我在研究如何对图像进行二维快速傅里叶变换，求指点

我发现了一个给图片加不可见的水印的方法， 据说这种“盲水印”不影响用户体验，又能追溯图片来源和传播过程，在捍卫版权和知识付费越来越看重的现在，这个东西应该蛮有前途的。我就在想，我没有Matlab2010a这软件，我无法体验给图片加密，我试试找找能否用PHP实现在线加盲水印的办法，希望能做成WEB应用。

我得到的线索来自这里：
解密：阿里巴巴公司根据截图查到泄露信息的员工的技术是什么
https://www.zhihu.com/question/50735753
解密：阿里巴巴公司根据截图查到泄露信息的员工的技术是什么
http://www.phpchina.com/portal.php?mod=view&aid=40242

大概技术细节是这两张图来解释：
加水印：
 然后解水印：

里面关键的部分是对图片进行“二维快速傅里叶变换”，英文叫 Two-Dimensional Fast Fourier Transform，在技术文档里通常简称为FFT。


我阅读了Fuqiang Liu 的Matlab2010a下的源代码，也去读了《如果看了此文你还不懂傅里叶变换，那就过来掐死我吧【完整版】  》，初步了解了快速傅里叶变换是怎么回事。

我找到以下关于FFT的PHP代码实现相关链接：
https://www.phpclasses.org/package/6193-PHP-Compute-the-Fast-Fourier-Transform-of-sampled-data.html ，这个类库能Compute the Fast Fourier Transform of sampled data，但我不知道怎么使用，有样本代码教我怎么做FFT和inverse FFT。
http://www.jasonbailey.net/stuff/php-fast-fourier-transform-fft-brighton-php-october-2013/

我也尝试找PHP是否支持 快速傅里叶变换，找到有图像组件
http://php.net/manual/zh/book.imagick.php下的 Imagick::forwardFourierTransformImage是支持 “离散傅里叶变换”，英文是discrete Fourier transform (DFT) ，但不知道与“快速傅里叶变换”Fast Fourier Transform（FFT）有什么差别。

这里还有一个 Two-Dimensional Fast Fourier Transform（二维快速傅里叶变换） 的文章，
http://imagejdocu.tudor.lu/doku.php?id=gui:process:fft 也是基于 Matlab在讲。

现在问题是，用php里的imagecopy函数合并背景图片和水印到同一张照片的功能我测试通过了，如何用FFT的类库对图片作FFT和inverse FFT操作？求有图像处理经验的的高手指导一下，我已经试过用英文搜索了，如 php Fourier image / php fft convert image / ,找到这些链接，http://www.fmwconcepts.com/misc_tests/FFT_tests/
http://www.roborealm.com/help/FFT.php
都不知道如何使用傅里叶变换对图片进行操作。

盲水印有人用clojure写了一个，还公开了源代码：一个智慧的水印 - Share - Clojure China http://clojure-china.org/t/topic/545

下面是Fuqiang Liu 的Matlab2010a下的源代码：

%%傅里叶变换加水印源代码
%% 运行环境Matlab2010a 
clc;clear;closeall;
alpha=1;

%% read data
im=double(imread('gl1.jpg'))/255;
mark=double(imread('watermark.jpg'))/255;
figure,imshow(im),title('original image');
figure,imshow(mark),title('watermark');

%% encode mark
imsize=size(im);
%random
TH=zeros(imsize(1)*0.5,imsize(2),imsize(3));
TH1=TH;
TH1(1:size(mark,1),1:size(mark,2),:)=mark;
M=randperm(0.5*imsize(1));
N=randperm(imsize(2));
save('encode.mat','M','N');
fori=1:imsize(1)*0.5
forj=1:imsize(2)
TH(i,j,:)=TH1(M(i),N(j),:);
end
end
% symmetric
mark_=zeros(imsize(1),imsize(2),imsize(3));
mark_(1:imsize(1)*0.5,1:imsize(2),:)=TH;
fori=1:imsize(1)*0.5
forj=1:imsize(2)
mark_(imsize(1)+1-i,imsize(2)+1-j,:)=TH(i,j,:);
end
end
figure,imshow(mark_),title('encoded watermark');
%imwrite(mark_,'encoded watermark.jpg');

%% add watermark
FA=fft2(im);
figure,imshow(FA);title('spectrum of original image');
FB=FA+alpha*double(mark_);
figure,imshow(FB);title('spectrum of watermarked image');
FAO=ifft2(FB);
figure,imshow(FAO);title('watermarked image');
%imwrite(uint8(FAO),'watermarked image.jpg');
RI=FAO-double(im);
figure,imshow(uint8(RI));title('residual');
%imwrite(uint8(RI),'residual.jpg');
xl=1:imsize(2);
yl=1:imsize(1);
[xx,yy]=meshgrid(xl,yl);
figure,plot3(xx,yy,FA(:,:,1).^2+FA(:,:,2).^2+FA(:,:,3).^2),title('spectrum of original image');
figure,plot3(xx,yy,FB(:,:,1).^2+FB(:,:,2).^2+FB(:,:,3).^2),title('spectrum of watermarked image');
figure,plot3(xx,yy,FB(:,:,1).^2+FB(:,:,2).^2+FB(:,:,3).^2-FA(:,:,1).^2+FA(:,:,2).^2+FA(:,:,3).^2),title('spectrum of watermark');

%% extract watermark
FA2=fft2(FAO);
G=(FA2-FA)/alpha;
GG=G;
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(M(i),N(j),:)=G(i,j,:);
end
end
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(imsize(1)+1-i,imsize(2)+1-j,:)=GG(i,j,:);
end
end
figure,imshow(GG);title('extracted watermark');
%imwrite(uint8(GG),'extracted watermark.jpg');

%% MSE and PSNR
C=double(im);
RC=double(FAO);
MSE=0;PSNR=0;
fori=1:imsize(1)
forj=1:imsize(2)
MSE=MSE+(C(i,j)-RC(i,j)).^2;
end
end
MSE=MSE/360.^2;
PSNR=20*log10(255/sqrt(MSE));
MSE
PSNR

%% attack test
%% attack by smearing
%A = double(imread('gl1.jpg'));
%B = double(imread('attacked image.jpg'));
attack=1-double(imread('attack.jpg'))/255;
figure,imshow(attack);
FAO_=FAO;
fori=1:imsize(1)
forj=1:imsize(2)
ifattack(i,j,1)+attack(i,j,2)+attack(i,j,3)>0.5
FAO_(i,j,:)=attack(i,j,:);
end
end
end
figure,imshow(FAO_);
%extract watermark
FA2=fft2(FAO_);
G=(FA2-FA)*2;
GG=G;
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(M(i),N(j),:)=G(i,j,:);
end
end
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(imsize(1)+1-i,imsize(2)+1-j,:)=GG(i,j,:);
end
end
figure,imshow(GG);title('extracted watermark');

%% attack by cutting
s2=0.8;
FAO_=FAO;
FAO_(:,s2*imsize(2)+1:imsize(2),:)=FAO_(:,1:int32((1-s2)*imsize(2)),:);
figure,imshow(FAO_);
%extract watermark
FA2=fft2(FAO_);
G=(FA2-FA)*2;
GG=G;
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(M(i),N(j),:)=G(i,j,:);
end
end
fori=1:imsize(1)*0.5
forj=1:imsize(2)
GG(imsize(1)+1-i,imsize(2)+1-j,:)=GG(i,j,:);
end
end
figure,imshow(GG);title('extracted watermark');


%%小波变换加水印，解水印大家按照加的思路逆过来就好
clc;clear;closeall;
%% read data
im=double(imread('gl1.jpg'))/255;
mark=double(imread('watermark.jpg'))/255;
figure,imshow(im),title('original image');
figure,imshow(mark),title('watermark');
%% RGB division
im=double(im);
mark=double(mark);
imr=im(:,:,1);
markr=mark(:,:,1);
img=im(:,:,2);
markg=mark(:,:,2);
imb=im(:,:,3);
markb=mark(:,:,3);
%% parameter
r=0.04;
g=0.04;
b=0.04;
%% wavelet tranform and add watermark
% for red
[Cwr,Swr]=wavedec2(markr,1,'haar');
[Cr,Sr]=wavedec2(imr,2,'haar');
% add watermark
Cr(1:size(Cwr,2)/16)=... 
Cr(1:size(Cwr,2)/16)+r*Cwr(1:size(Cwr,2)/16);
k=0;
whilek<=size(Cr,2)/size(Cwr,2)-1
Cr(1+size(Cr,2)/4+k*size(Cwr,2)/4:size(Cr,2)/4+... 
(k+1)*size(Cwr,2)/4)=Cr(1+size(Cr,2)/4+... 
k*size(Cwr,2)/4:size(Cr,2)/4+(k+1)*size(Cwr,2)/4)+... 
r*Cwr(1+size(Cwr,2)/4:size(Cwr,2)/2);
Cr(1+size(Cr,2)/2+k*size(Cwr,2)/4:size(Cr,2)/2+... 
(k+1)*size(Cwr,2)/4)=Cr(1+size(Cr,2)/2+... 
k*size(Cwr,2)/4:size(Cr,2)/2+(k+1)*size(Cwr,2)/4)+... 
r*Cwr(1+size(Cwr,2)/2:3*size(Cwr,2)/4);
Cr(1+3*size(Cwr,2)/4+k*size(Cwr,2)/4:3*size(Cwr,2)/4+... 
(k+1)*size(Cwr,2)/4)=Cr(1+3*size(Cr,2)/4+... 
k*size(Cwr,2)/4:3*size(Cr,2)/4+(k+1)*size(Cwr,2)/4)+... 
r*Cwr(1+3*size(Cwr,2)/4:size(Cwr,2));
k=k+1;
end;
Cr(1:size(Cwr,2)/4)=Cr(1:size(Cwr,2)/4)+r*Cwr(1:size(Cwr,2)/4);

% for green
[Cwg,Swg]=WAVEDEC2(markg,1,'haar');
[Cg,Sg]=WAVEDEC2(img,2,'haar');
Cg(1:size(Cwg,2)/16)=... 
Cg(1:size(Cwg,2)/16)+g*Cwg(1:size(Cwg,2)/16);
k=0;
whilek<=size(Cg,2)/size(Cwg,2)-1
Cg(1+size(Cg,2)/4+k*size(Cwg,2)/4:size(Cg,2)/4+... 
(k+1)*size(Cwg,2)/4)=Cg(1+size(Cg,2)/4+... 
k*size(Cwg,2)/4:size(Cg,2)/4+(k+1)*size(Cwg,2)/4)+... 
g*Cwg(1+size(Cwg,2)/4:size(Cwg,2)/2);
Cg(1+size(Cg,2)/2+k*size(Cwg,2)/4:size(Cg,2)/2+... 
(k+1)*size(Cwg,2)/4)=Cg(1+size(Cg,2)/2+... 
k*size(Cwg,2)/4:size(Cg,2)/2+(k+1)*size(Cwg,2)/4)+... 
g*Cwg(1+size(Cwg,2)/2:3*size(Cwg,2)/4);
Cg(1+3*size(Cg,2)/4+k*size(Cwg,2)/4:3*size(Cg,2)/4+... 
(k+1)*size(Cwg,2)/4)=Cg(1+3*size(Cg,2)/4+... 
k*size(Cwg,2)/4:3*size(Cg,2)/4+(k+1)*size(Cwg,2)/4)+... 
g*Cwg(1+3*size(Cwg,2)/4:size(Cwg,2));
k=k+1;
end;
Cg(1:size(Cwg,2)/4)=Cg(1:size(Cwg,2)/4)+g*Cwg(1:size(Cwg,2)/4);

% for blue
[Cwb,Swb]=WAVEDEC2(markb,1,'haar');
[Cb,Sb]=WAVEDEC2(imb,2,'haar');
Cb(1:size(Cwb,2)/16)+b*Cwb(1:size(Cwb,2)/16);
k=0;
whilek<=size(Cb,2)/size(Cwb,2)-1
Cb(1+size(Cb,2)/4+k*size(Cwb,2)/4:size(Cb,2)/4+... 
(k+1)*size(Cwb,2)/4)=Cb(1+size(Cb,2)/4+... 
k*size(Cwb,2)/4:size(Cb,2)/4+(k+1)*size(Cwb,2)/4)+... 
g*Cwb(1+size(Cwb,2)/4:size(Cwb,2)/2);
Cb(1+size(Cb,2)/2+k*size(Cwb,2)/4:size(Cb,2)/2+... 
(k+1)*size(Cwb,2)/4)=Cb(1+size(Cb,2)/2+... 
k*size(Cwb,2)/4:size(Cb,2)/2+(k+1)*size(Cwb,2)/4)+... 
b*Cwb(1+size(Cwb,2)/2:3*size(Cwb,2)/4);
Cb(1+3*size(Cb,2)/4+k*size(Cwb,2)/4:3*size(Cb,2)/4+... 
(k+1)*size(Cwb,2)/4)=Cb(1+3*size(Cb,2)/4+... 
k*size(Cwb,2)/4:3*size(Cb,2)/4+(k+1)*size(Cwb,2)/4)+... 
b*Cwb(1+3*size(Cwb,2)/4:size(Cwb,2));
k=k+1;
end;
Cb(1:size(Cwb,2)/4)=Cb(1:size(Cwb,2)/4)+b*Cwb(1:size(Cwb,2)/4);
%% image reconstruction
imr=WAVEREC2(Cr,Sr,'haar');
img=WAVEREC2(Cg,Sg,'haar');
imb=WAVEREC2(Cb,Sb,'haar');
imsize=size(imr);
FAO=zeros(imsize(1),imsize(2),3);
fori=1:imsize(1);
forj=1:imsize(2);
FAO(i,j,1)=imr(i,j);
FAO(i,j,2)=img(i,j);
FAO(i,j,3)=imb(i,j);
end
end
figure,imshow(FAO);title('watermarked image');