Thread Subject: unstable and un-convergence total variation regularization (with the code)

HI,everyone:

I am working to solve a linear ill-posed problem Au=f using total variation regularization as follows:

min || A*u - f ||^2_L^2 + lambda*TV(u)

However, the TV regularized solution is not stable. When the lambda is large, the solution contains large oscillation. When the lambda is small, the solution remains the same as the direct matrix inversion solution,i.e. u=f\A. Besides, the solution doesn't converge. Whatever lambda is, it is either one of the aforemention cases.

This really puzzles me, because the same code works well for denoising. will anyone give me some hints?
Thanks a lot!

Code:

% solve the WING problem using TV regularization:
% min || A*u - f ||^2_L^2 + lambda*TV(u)
%% setup problem
n=64;% length of tested data.
t1=1/3;
t2=2/3;

% Set up matrix A.
A = zeros(n,n); h = 1/n;
sti = ((1:n)-0.5)*h;
for i=1:n
  A(i,:) = h*sti.*exp(-sti(i)*sti.^2);
end
% set up the idea solution;
I = find(t1 < sti & sti < t2);
u0 = zeros(n,1); u0(I) = sqrt(h)*ones(length(I),1);
% setup the right-hand
f = sqrt(h)*0.5*(exp(-sti*t1^2)' - exp(-sti*t2^2)')./sti';

%% solve by TV regularized problem
% parmeters setup
ep2 = 1e-3;
dt = 0.02; % time step
lambda = 1;
nx = size(f,1);
NumSteps=400; % iteration number

u=zeros(size(f));% set initial value

for i=1:NumSteps,
   % estimate derivatives
u_x = (u([2:nx nx],:)-u([1 1:nx-1],:))/2;
u_xx = u([2:nx nx],:)+u([1 1:nx-1],:)-2*u;
   % compute flow
    Num = ep2.*u_xx;
    Den = (ep2+u_x.^2).^(3/2);
    u_tv = Num./Den;
    u_fidelity = 2*A'*(f-A*u);
   % evolve image by dt
    u=u+dt*(lambda*u_tv+u_fidelity);
end
%% display
figure(2);plot(u,'r');hold on;plot(u0,'-.k');hold off;legend('tv','original');

"sheng fang" <maelstromer@gmail.com> wrote in message <gc44nd$ejf$1@fred.mathworks.com>...

>
> This really puzzles me, because the same code works well for denoising. will anyone give me some hints?
> Thanks a lot!

Have you studied the stability of your discretization scheme? Notably selecting dt by Courant-Friedich-Lecy and/or discretize the TV term by upwind scheme?

Bruno

In complement, see for example:

SIGAL GOTTLIEB, CHI-WANG SHU
TOTAL VARIATION DIMINISHING RUNGE-KUTTA SCHEMES

MATHEMATICS OF COMPUTATION
Volume 67, Number 221, 73-85, January 1998.

Bruno

"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <gc4dgm$4nu$1@fred.mathworks.com>...
> In complement, see for example:
>
> SIGAL GOTTLIEB, CHI-WANG SHU
> TOTAL VARIATION DIMINISHING RUNGE-KUTTA SCHEMES
>
> MATHEMATICS OF COMPUTATION
> Volume 67, Number 221, 73-85, January 1998.
>
> Bruno
Hi, Bruno:

Thank you very much!
I download the paper and I am working on it!

Sheng Fang