12. y = y';%y should be a column vector
13. end
14. [M,N] = size(A);%传感矩阵A为M*N矩阵
15. theta = zeros(N,1);%用来存储恢复的theta(列向量)
16. At = zeros(M,t);%用来迭代过程中存储A被选择的列
17. Pos_theta = zeros(1,t);%用来迭代过程中存储A被选择的列序号
18. r_n = y;%初始化残差(residual)为y
19. for ii=1:
t%迭代t次,t为输入参数
20. product = A'*r_n;%传感矩阵A各列与残差的内积
21. [val,pos] = max(abs(product));%找到最大内积绝对值,即与残差最相关的列
22. At(:
ii) = A(:
pos);%存储这一列
23. Pos_theta(ii) = pos;%存储这一列的序号
24. A(:
pos) = zeros(M,1);%清零A的这一列,其实此行可以不要,因为它与残差正交
25. %y=At(:
1:
ii)*theta,以下求theta的最小二乘解(Least Square)
26. theta_ls = (At(:
1:
ii)'*At(:
1:
ii))^(-1)*At(:
1:
ii)'*y;%最小二乘解
27. %At(:
1:
ii)*theta_ls是y在At(:
1:
ii)列空间上的正交投影
28. r_n = y - At(:
1:
ii)*theta_ls;%更新残差
29. end
30. theta(Pos_theta)=theta_ls;%恢复出的theta
31.end
3、OMP单次重构测试代码(CS_Reconstuction_Test.m)
代码中,直接构造一个K稀疏的信号,所以稀疏矩阵为单位阵。
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1.%压缩感知重构算法测试
2.clear all;close all;clc;
3.M = 64;%观测值个数
4.N = 256;%信号x的长度
5.K = 10;%信号x的稀疏度
6.Index_K = randperm(N);
7.x = zeros(N,1);
8.x(Index_K(1:
K)) = 5*randn(K,1);%x为K稀疏的,且位置是随机的
9.Psi = eye(N);%x本身是稀疏的,定义稀疏矩阵为单位阵x=Psi*theta
10.Phi = randn(M,N);%测量矩阵为高斯矩阵
11.A = Phi * Psi;%传感矩阵
12.y = Phi * x;%得到观测向量y
13.%% 恢复重构信号x
14.tic
15.theta = CS_OMP(y,A,K);
16.x_r = Psi * theta;% x=Psi * theta
17.toc
18.%% 绘图
19.figure;
20.plot(x_r,'k.-');%绘出x的恢复信号
21.hold on;
22.plot(x,'r');%绘出原信号x
23.hold off;
24.legend('Recovery','Original')
25.fprintf('\n恢复残差:
');
26.norm(x_r-x)%恢复残差
运行结果如下:
(信号为随机生成,所以每次结果均不一样)
1)图:
2)CommandWindows
Elapsedtimeis0.849710seconds.
恢复残差:
ans=
5.5020e-015
4、测量数M与重构成功概率关系曲线绘制例程代码
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1.%压缩感知重构算法测试CS_Reconstuction_MtoPercentage.m
2.% 绘制参考文献中的Fig.1
3.% 参考文献:
Joel A. Tropp and Anna C. Gilbert
4.% Signal Recovery From Random Measurements Via Orthogonal Matching
5.% Pursuit,IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 12,
6.% DECEMBER 2007.
7.% Elapsed time is 1171.606254 seconds.(@20150418night)
8.clear all;close all;clc;
9.%% 参数配置初始化
10.CNT = 1000;%对于每组(K,M,N),重复迭代次数
11.N = 256;%信号x的长度
12.Psi = eye(N);%x本身是稀疏的,定义稀疏矩阵为单位阵x=Psi*theta
13.K_set = [4,12,20,28,36];%信号x的稀疏度集合
14.Percentage = zeros(length(K_set),N);%存储恢复成功概率
15.%% 主循环,遍历每组(K,M,N)
16.tic
17.for kk = 1:
length(K_set)
18. K = K_set(kk);%本次稀疏度
19. M_set = K:
5:
N;%M没必要全部遍历,每隔5测试一个就可以了
20. PercentageK = zeros(1,length(M_set));%存储此稀疏度K下不同M的恢复成功概率
21. for mm = 1:
length(M_set)
22. M = M_set(mm);%本次观测值个数
23. P = 0;
24. for cnt = 1:
CNT %每个观测值个数均运行CNT次
25. Index_K = randperm(N);
26. x = zeros(N,1);
27. x(Index_K(1:
K)) = 5*randn(K,1);%x为K稀疏的,且位置是随机的
28. Phi = randn(M,N);%测量矩阵为高斯矩阵
29. A = Phi * Psi;%传感矩阵
30. y = Phi * x;%得到观测向量y
31. theta = CS_OMP(y,A,K);%恢复重构信号theta
32. x_r = Psi * theta;% x=Psi * theta
33. if norm(x_r-x)<1e-6%如果残差小于1e-6则认为恢复成功
34. P = P + 1;
35. end
36. end
37. PercentageK(mm) = P/CNT*100;%计算恢复概率
38. end
39. Percentage(kk,1:
length(M_set)) = PercentageK;
40.end
41.toc
42.save MtoPercentage1000 %运行一次不容易,把变量全部存储下来
43.%% 绘图
44.S = ['-ks';'-ko';'-kd';'-kv';'-k*'];
45.figure;
46.for kk = 1:
length(K_set)
47. K = K_set(kk);
48. M_set = K:
5:
N;
49. L_Mset = length(M_set);
50. plot(M_set,Percentage(kk,1:
L_Mset),S(kk,:
));%绘出x的恢复信号
51. hold on;
52.end
53.hold off;
54.xlim([0 256]);
55.legend('K=4','K=12','K=20','K=28','K=36');
56.xlabel('Number of measurements(M)');
57.ylabel('Percentage recovered');
58.title('Percentage of input signals recovered correctly(N=256)(Gaussian)');
本程序在联想ThinkPadE430C笔记本(4GBDDR3内存,i5-3210)上运行共耗时1171.606254秒,程序中将所有数据均通过“saveMtoPercentage1000”存储了下来,以后可以再对数据进行分析,只需“loadMtoPercentage1000”即可。
程序运行结果比文献[1]的Fig.1要好,原因不详。
本程序运行结果:
文献[1]中的Fig.1:
5、信号稀疏度K与重构成功概率关系曲线绘制例程代码
[plain] viewplaincopy
1.%压缩感知重构算法测试CS_Reconstuction_KtoPercentage.m
2.% 绘制参考文献中的Fig.2
3.% 参考文献:
Joel A. Tropp and Anna C. Gilbert
4.% Signal Recovery From Random Measurements Via Orthogonal Matching
5.% Pursuit,IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 12,
6.% DECEMBER 2007.
7.% Elapsed time is 1448.966882 seconds.(@20150418night)
8.clear all;close all;clc;
9.%% 参数配置初始化
10.CNT = 1000;%对于每组(K,M,N),重复迭代次数
11.N = 256;%信号x的长度
12.Psi = eye(N);%x本身是稀疏的,定义稀疏矩阵为单位阵x=Psi*theta
13.M_set = [52,100,148,196,244];%测量值集合
14.Percentage = zeros(length(M_set),N);%存储恢复成功概率
15.%% 主循环,遍历每组(K,M,N)
16.tic
17.for mm = 1:
length(M_set)
18. M = M_set(mm);%本次测量值个数
19. K_set = 1:
5:
ceil(M/2);%信号x的稀疏度K没必要全部遍历,每隔5测试一个就可以了
20. PercentageM = zeros(1,length(K_set));%存储此测量值M下不同K的恢复成功概率
21. for kk = 1:
length(K_set)
22. K = K_set(kk);%本次信号x的稀疏度K
23. P = 0;
24. for cnt = 1:
CNT %每个观测值个数均运行CNT次
25. Index_K = randperm(N);
26. x = zeros(N,1);
27. x(Index_K(1:
K)) = 5*randn(K,1);%x为K稀疏的,且位置是随机的
28. Phi = randn(M,N);%测量矩阵为高斯矩阵
29. A = Phi * Psi;%传感矩阵
30. y = Phi * x;%得到观测向量y
31. theta = CS_OMP(y,A,K);%恢复重构信号theta
32. x_r = Psi * theta;% x=Psi * theta
33. if norm(x_r-x)<1e-6%如果残差小于1e-6则认为恢复成功
34. P = P + 1;
35. end
36. end
37. PercentageM(kk) = P/CNT*100;%计算恢复概率
38. end
39. Percentage(mm,1:
length(K_set)) = PercentageM;
40.end
41.toc
42.save KtoPercentage1000test %运行一次不容易,把变量全部存储下来
43.%% 绘图
44.S = ['-ks';'-ko';'-kd';'-kv';'-k*'];
45.figure;
46.for mm = 1:
length(M_set)
47. M = M_set(mm);
48. K_set = 1:
5:
ceil(M/2);
49. L_Kset = length(K_set);
50. plot(K_set,Percentage(mm,1:
L_Kset),S(mm,:
));%绘出x的恢复信号
51. hold on;
52.end
53.hold off;
54.xlim([0 125]);
55.legend('M=52','M=100','M=148','M=196','M=244');
56.xlabel('Sparsity level(K)');
57.ylabel('Percentage recovered');
58.title('Percentage of input signals recovered correctly(N=256)(Gaussian)');
本程序在联想ThinkPadE430C笔记本(4GBDDR3内存,i5-3210)上运行共耗时1448.966882秒,程序中将所有数据均通过“saveKtoPercentage1000”存储了下来,以后可以再对数据进行分析,只需“loadKtoPercentage1000”即可。
程序运行结果比文献[1]的Fig.2要好,原因不详。
本程序运行结果:
文献[1]中的Fig.2:
6、参考文献
【1】JoelA.TroppandAnnaC.Gilbert.SignalRecoveryFromRandomMeasurementsViaOrthogonalMatchingPursuit[J].IEEETransactionsonInformationTheory,VOL.53,NO.12,DECEMBER2007.
【2】Y.C.Pati,R.Rezaiifar,andP.S.Krishnaprasad.OrthogonalMatchingPursuit-RecursiveFunctionApproximationwithApplicationstowaveletdecomposition,Proc.27thAnnu.AsilomarConf.Signals,Systems,andComputers,PacificGrove,CA,Nov.1993,vol.1,pp40-44.
【3】沙威.CS_OMP,http:
//www.eee.hku.hk/~wsha/Freecode/Files/CS_OMP.zip