Lsqnonlin optimization unexpected behavior

Hello,

I am experimenting with lsqnonlin, first with a very simple problem: make lsqnonlin find translation params. I have an "original" image I, appy translations and then try to make lsqnonlin find them: results of lsqnonlin are in txy_sol_lsqnonlin.

The problem is that it almost immediately stops and always gives me useless results i.e. the same init values that I give it + maybe 0.5 or something like if it can't do more than a few steps and immediately stop searching anymore... Please help me find the problem.

The full code is :

I = phantom(256);
tx=22;
ty=5;
I_tr = imtranslate(I,[tx ty]);
fun = @(txy) [reshape( imtranslate(I,[txy(1) txy(2)])-I_tr,[],1) ];

lsqnonlinoptions=optimset('Algorithm','Levenberg-Marquardt','MaxIter',1000,'Display','iter');
tx_init = 0;
ty_init = 0;
[txy_sol_lsqnonlin,resnorm,residual,exitflag,output]  = lsqnonlin(fun,[tx_init, ty_init],[],[],lsqnonlinoptions);




% the Output in Matlab console is:
% 
%                                         First-Order                    Norm of 
%  Iteration  Func-count    Residual       optimality      Lambda           step
%      0           3         5058.12             410         0.01
%      1           6         4752.75            58.7        0.001       0.572352
%      2           9         4738.31            1.21       0.0001       0.177726
%      3          12          4738.3         0.00463        1e-05     0.00236374
%      4          15          4738.3        2.41e-05        1e-06    1.27094e-05
%      
%     So it seems clear that it thinks it has reached an optimum but why? 

[AD]

I tried to change tolerance values with 'TolX' but it does not seem to be related to this ...

Answer 1

[Alan Weiss]

lsqnonlin is a gradient-based solver. It first attempts to estimate the local gradient by small finite difference steps. If your function is locally constant, such as a step function, then lsqnonlin sees zero gradient and stops.

To have your function not look like a step function, perhaps you can interpolate it using interp2, for example.

Alan Weiss

MATLAB mathematical toolbox documentation