Porting a MATLAB code, Numeric vs Analytic derivatives.

[xeo...@gmail.com]

Hi,

I'm new to the Ceres and non linear least square in general. I've been given the task of porting this piece of MATLAB code. After reading tutorials I was wondering if it's possible to solve this using only numeric derivatives, or since I have the Jacobian, Maybe I should go with analytic derivatives.

Any help will be greatly appreciated!

function [f, J] = Cost(I, LD, lambda)
% The Cost function
% INPUT:
%     I: MxN matrix for the intensity
%     LD: 3xM matrix for normalized unit directions
%     lambda: Mx1 vector for the strength
% OUTPUT:
%     f: (M*N)x1 vector for intensity error
%     J: (M*N)xM matrix for Jacobian of 'f' over 'lambda'.

[M, N] = size(I);
L = repmat(lambda', [3 1]) .* LD;

f = L'*(L'\I) - I;
f = f(:);

if (nargout > 1)

   J = zeros(M*N, M);
   LLinv = inv(L*L');
   for i = 1:M
       dL = zeros(3, M);
       dL(:,i) = LD(:,i);
       df = dL'*LLinv*L - L'*LLinv*(dL*L'+L*dL')*LLinv*L + L'*LLinv*dL;
       Ji = df * I;
       J(:,i) = Ji(:);
    end
end


% Solve
lambda0 = ones(M, 1);
fcn = @(lambda)Cost(I, LD, lambda);
optimOpts = struct('Jacobian', 'on');
lambda = NonlinearLeastSquares(fcn, lambda0, [], [], optimOpts);

[Sameer Agarwal]

Please consider providing a mathematical description of your cost function.

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[xeo...@gmail.com]

Sure, I have a series of lighting vector directions (LD) and measured pixels intensities (I) and I would like to find the unknown lighting strength (lambda) parameters from the measured pixels intensities. The cost function can be in from of 

where M is the number of images and N number of pixels. By assuming the directional lighting model, the strength (lambda) can be estimated

where

the author suggested that the minimization over "bj " inside the sum can be analytically solved, resulting in a cost function below

Thank you for your help.

[Sameer Agarwal]

you could try not following the authors' advice and keep your life simpler and not optimize out b_j.
sure it reduces the size of the optimization problem, but it makes it more non-linear and makes derivatives a problem - it is possible to derive the derivatives, but it will take some work.

use analytical derivatives if possible.

Sameer

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