压缩感知（十）

压缩感知仿真验证

一维信号重建实验

clear; 
close all;
 
choice_transform=1;       
choice_Phi=0;             
n = 512;
t = [0: n-1];
f = cos(2*pi/256*t) + sin(2*pi/128*t);   % 
n = length(f);
a = 0.2;                    
m = double(int32(a*n));

switch choice_transform
    case 1
        ft = dct(f);
        disp(\'ft = dct(f)\')
    case 0
        ft = fft(f);
        disp(\'ft = fft(f)\')
end
 
disp([\'ÐÅºÅÏ¡Êè¶È£º\',num2str(length(find((abs(ft))>0.1)))])
figure(\'name\', \'A Tone Time and Frequency Plot\');
subplot(2, 1, 1);
plot(f);
xlabel(\'Time (s)\'); 
% ylabel(\'f(t)\');
subplot(2, 1, 2);
 
switch choice_transform
    case 1
        plot(ft)
        disp(\'plot(ft)\')
    case 0
        plot(abs(ft));
        disp(\'plot(abs(ft))\')
end
xlabel(\'Frequency (Hz)\'); 
% ylabel(\'DCT(f(t))\');

switch choice_Phi
    case 1
        Phi = PartHadamardMtx(m,n);       
    case 0
        Phi = sqrt(1/m) * randn(m,n);    
end
f2 = (Phi * f\')\';       
% f2 = f(1:2:n);
 
switch choice_transform
    case 1
        Psi = dct(eye(n,n));           
        disp(\'Psi = dct(eye(n,n));\')
    case 0
        Psi = inv(fft(eye(n,n)));      
        disp(\'Psi = inv(fft(eye(n,n)));\')
end
 
A = Phi * Psi;                    % A = Phi * Psi
cvx_begin;
    variable x(n) complex;
%     variable x(n) ;
    minimize(norm(x,1));
    subject to
      A*x == f2\';
cvx_end;
 
figure;
subplot(2,1,2);
switch choice_transform
    case 1
        plot(real(x));
        disp(\'plot(real(x))\')
    case 0
        plot(abs(x));
        disp(\'plot(abs(x))\')
end
 
title(\'Using L1 Norm£¨Frequency Domain£©\');
 
%  ylabel(\'DCT(f(t))\'); xlabel(\'Frequency (Hz)\');
switch choice_transform
    case 1
        sig = dct(real(x));
        disp(\'sig = dct(real(x))\')
    case 0
        sig = real(ifft(full(x)));
        disp(\'sig = real(ifft(full(x)))\')
end
subplot(2,1,1);
plot(f)
hold on;plot(sig);hold off
title(\'Using L1 Norm (Time Domain)\');
% ylabel(\'f(t)\'); xlabel(\'Time (s)\');
legend(\'Original\',\'Recovery\')

for K = 1:100
    theta = CS_OMP(f2,A,K);
    %     figure;plot(dct(theta));title([\'K=\',num2str(K)])
    switch choice_transform
        case 1
            re(K) = norm(f\'-(dct(theta)));
        case 0
            re(K) = norm(f\'-real(ifft(full(theta))));
    end
end
theta = CS_OMP(f2,A,find(re==min(min(re))));
disp([\'×î¼ÑÏ¡Êè¶ÈK=\',num2str(find(re==min(min(re))))]);
% theta = CS_OMP(f2,A,10);
figure;subplot(2,1,2);
switch choice_transform
    case 1
        plot(theta);
        disp(\'plot(theta)\')
    case 0
        plot(abs(theta));
        disp(\'plot(abs(theta))\')
end
 
title([\'Using OMP(Frequence Domain)  K=\',num2str(find(re==min(min(re))))])
 
switch choice_transform
    case 1
        sig2 = dct(theta);
        disp(\'sig2 = dct(theta)\')
    case 0
        sig2 = real(ifft(full(theta)));
        disp(\'sig2 = real(ifft(full(theta)))\')
end
 
subplot(2,1,1);plot(f);hold on;
plot(sig2)
hold off;
title([\'Using OMP(Time Domain)  K=\',num2str(find(re==min(min(re))))]);
legend(\'Original\',\'Recovery\')

一维信号仿真结果

如上图所示，为原始信号f = cos(2*pi/256*t) + sin(2*pi/128*t)，及其频域图（频域稀疏）。

取原信号的20%，使用L1范数算法对原信号进行恢复重建，效果如上图所示，蓝色为原始信号，红色为恢复信号。

取原信号的20%，使用OMP算法对原信号进行恢复重建，效果如上图所示，蓝色为原始信号，红色为恢复信号。

取原信号的30%，使用L1范数算法对原信号进行恢复重建，效果如上图所示，蓝色为原始信号，红色为恢复信号。

取原信号的30%，使用OMP算法对原信号进行恢复重建，效果如上图所示，蓝色为原始信号，红色为恢复信号。

小结：压缩感知原本就是为了信号（非图像）采集而生，所以在信号采集上有很强的实用性，甚至只需要原信号10~20%的信息，就可以复原出原信号的大部分特性。