry property and its implications for compressed sensing.pdf


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/Refactoring articles/ROMP Signal Recovery From Incomplete and Inaccurate Measurements Via Regularized Orthogonal Matching Pursuit.pdf:
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/Refactoring articles/SAMP  Sparsity Adaptive Matching Pursuit Algotithm For Practical Compressive Sensing.pdf:
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/Refactoring articles/SP Subspace Pursuit for Compressive Sensing Signal Reconstruction.pdf:
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/Refactoring articles/Signal Space CoSaMP for Sparse Recovery With Redundant Dictionaries With Redundant  Dictionary.pdf:
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/Refactoring articles/StOMP.pdf:
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/Refactoring articles/本文件夹中大多为重构相关文章.txt:
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/Some examples/Demo_CS_BP.m:
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/Some examples/Demo_CS_CoSaMP.m:
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/Some examples/Demo_CS_CoSaMP_2.m:
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/Some examples/Demo_CS_ROMP.m:
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/Some examples/Demo_CS_SAMP.m:
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/Some examples/Demo_CS_SP.m:
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/Some examples/Demo_CS_SP_2.m:
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/Some examples/lena.bmp:
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/Some examples/一些一维和二维信号CS的Demo.txt:
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/Sparse basis/DHartleyTmtx.m:
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/Sparse basis/DWT.m:
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/Sparse basis/DWangTmtx.m:
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https://raw.githubusercontent.com/tyl-stu/Compressed-sensing-code/00415dd00972e40f59eb122428c4ff2a14ce5261/Sparse basis/DWangTmtx.m


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/Sparse basis/other simple basis.txt:
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 1 | 
 2 | %% 注：以下均是 matlab 程序，其中 N 为信号长度
 3 | 
 4 | 1、傅里叶变换基  
 5 |  Psi = fft(eye(N))/sqrt(N);
 6 | 
 7 | 2、余弦稀疏基
 8 |  Psi = dctmtx(N);
 9 | 
10 | 3、离散傅里叶变换基
11 |  Psi = dftmtx(N)/sqrt(N);
12 | 
13 | 4、单位阵
14 | Psi = eye(N);


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/Sparse basis/本文件夹中是为稀疏基的实现.txt:
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