fft2 function with reveresed matrix matlab

Question

I'm reading an article 'Deriving intrinsic images from image sequences'. I'm trying to get results from the article.

fnr is the reversed filter of fn:fn(x,y)=fnr(-x,-y). now I use fourier and pseudo-inverse to solve this equation and find

g : g*(sigma(fnr*fn)=delta

(sigma over n , '*' is convulotion everywhere),

%% fnt_fn_conv=sigma(fnr*fn)
fftr = fft2(fnr_fn_conv); 

 % find g
 fftrp = pinv(fftr);
 G = delta_fft x fftrp; (matrix mul)
 g = ifft2(G);

My problem is that g*(sigma(fnr*fn) **!=** delta, so g is wrong ! How should I build the fnr ? is fnr=rot90(fn, 2) sufficient, or I have to change the center of fnr with zero padding so fourier function will work correctly ?

Update :

The filtered images looks fine... I used 2 derivative filters ( horizental and vertical).

% given dx dy reconstruct image
% here we do the convolutions using FFT2


function [im,invKhat,k]=reconsEdge3(dx,dy,invKhat)



im=zeros(size(dx));
[sx,sy]=size(dx);
mxsize=max(sx,sy);

if ~exist('invKhat')
  [invK,k]=invDel2(2*mxsize);
  invKhat=fft2(invK);   %fourier for convolotion  G
end

%%
%% sigma fnr*r_n for 2 filters
imX=conv2(dx,fliplr([0 1 -1]),'same');
imY=conv2(dy,flipud([0;1;-1]),'same');

imS=imX+imY;  sigma

%%

imShat=fft2(imS,2*mxsize,2*mxsize);
im=real(ifft2(invKhat.*imShat));       %r
im=im(mxsize+1:mxsize+sx,mxsize+1:mxsize+sy);




function [invK,K]=invDel2(isize)

 %sigma(7) fnr*fn 
 K=zeros(isize);
 K(isize/2,isize/2)=-4;
 K(isize/2+1,isize/2)=1;
 K(isize/2,isize/2+1)=1;
 K(isize/2-1,isize/2)=1;
 K(isize/2,isize/2-1)=1;

 Khat=fft2(K);  %fourier for sigma(7)

 %add 1 to inverse without zeros
 I=find(Khat==0);   % find all zeros in Khat
 Khat(I)=1;     % put 1 where all zeros 
 invKhat=1./Khat;   % inverse
 invKhat(I)=0;  %dec the 1 added
 invK=ifft2(invKhat);
 invK=-real(invK);
 % invK and delta are not in the same dim, so we convolve
 % and then use transform fourier for invK in reconsEdge3
 invK=conv2(invK,[1 0 0;0 0 0;0 0 0],'same');

Note: conv2(invKhat,K)!=delta why ?!

Answer 1

https://imgproc2014.github.io/shadow_removal/

Implemented the article and finished the project !

Thanks anyways...