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 array
im = 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, roy@mail.ustc.edu.cn, 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 image
img=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 column


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_cosamp(img_cs_1d(:,i),Theta_1d,height); sparse_rec_1d(:,i)=column_rec';           % sparse representation
end
img_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)/4

n=length(y);                           % length of measurements
s=floor(n/4);                                 % sparsity                  
r_n=y;                                 % initial residuals

sig_pos_lt=[];                         % significant pos for last time iteration

for 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.