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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 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=0.7  #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, roy@mail.ustc.edu.cn, 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 column

Phi = 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 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_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)/4

hat_x_tp=zeros(m,1);         % initialization with the size of original
s=floor(length(y)/4);        % sparsity
u=0.5;                       % impact factor

% T_Mat=T_Mat/sqrt(sum(sum(T_Mat.^2))); % normalizae the whole matrix

for 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;          % update

end

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.

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