How to use regularization with lsqnonlin function

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AD 2020-11-14

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https://ww2.mathworks.cn/matlabcentral/answers/648078-how-to-use-regularization-with-lsqnonlin-function

编辑： AD 2020-11-15

采纳的回答： Bruno Luong

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Hi,

I would like to use lsqnonlin functinon with regularization (e.g. Tikhonov regularization). However, to my big surprise there is no such option it seems!

I like lsqnonlin because I can specify a cost function with an anonymous function in a convenient way.

Please, tell me there is a way to add regularization or another function that does the same thing with regularization?

Also please note that my fonction takes as input a matrix and convolves with a small kernel k:

fun = @(x)conv2(x,k,'same')-y;
lsqnonlinoptions=optimset('Algorithm','Levenberg-Marquardt','MaxIter',maxIterArg);
x_lsq =lsqnonlin(fun,x0,[],[],lsqnonlinoptions);

采纳的回答

Bruno Luong 2020-11-14

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https://ww2.mathworks.cn/matlabcentral/answers/648078-how-to-use-regularization-with-lsqnonlin-function#answer_544768

编辑：Bruno Luong 2020-11-14

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You are free to incorporation anything in the objective function, e.g., l^2 regularization

regcoef = some_small_coef;
fun = @(x) [reshape(conv2(x,k,'same')-y,[],1); regcoef*x(:)];

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AD 2020-11-15

编辑：AD 2020-11-15

在 MATLAB Online 中打开

Also I noticed lsqnonlin works with the conv2 problem I showed in the OP, but when I try:

fun = @(theta) reshape(imrotate(x,theta,'crop')-y,[],1);

it doesnt find the rotation angle and seem to just output the same x as I gave him (with very minor rotation maybe 1.5 degrees whereas I gave an y with 20 degrees) . Note that x is the ground truth image (no rotation) and y is the image with rotation see full code below)

any idea why it does not work with rotation? (Although I know there are other methods for estimating params but I am curious about lsqnonlin )

full function code so you can try it yourself:

function lsq_rec_estimRot_withReg(x,y,maxIterArg)
fun = @(theta) reshape(imrotate(x,theta,'crop')-y,[],1);
 
theta0 = randn();%initial guess
lsqnonlinoptions=optimset('Algorithm','Levenberg-Marquardt','MaxIter',maxIterArg);
theta_hat =lsqnonlin(fun,theta0,[],[],lsqnonlinoptions);
%% disp inside func for now
figure;
subplot(131);
imshow(x);
title('x');
subplot(132);
imshow(y);
title('y');
subplot(133);
imshow(imrotate(x,theta_hat,'crop'));
title(['x_{hat}, rot angle:' num2str(theta_hat) ] );
end

Bruno Luong 2020-11-15

I have no image-processing tbox so I can't try, and I guess I know the reason. But I stop to answer your many questions that drift from the original question.

AD 2020-11-15

编辑：AD 2020-11-15

Ok, - sorry for the many questions, its just that I am trying things - shall I post another question with this? It's just for getting a better intuition of things... I know there are algorithms for image re-alignment.

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