基于matlab的低秩结构重构算法仿真实现，对比ALM，IT，APG，ADMM

目录

​​一、理论基础​​

​​二、核心程序​​

​​三、测试结果​​

一、理论基础

       从数学上讲, 矩阵的秩反应了矩阵的固有属性, 矩阵的低秩性是指矩阵的秩相对于矩阵的行数和列数而言很小.低秩矩阵稀疏分解模型是将已知矩阵M (M∈Rm×n) 分解为一个低秩矩阵部分L (L∈Rm×n, rank (L) ≤m, n) 和一个稀疏矩阵部分S (S∈Rm×n) , 即M=L+S.

增广拉格朗日乘子法（ALM）                  

迭代阈值算法（IT）

加速近端梯度算法(APG)

交替方向乘子算法(ADMM)

二、核心程序

增广拉格朗日乘子法（ALM）                  

clc;
clear;
close all;
warning off;
addpath 'func\'

I1=imresize(imread('1.jpg'),[256,256]);
I2=imresize(imread('2.jpg'),[256,256]);

Init = I2;  


figure;
subplot(121);
imshow(Init);
title('原始图'); 


%original image
x                  = double(rgb2gray(Init));

tic;
[A_hat,E_hat,iter] = func_ALM(x,0.1,0.05,100);
toc;

subplot(122);
imshow(A_hat,[]);
title('ALM处理结果');      

迭代阈值算法（IT）

clc;
clear;
close all;
warning off;
addpath 'func\'

I1=imresize(imread('1.jpg'),[256,256]);
I2=imresize(imread('2.jpg'),[256,256]);

Init = I2;  


figure;
subplot(121);
imshow(Init);

tic;
title('原始图'); 
%迭代阈值算法
Im   = rgb2gray(Init); 
[x,y]= size(Im);                    
b    = double(Im);                    
zd   = double(max(Im));                  
zx   = double(min(Im));              
T    = double((zd+zx))/2;                    
count= double(0);                   
while 1                   % 迭代最佳阈值分割算法  
    count=count+1;  
    S0=0.0; n0=0.0;                    
    S1=0.0; n1=0.0;                  
    for i=1:x 
        for j=1:y 
            if double(Im(i,j))>=T 
               %大于阈域值图像点灰度值累加 
               S1=S1+double(Im(i,j));  
               %大于阈域值图像点个数累加 
               n1=n1+1;                
            else  
                %小于阈域值图像点灰度值累加 
                S0=S0+double(Im(i,j));  
                %小于阀域值图像点个数累加 
                n0=n0+1;                
            end  
        end  
    end   
    %求小于阀域值均值 
    T0=S0/n0; 
    %求大于阀域值均值 
    T1=S1/n1; 
     %迭代至前后两次阀域值相差几乎为0时
    if abs(T-((T0+T1)/2))<0.1    
        break; 
    else 
       %在阈值T下，迭代阈值的计算过程
       T=(T0+T1)/2;                   
   end  
end   
toc
T 
i1=im2bw(Im,T/255);               % 图像在最佳阈值下二值化  
subplot(122)
imshow(i1)  
title(['迭代后最终阈值输出：',num2str(T)]);      

加速近端梯度算法(APG)

clc;
clear;
close all;
warning off;
addpath 'func\'

I1=imresize(imread('1.jpg'),[256,256]);
I2=imresize(imread('2.jpg'),[256,256]);

Init = I2;  


figure;
subplot(131);
imshow(Init);
title('原始图'); 
tic
[A_hat,E_hat] = func_APG(double(rgb2gray(Init)),0.05,1000); 
toc
subplot(132);
imshow(A_hat,[]);
title('low-rank结果');
subplot(133);
imshow(E_hat);
title('处理误差');      

交替方向乘子算法(ADMM)

clc;
clear;
close all;
warning off;
addpath 'func\'

I1=imresize(imread('1.jpg'),[256,256]);
I2=imresize(imread('2.jpg'),[256,256]);

Init = I2;  


figure;
subplot(121);
imshow(Init);
title('原始图'); 


 
%original image
tic
x     = double(rgb2gray(Init));
[M,N] = size(x);
%FFT2
[H_FFT,HC_FFT]=func_hfft(x);

A     = @(x) real(ifft2(H_FFT.*fft2(x)));
AT    = @(x) real(ifft2(HC_FFT.*fft2(x)));

global calls;
calls = 0;
A     = @(x) callcounter(A,x);
AT    = @(x) callcounter(AT,x);
%观测矩阵
Ax    = A(x);
Psig  = norm(Ax,'fro')^2/(M*N);
y     = Ax;
%parameters
lambda     = 2.0e-2; 
mu         = lambda/10;
outeriters = 500;
tol        = 1e-5;
%TV
Phi_TV     = @(x) TVnorm(x);
chambolleit= 10;
Psi_TV     = @(x,th)  mex_vartotale(x,th,'itmax',chambolleit); 
alpha      = 1.2;
filter_FFT = 1./(abs(H_FFT).^2 + mu*alpha);
invLS      = @(x) real(ifft2(filter_FFT.*fft2( x )));
invLS      = @(x) callcounter(invLS,x);

[x_salsa, numA, numAt, objective1, distance, times1, mses1] = func_ADMM(y, A, lambda, 'MU', mu, 'AT', AT,  'StopCriterion', 1,'True_x', x, 'ToleranceA', tol, 'MAXITERA', outeriters,  'Psi', Psi_TV, 'Phi', Phi_TV,  'TVINITIALIZATION', 1, 'TViters', 10,'LS', invLS, 'VERBOSE', 0);
     
toc
subplot(122); 
imshow(x_salsa,[]);
title('admm处理结果');      

三、测试结果