IRLS（iteratively reweighted least squares）演算法

（本文給出的程式碼未進行優化，只是為了說明演算法流程 ，所以執行速度不是很快）
IRLS（iteratively reweighted least squares）演算法是壓縮感知重建演算法當中的一個基本演算法。主要是為了解決

程式碼

要利用python實現，電腦必須安裝以下程式

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

python程式碼

#coding: utf-8
'''
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# DCT基作為稀疏基，重建演算法為OMP演算法 ，影象按列進行處理
# email:
 
[email protected], 
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
'''
#匯入整合庫
import math

# 匯入所需的第三方庫檔案
import  numpy as np    #對應numpy包
from PIL import Image  #對應pillow包


#讀取影象，並變成numpy型別的 array
im = np.array(Image.open('lena.bmp'))
#print (im.shape, im.dtype)uint8

#生成高斯隨機測量矩陣
sampleRate = 0.7
 
  #取樣率
Phi = np.random.randn(256, 256)
u, s, vh = np.linalg.svd(Phi)
Phi = u[:256*sampleRate,] #將測量矩陣正交化

#生成稀疏基DCT矩陣
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)

#隨機測量
img_cs_1d=np.dot(Phi,im)

#IRLS演算法函式
def cs_irls(y,T_Mat):   
    L=math.floor((y.shape[0])/4)
    hat_x_tp=np.dot(T_Mat.T ,y)
    epsilong=1
    p=1 # solution for l-norm p
    times=1
    while (epsilong>10e-9) and (times<L):  #迭代次數
        weight=(hat_x_tp**2+epsilong)**(p/2-1)
        Q_Mat=np.diag(1/weight)
        #hat_x=Q_Mat*T_Mat'*inv(T_Mat*Q_Mat*T_Mat')*y
        temp=np.dot(np.dot(T_Mat,Q_Mat),T_Mat.T)
        temp=np.dot(np.dot(Q_Mat,T_Mat.T),np.linalg.inv(temp))
        hat_x=np.dot(temp,y)        
        if(np.linalg.norm(hat_x-hat_x_tp,2) < np.sqrt(epsilong)/100):
            epsilong = epsilong/10
        hat_x_tp=hat_x
        times=times+1
    return hat_x



#重建
sparse_rec_1d=np.zeros((256,256))   # 初始化稀疏係數矩陣    
Theta_1d=np.dot(Phi,mat_dct_1d)   #測量矩陣乘上基矩陣
for i in range(256):
    print('正在重建第',i,'列。。。')
    column_rec=cs_irls(img_cs_1d[:,i],Theta_1d)  #利用IRLS演算法計算稀疏係數
    sparse_rec_1d[:,i]=column_rec;        
img_rec=np.dot(mat_dct_1d,sparse_rec_1d)          #稀疏係數乘上基矩陣

#顯示重建後的圖片
image2=Image.fromarray(img_rec)
image2.show()

matlab程式碼

%matlab版本用的R2010b
function Demo_CS_IRLS()
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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: R. Chartrand and W. Yin, “Iteratively Reweighted Algorithms
% for Compressed Sensing,” 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/3),width);  % only keep one third of the original data  
Phi = Phi./repmat(sqrt(sum(Phi.^2,1)),[floor(height/3),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_irls(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),imagesc(img_rec_1d),title(strcat('1d rec img ',num2str(psnr),'dB'))

%****************************************
function hat_x=cs_irls(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

hat_x_tp=T_Mat'*y; 
epsilong=1;
p=1;                             % solution for l-norm p
times=1;
while (epsilong>10e-9) && (times<length(y)/4)
    weight=(hat_x_tp.^2+epsilong).^(p/2-1); 
    Q_Mat=diag(1./weight,0); 
    hat_x=Q_Mat*T_Mat'*inv(T_Mat*Q_Mat*T_Mat')*y;
        if(norm(hat_x-hat_x_tp,2) < sqrt(epsilong)/100)
        epsilong=epsilong/10;
    end
        hat_x_tp=hat_x;
    times=times+1;
end

參考文獻

1、R. Chartrand and W. Yin, “Iteratively Reweighted Algorithms for Compressed Sensing,” 2008.

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