B - Spline Surface Fitting on a gridded data

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Davide Mastrodicasa
Hello everyone. I'm new on the topic of Spline interpolation. My problem is:
  • I have a x vector containing the value of the grid along the x direction.
  • I have a y vector containing the value of the grid along the y direction. This vector could have a different legth with respect to the x vector (Rectangular grid)
  • I have a z(x,y) matrix containing the data on this grid.
I would like to fit the z(x,y) surface using Multivariante B-splines with a certain knots along x and y directions (knotsx,knotsy) and a certain degree of the spline along x and y directions (k_degree_x k_degree_y). From a simple research the Matlab function to use is:
spap2({knorl1,...,knorlm},k,{x1,...,xm},y)
and, in my case:
sp=spap2({knotsx,knotsy},[k_degree_x k_degree_y],{x,y},z)
Is it right? Am I missing something so far?
For now in my code I still let aptknt to decide the knots of the spline.
Here there is the complete code based on a simple function that I'm trying to fit:
x = linspace(-2,2,100);
y = linspace(-5,5,500);
k_degree_x=4;
k_degree_y=4;
knots_mult_x=3;
knots_mult_y=3;
x_knot = aptknt(x,knots_mult_x); % returns a knot sequence suitable for interpolation at the data sites tau by splines of order k.
y_knot = aptknt(y,knots_mult_x);
[xx, yy] = meshgrid(x,y);
z = exp(-(xx.^2+yy.^2));
figure
surf(x,y,z);
figure
sp=spap2({x_knot,y_knot},[k_degree_x k_degree_y],{x,y},z');
fnplt(sp);
What I want to have as Output is the coefficient matrix of the B-spline function that i think is sp.coefs. From the Mathematical formulation that I found online about B-splines this matrix should have the dimension: k_degree_x+knotsx+1 x k_degree_y+knotsy+1.
where g and h are the length of the knots vectors in, respectively, x and y direction and k and l are, respectively, the degree of the splines in x and y direction.
My question is: Why from Matlab output i retrive a sp.coefs having the dimension of the input grid? In my case length(x)=500 X length(y)=100?
Can you send me some documentation, if you have, about what is implemented in Matlab with spap2?
Thank you in advance,
Davide

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