function u=ATV_ROF(f,mu)
%============================================
% Anisotropic ROF denoising
% This function performs the minimization of
% u=arg min |D_x u|+|D_y u|+0.5*mu*||u-f||_2^2
% by the Split Bregman Iteration
% f = noisy image
% mu = regularization parameter
% Reference:Goldstein T , Osher S . The Split Bregman Method for
% L1-Regularized Problems[J].
%SIAM Journal on Imaging Sciences, 2009, 2(2):323-343.
%============================================
[M,N]=size(f);
f=double(f);
dx=zeros(M,N);
dy=zeros(M,N);
bx=zeros(M,N);
by=zeros(M,N);
u=f;
Z=zeros(M,N);
Mask=zeros(M,N);
Mask(1,1) = 1;
FMask=fft2(Mask);
lambda=0.2;
%Fourier Laplacian mask initialization
D = zeros(M,N);
D([end,1,2],[end,1,2]) = [0,1,0;1,-4,1;0,1,0];
FD=fft2(D);
%Fourier constant initialization
FW=((mu/lambda)*abs(FMask).^2-real(FD)).^-1;
FF=(mu/lambda)*conj(FMask).*fft2(f);
err=100;
tol=1e-8;
while err>tol
tx=dx-bx;
ty=dy-by;
up=u;
%Update u
u=real(ifft2(FW.*(FF-fft2(tx-tx(:,[1,1:N-1])+ty-ty([1,1:M-1],:)))));
ux=u-u(:,[1,1:N-1]);
uy=u-u([1,1:M-1],:);
%Update dx
tmpx=ux+bx;
dx=sign(tmpx).*max(Z,abs(tmpx)-1/lambda);
%Update dy
tmpy=uy+by;
dy=sign(tmpy).*max(Z,abs(tmpy)-1/lambda);
%Update bx and by
bx=tmpx-dx;
by=tmpy-dy;
err=sum(sum((up-u).^2));
end
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