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Orthogonal Least Squares （OLS）算法流程

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实验

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）

python代码

#coding: utf-8
'''
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# DCT基作为稀疏基，重建算法为OMP算法 ，图像按列进行处理
# email:ncuzhangben@qq.com,
#%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
'''
#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'))
#print (im.shape, im.dtype)uint8

#生成高斯随机测量矩阵
sampleRate=0.7  #采样率
Phi=np.random.randn(256*sampleRate,256)

#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)

#OLS算法函数（未完成待修改）
def cs_ols(y,D):
    L=math.floor(3*(y.shape[0])/4)
    residual=y  #初始化残差
    index=np.zeros((L),dtype=int)
    for i in range(L):
        index[i]= -1
    result=np.zeros((256))
    for j in range(L):  #迭代次数
        pos_temp=np.argsort(np.fabs(np.dot(np.linalg.pinv(D[:,pos]),y)))#对应步骤2

        product=np.fabs(np.dot(D.T,residual))
        pos=np.argmax(product)  #最大投影系数对应的位置
        index[j]=pos
        my=np.linalg.pinv(D[:,index>=0])
        a=np.dot(my,y)
        residual=y-np.dot(D[:,index>=0],a)
    result[index>=0]=a
    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_ols(img_cs_1d[:,i],Theta_1d)  #利用OMP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_OLS()
%------------ 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(0.5*height),width);  % only keep one third of the original data
Phi = Phi./repmat(sqrt(sum(Phi.^2,1)),[floor(0.5*height),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); % height*width的0矩阵
Theta_1d=Phi*mat_dct_1d;%测量矩阵乘上基矩阵
for i=1:width
    column_rec=OLS(img_cs_1d(:,i),Theta_1d);%算法的目的是得到稀疏系数
    sparse_rec_1d(:,i)=column_rec';           % sparse representation 稀疏系数
end
img_rec_1d=mat_dct_1d*sparse_rec_1d;          % inverse transform 稀疏系数乘上基矩阵

%------------ show the results --------------------
figure(1)
subplot(2,2,1),imshow(uint8(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'));

%************************************************************************%
function [s, residual] = OLS(y, A, err)

% Orthogonal Least Squares [1] for Sparse Signal Reconstruction

% Input
% A = N X d dimensional measurment matrix
% y = N dimensional observation vector
% m = sparsity of the underlying signal

% Output
% s = estimated sparse signal
% r = residual 

% [1] T. Blumensath, M. E. Davies; "On the Difference Between Orthogonal
% Matching Pursuit and Orthogonal Least Squares", manuscript 2007

if nargin < 3
     err = 1e-5;
end

n1=length(y);
m=floor(3*n1/4);  

s = zeros(size(A,2),1);
r(:,1) = y; L = []; Psi = [];
normA=(sum(A.^2,1)).^0.5;
NI = 1:size(A,2);

for i = 2:m+1
    DR = A'*r(:,i-1);
    [v, I] = max(abs(DR(NI))./normA(NI)');
    INI = NI(I);
    L = [L' INI']';
    NI(I)=[];
    Psi = A(:,L);
    x = Psi\y;
    yApprox = Psi*x;
    r(:,i) = y - yApprox;
    if norm(r(:,end)) < err
        break;
    end
end
s(L) = x;
residual = r(:,end);

参考文章

1、Blumensath T, Davies M E. On the difference between orthogonal matching pursuit and orthogonal least squares[J]. 2007.

2、Hashemi A, Vikalo H. Sparse Linear Regression via Generalized Orthogonal Least-Squares[J]. arXiv preprint arXiv:1602.06916, 2016.

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