压缩感知重构算法之CoSaMP算法python实现

算法流程

算法分析

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CoSaMP算法,图像按列进行处理# 参考文献: D. Deedell andJ. Tropp, “COSAMP: Iterative Signal Recovery from#Incomplete and Inaccurate Samples,” 2008.#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%#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=0.5  #采样率Phi=np.random.randn(256*sampleRate,256)# 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)#CoSaMP算法函数def cs_CoSaMP(y,D):         S=math.floor(y.shape[0]/4)  #稀疏度        residual=y  #初始化残差    pos_last=np.array([],dtype=np.int64)    result=np.zeros((256))    for j in range(S):  #迭代次数        product=np.fabs(np.dot(D.T,residual))               pos_temp=np.argsort(product)        pos_temp=pos_temp[::-1]#反向，得到前面L个大的位置        pos_temp=pos_temp[0:2*S]#对应步骤3        pos=np.union1d(pos_temp,pos_last)           result_temp=np.zeros((256))        result_temp[pos]=np.dot(np.linalg.pinv(D[:,pos]),y)        pos_temp=np.argsort(np.fabs(result_temp))        pos_temp=pos_temp[::-1]#反向，得到前面L个大的位置        result[pos_temp[:S]]=result_temp[pos_temp[:S]]        pos_last=pos_temp        residual=y-np.dot(D,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_CoSaMP(img_cs_1d[:,i],Theta_1d)  #利用CoSaMP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

function Demo_CS_CoSaMP()%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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: D. Deedell andJ. Tropp, “COSAMP: Iterative Signal Recovery from% Incomplete and Inaccurate Samples,” 2008.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------------ read in the image --------------img=imread('lena.bmp');     % testing imageimg=double(img);[height,width]=size(img);%------------ form the measurement matrix and base matrix ---------------Phi=randn(floor(height/2),width);  % only keep one third of the original data  Phi = Phi./repmat(sqrt(sum(Phi.^2,1)),[floor(height/2),1]); % normalize each columnmat_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_cosamp(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_cosamp(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)/4n=length(y);                           % length of measurementss=floor(n/4);                                 % sparsity                  r_n=y;                                 % initial residualssig_pos_lt=[];                         % significant pos for last time iterationfor times=1:s                          % number of iterations    product=abs(T_Mat'*r_n);    [val,pos]=sort(product,'descend');    sig_pos_cr=pos(1:2*s);             % significant pos for curretn iteration    sig_pos=union(sig_pos_cr,sig_pos_lt);    Aug_t=T_Mat(:,sig_pos);            % current selected entries of T_Mat     aug_x_cr=zeros(m,1);                   aug_x_cr(sig_pos)=(Aug_t'*Aug_t)^(-1)*Aug_t'*y;  % temp recovered x (sparse)    [val,pos]=sort(abs(aug_x_cr),'descend');    hat_x=zeros(1,m);    hat_x(pos(1:s))=aug_x_cr(pos(1:s));% recovered x with s sparsity      sig_pos_lt=pos(1:s);               % refresh the significant positions    r_n=y-T_Mat*hat_x';end

参考文献

1、D. Deedell andJ. Tropp, “COSAMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples,” 2008.