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;close all;
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');
for i=1:imsize(1)*0.5
    for j=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;
for i=1:imsize(1)*0.5
    for j=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;
for i=1:imsize(1)*0.5
    for j=1:imsize(2)
        GG(M(i),N(j),:)=G(i,j,:);
    end
end
for i=1:imsize(1)*0.5
    for j=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;
for i=1:imsize(1)
    for j=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;
for i=1:imsize(1)
    for j=1:imsize(2)
        if attack(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;
for i=1:imsize(1)*0.5
    for j=1:imsize(2)
        GG(M(i),N(j),:)=G(i,j,:);
    end
end
for i=1:imsize(1)*0.5
    for j=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;
for i=1:imsize(1)*0.5
    for j=1:imsize(2)
        GG(M(i),N(j),:)=G(i,j,:);
    end
end
for i=1:imsize(1)*0.5
    for j=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;close all;
%% 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; 
while k<=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; 
while k<=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; 
while k<=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); 
for i=1:imsize(1); 
for j=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');