# 3D quadratic fitting to 27 voxel points, extend Matlab's findpeak function to 3rd dimention

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hamed amini il 7 Set 2019
the function below from Matlab's findpeak, fits a 2D quadtratic function u(x,y) to 9 pixels, I would like to extend that to 3D. i.e. u(x,y,z) for a 3x3x3 matrix.
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% simplified findpeak function below
[x_max,y_max] = simple_findpeak(u)
% fit a 2nd order polynomial to 9 points
% u is a 3x3 matrix
u = u(:);
x = [-1 -1 -1 0 0 0 1 1 1]';
y = [-1 0 1 -1 0 1 -1 0 1]';
% u(x,y) = A(1) + A(2)*x + A(3)*y + A(4)*x*y + A(5)*x^2 + A(6)*y^2
X = [ones(9,1), x, y, x.*y, x.^2, y.^2];
% u = X*A
A = X\u;
% get absolute maximum, where du/dx = du/dy = 0
x_max = (-A(3)*A(4)+2*A(6)*A(2)) / (A(4)^2-4*A(5)*A(6));
y_max = -1 / ( A(4)^2-4*A(5)*A(6))*(A(4)*A(2)-2*A(5)*A(3));
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