压缩感知重构算法之IHT算法python实现

IHT（iterative hard thresholding ）算法是压缩感知中一种非常重要的贪婪算法，它具有算法简单的有点，且易于实现，在实际中应用较多。本文给出了IHT算法的python和matlab代码（本文给出的代码未经过优化，所以重建质量不是非常好），以及完整的仿真过程。

算法流程

python代码

要利用python实现，电脑必须安装以下程序

• python （本文用的python版本为3.5.1）
• numpy python包（本文用的版本为1.10.4）
• scipy python包（本文用的版本为0.17.0）
• pillow python包（本文用的版本为3.1.1）

#coding:utf-8#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%# DCT基作为稀疏基，重建算法为IHT算法,图像按列进行处理#  参考文献: Carrillo R E, Polania L F, Barner K E. Iterative hard thresholding for compressed sensing #with partially known support[C]#//Acoustics, Speech and Signal Processing (ICASSP), #2011 IEEE International Conference on. IEEE, 2011: 4028-4031.# #%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%#导入集成库import math# 导入所需的第三方库文件import  numpy as np    #对应numpy包from PIL import Image  #对应pillow包#读取图像，并变成numpy类型的 arrayim = np.array(Image.open('lena.bmp'))#图片大小256*256#生成高斯随机测量矩阵sampleRate=0.7  #采样率Phi=np.random.randn(256,256)u, s, vh = np.linalg.svd(Phi)Phi = u[:256*sampleRate,] #将测量矩阵正交化#生成稀疏基DCT矩阵mat_dct_1d=np.zeros((256,256))v=range(256)for k in range(0,256):      dct_1d=np.cos(np.dot(v,k*math.pi/256))    if k>0:        dct_1d=dct_1d-np.mean(dct_1d)    mat_dct_1d[:,k]=dct_1d/np.linalg.norm(dct_1d)#随机测量img_cs_1d=np.dot(Phi,im)#IHT算法函数def cs_IHT(y,D):        K=math.floor(y.shape[0]/3)  #稀疏度        result_temp=np.zeros((256))  #初始化重建信号       u=0.5  #影响因子    result=result_temp    for j in range(K):  #迭代次数        x_increase=np.dot(D.T,(y-np.dot(D,result_temp)))    #x=D*(y-D*y0)        result=result_temp+np.dot(x_increase,u) #   x(t+1)=x(t)+D*(y-D*y0)        temp=np.fabs(result)        pos=temp.argsort()         pos=pos[::-1]#反向，得到前面L个大的位置        result[pos[K:]]=0        result_temp=result           return  result#重建sparse_rec_1d=np.zeros((256,256))   # 初始化稀疏系数矩阵    Theta_1d=np.dot(Phi,mat_dct_1d)   #测量矩阵乘上基矩阵for i in range(256):    print('正在重建第',i,'列。。。')    column_rec=cs_IHT(img_cs_1d[:,i],Theta_1d)  #利用IHT算法计算稀疏系数    sparse_rec_1d[:,i]=column_rec;        img_rec=np.dot(mat_dct_1d,sparse_rec_1d)          #稀疏系数乘上基矩阵#显示重建后的图片image2=Image.fromarray(img_rec)image2.show()

matlab代码

%代码在matlab2010b测试通过function Demo_CS_IHT()%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% the DCT basis is selected as the sparse representation dictionary% instead of seting the whole image as a vector, I process the image in the% fashion of column-by-column, so as to reduce the complexity.% Author: Chengfu Huo, [e-mail], http://home.ustc.edu.cn/~roy% Reference: T. Blumensath and M. Davies, “Iterative Hard Thresholding for% Compressed Sensing,” 2008.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------------ read in the image --------------img=imread('lena.bmp');     % 256*256大小img=double(img);[height,width]=size(img);sampleRate=0.7; %采样率%------------ form the measurement matrix and base matrix ---------------%Phi=randn(floor(height/3),width);  % only keep one third of the original data  %Phi = Phi./repmat(sqrt(sum(Phi.^2,1)),[floor(height/3),1]); % normalize each columnPhi = orth(rand(256, 256));Phi=Phi(1:256*sampleRate, :);mat_dct_1d=zeros(256,256);  % building the DCT basis (corresponding to each column)for k=0:1:255     dct_1d=cos([0:1:255]'*k*pi/256);    if k>0        dct_1d=dct_1d-mean(dct_1d);     end;    mat_dct_1d(:,k+1)=dct_1d/norm(dct_1d);end%--------- projection ---------img_cs_1d=Phi*img;          % treat each column as a independent signal%-------- recover using omp ------------sparse_rec_1d=zeros(height,width);            Theta_1d=Phi*mat_dct_1d;for i=1:width    column_rec=cs_iht(img_cs_1d(:,i),Theta_1d,height);    sparse_rec_1d(:,i)=column_rec';           % sparse representationendimg_rec_1d=mat_dct_1d*sparse_rec_1d;          % inverse transform%------------ show the results --------------------figure(1)subplot(2,2,1),imagesc(img),title('original image')subplot(2,2,2),imagesc(Phi),title('measurement mat')subplot(2,2,3),imagesc(mat_dct_1d),title('1d dct mat')psnr = 20*log10(255/sqrt(mean((img(:)-img_rec_1d(:)).^2)));subplot(2,2,4),imshow(uint8(img_rec_1d));title(strcat('PSNR=',num2str(psnr),'dB'));disp('over')%************************************************************************%function hat_x=cs_iht(y,T_Mat,m)% y=T_Mat*x, T_Mat is n-by-m% y - measurements% T_Mat - combination of random matrix and sparse representation basis% m - size of the original signal% the sparsity is length(y)/4hat_x_tp=zeros(m,1);         % initialization with the size of original s=floor(length(y)/4);        % sparsityu=0.5;                       % impact factor% T_Mat=T_Mat/sqrt(sum(sum(T_Mat.^2))); % normalizae the whole matrixfor times=1:s    x_increase=T_Mat'*(y-T_Mat*hat_x_tp);    hat_x=hat_x_tp+u*x_increase;    [val,pos]=sort((hat_x),'descend');  % why? worse performance with abs()    hat_x(pos(s+1:end))=0;   % thresholding, keeping the larges s elements    hat_x_tp=hat_x;          % updateend

参考文章

1、Carrillo R E, Polania L F, Barner K E. Iterative hard thresholding for compressed sensing with partially known support[C]//Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference on. IEEE, 2011: 4028-4031.