低阶秩序图像修复模型
低阶秩图像修复模型，假设图像的数据维度秩（rank）较低，那么，可以利用该性质实现图像修复，具体模型如下：
通过将迹范数转化为核范数，将非凸优化问题转化为凸优化问题，如下：
通过迭代求解上述凸优化问题可以实现图像修复：
% demo2.m
% Date: 2015/6/12
% Author: HSW
% HARBIN INSTITUTE OF TECHNOLOGY
% set matlab
close all;
clear all;
clc;
% add path
addpath(genpath('DamagedImage\'));
addpath(genpath('ResultsImage\'));
addpath(genpath('Solvers\'));
addpath(genpath('TestImage\'));
% Set True rank of Image
Rank = 40;
% read image
Img = imread('TestImage2\Lowrank.bmp');
Img = imresize(Img,[256,256],'bicubic');
% Img = imread('TestImage2\mat.bmp');
% Img = imread('TestImage2\rattan.tif');
% Img = imread('TestImage2\Build.bmp'); %可以达到95%
% Img = imread('TestImage\barba256.png');
% Img = imread('D82.gif');
% Img = imresize(Img,[256,256],'bicubic');
if size(Img,3) == 3
OrigImg = rgb2gray(Img);
elseif size(Img,3) == 4
OrigImg = mean(Img,3);
else
OrigImg = Img;
end
OrigImg = OrigImg';
OrigImg = double(OrigImg);
[U,S,V] = svd(OrigImg);
OrigRank = length(find(S ~= 0));
S(Rank+1:end,:) = 0; %注意这里
LowRankImg = U*S*V';
figure;
subplot(1,3,1)
imshow(OrigImg/255,[]);
title(['Image of Rank = ', num2str(OrigRank)]);
subplot(1,3,2);
imshow(LowRankImg/255,[]);
title(['Image of Rank = ',num2str(Rank)]);
subplot(1,3,3);
imshow(abs(OrigImg - LowRankImg)/255,[]);
title('差值图像');
% generate damaged image
% 随机缺失
Ratio = 0.9;
Mask = rand(size(LowRankImg));
Mask = (Mask >= Ratio);
% 读入掩膜
% Mask = imread('Mask3.png');
% Mask = double(rgb2gray(Mask)) <= 2;
% Mask = Mask';
% Mask = flipud(Mask);
InImg = LowRankImg.*Mask;
InterpImg = ImgInterp2(InImg,Mask,1:size(LowRankImg,1),1:size(LowRankImg,2));
% parameters
muMin = 1e-8;
eta = 0.25;
kao = 1;
BregmanMaxIterNum = 100;
MaxIterNum = 50;
xtol = 1e-10;
ResultImg = FPCBregman(InImg,Mask,InterpImg,muMin,eta,kao,BregmanMaxIterNum,MaxIterNum,xtol);
ResultImg = max(0,min(ResultImg,255));
PSNRin = 20*log10(255/sqrt(mean((InImg(:)-LowRankImg(:)).^2)));
PSNRinterp = 20*log10(255/sqrt(mean((InterpImg(:)-LowRankImg(:)).^2)));
PSNRout = 20*log10(255/sqrt(mean((ResultImg(:)-LowRankImg(:)).^2)));
figure;
subplot(1,3,1);
imshow(InImg/255,[]);
title(['缺损图像 PSNR = ',num2str(PSNRin)]);
subplot(1,3,2);
imshow(InterpImg/255,[]);
title(['插值图像 PSNR = ',num2str(PSNRinterp)]);
subplot(1,3,3);
imshow(ResultImg/255,[]);
title(['修复结果 PSNR = ',num2str(PSNRout)]);
figure;
subplot(1,3,1);
imshow(InImg/255,[]);
title(['缺损图像 PSNR = ',num2str(PSNRin)]);
subplot(1,3,2);
imshow(ResultImg/255,[]);
title(['主要部分 PSNR = ',num2str(PSNRinterp)]);
subplot(1,3,3);
imshow((OrigImg-ResultImg)/255,[]);
title(['反光部分 PSNR = ',num2str(PSNRout)]);FPC_Bregman.m/FPC2.m/ImgInterp2.m/ssim.m文件
function ResultImg = FPC_Bregman(InImg,Mask,InterpImg,muMin,eta,kao,BregmanMaxIterNum,MaxIterNum,xtol)
% Inputs:
%
% Outputs:
%
%
bk = 0;
Imgk = InterpImg;
condition = 1;
k = 1;
while condition
k = k + 1;
Imgold = Imgk;
bk = InImg + (bk - Imgk.*Mask);
Imgk = FPC2(bk,Mask,eta,muMin,kao,MaxIterNum,xtol);
condition = (norm(Imgk - Imgold,'fro')/max(norm(Imgold,'fro'),norm(Imgk,'fro'))>= xtol) && k <= BregmanMaxIterNum;
end
ResultImg = Imgk;
end %function FPC_Bregman
function ResultImg = FPC2(b,Mask,eta,muMin,kao,MaxIterNum,xtol)
%初始化
X = 0;
L = 1;
muL(L) = eta*norm(b,2);
while muL(L) > muMin
L = L + 1;
muL(L) = max(muL(L-1)*eta,muMin);
end
for iter = 1:L
condition = 1;
k = 0;
while condition
Xold = X;
k = k + 1;
Y = X - kao*(Mask.*X - b);
[U,S,V] = svd(Y);
X = U*SNG(S,kao,muL(iter))*V';
condition = ((norm(X-Xold,'fro')/max(1,norm(X,'fro'))) >= xtol)&&(k <= MaxIterNum);
end %while
end %for
ResultImg = X;
end %ResultImg
function S = SNG(S,kao,mu)
S = sign(S).*max(abs(S) - kao*mu,0);
end
function y0=ImgInterp2(y0,I_nonzero,selx,sely,interp)
if (~exist('interp','var'))
interp='nearest';
end
loc_z=y0(selx,sely);
loc_mask=I_nonzero(selx,sely);
[X,Y]=meshgrid(selx,sely);
x=X(loc_mask==1);
y=Y(loc_mask==1);
z=loc_z(loc_mask==1);
xi=X(loc_mask==0);
yi=Y(loc_mask==0);
interp=lower(interp);
zi = griddata(x,y,z,xi,yi,interp);
zi(isnan(zi))=0.;
loc_z(loc_mask==0)=zi;
y0(selx,sely)=loc_z;
function [mssim, ssim_map] = ssim(img1, img2, K, window, L)
%
%Input : (1) img1: the first image being compared
% (2) img2: the second image being compared
% (3) K: constants in the SSIM index formula (see the above
% reference). defualt value: K = [0.01 0.03]
% (4) window: local window for statistics (see the above
% reference). default widnow is Gaussian given by
% window = fspecial('gaussian', 11, 1.5);
% (5) L: dynamic range of the images. default: L = 255
%
%Output: (1) mssim: the mean SSIM index value between 2 images.
% If one of the images being compared is regarded as
% perfect quality, then mssim can be considered as the
% quality measure of the other image.
% If img1 = img2, then mssim = 1.
% (2) ssim_map: the SSIM index map of the test image. The map
% has a smaller size than the input images. The actual size
% depends on the window size and the downsampling factor.
%
% ?sic Usage:
% Given 2 test images img1 and img2, whose dynamic range is 0-255
%
% [mssim, ssim_map] = ssim(img1, img2);
%
% ?vanced Usage:
% User defined parameters. For example
%
% K = [0.05 0.05];
% window = ones(8);
% L = 100;
% [mssim, ssim_map] = ssim(img1, img2, K, window, L);
%
%Visualize the results:
%
% mssim %Gives the mssim value
% imshow(max(0, ssim_map).^4) %Shows the SSIM index map
%========================================================================
if (nargin < 2 | nargin > 5)
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
if (size(img1) ~= size(img2))
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
[M N] = size(img1);
if (nargin == 2)
if ((M < 11) | (N < 11))
ssim_index = -Inf;
ssim_map = -Inf;
return
end
window = fspecial('gaussian', 11, 1.5); %
K(1) = 0.01; % default settings
K(2) = 0.03; %
L = 255; %
end
if (nargin == 3)
if ((M < 11) | (N < 11))
ssim_index = -Inf;
ssim_map = -Inf;
return
end
window = fspecial('gaussian', 11, 1.5);
L = 255;
if (length(K) == 2)
if (K(1) < 0 | K(2) < 0)
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
else
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
end
if (nargin == 4)
[H W] = size(window);
if ((H*W) < 4 | (H > M) | (W > N))
ssim_index = -Inf;
ssim_map = -Inf;
return
end
L = 255;
if (length(K) == 2)
if (K(1) < 0 | K(2) < 0)
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
else
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
end
if (nargin == 5)
[H W] = size(window);
if ((H*W) < 4 | (H > M) | (W > N))
ssim_index = -Inf;
ssim_map = -Inf;
return
end
if (length(K) == 2)
if (K(1) < 0 | K(2) < 0)
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
else
ssim_index = -Inf;
ssim_map = -Inf;
return;
end
end
img1 = double(img1);
img2 = double(img2);
% automatic downsampling
f = max(1,round(min(M,N)/256));
%downsampling by f
%use a simple low-pass filter
if(f>1)
lpf = ones(f,f);
lpf = lpf/sum(lpf(:));
img1 = imfilter(img1,lpf,'symmetric','same');
img2 = imfilter(img2,lpf,'symmetric','same');
img1 = img1(1:f:end,1:f:end);
img2 = img2(1:f:end,1:f:end);
end
C1 = (K(1)*L)^2;
C2 = (K(2)*L)^2;
window = window/sum(sum(window));
mu1 = filter2(window, img1, 'valid');
mu2 = filter2(window, img2, 'valid');
mu1_sq = mu1.*mu1;
mu2_sq = mu2.*mu2;
mu1_mu2 = mu1.*mu2;
sigma1_sq = filter2(window, img1.*img1, 'valid') - mu1_sq;
sigma2_sq = filter2(window, img2.*img2, 'valid') - mu2_sq;
sigma12 = filter2(window, img1.*img2, 'valid') - mu1_mu2;
if (C1 > 0 & C2 > 0)
ssim_map = ((2*mu1_mu2 + C1).*(2*sigma12 + C2))./((mu1_sq + mu2_sq + C1).*(sigma1_sq + sigma2_sq + C2));
else
numerator1 = 2*mu1_mu2 + C1;
numerator2 = 2*sigma12 + C2;
denominator1 = mu1_sq + mu2_sq + C1;
denominator2 = sigma1_sq + sigma2_sq + C2;
ssim_map = ones(size(mu1));
index = (denominator1.*denominator2 > 0);
ssim_map(index) = (numerator1(index).*numerator2(index))./(denominator1(index).*denominator2(index));
index = (denominator1 ~= 0) & (denominator2 == 0);
ssim_map(index) = numerator1(index)./denominator1(index);
end
mssim = mean2(ssim_map);
结果如下，没有调整参数，如果调整参数结果更好：
原始图像如下：
参考文献：
Tensor Completion for Estimating Missing Values in Visual Data
