how are obtain the coefs in a bivariate 2nd orden ppform spline?

hello everyone!. I would like to ask a question regarding the piecewise polynomial splines that matlab uses (ppform). it is more a concern that I dont fully understand the mathematics behind it. More specifically, talking about the coefficients in multivariate ppform splines.

When dealing with univariate I know that this coefficients follow a simple algorythm of :

for a k order spline, at the j piece, and it'scorresponding breakpoint. Coefs here are stored according to the Cij values.

But, when using multivariate splines, the coefs are calculated by a tensor product, and here is where my mind get lost. I have tried the following in order to simplify things and try to understand what is really doing:

[xx,yy] = meshgrid(0:1); %create the grid

z = xx+yy; % Just a simple add function

korder = {2,2}; % 2nd order splines in both axes

x = 0:1; y = 0:1;

knots = {x,y};

s = spapi(korder,knots,z); %create the B-form spline

g = fn2fm(s,'pp'); %transform into ppform

mesh(x,y,z) %visualization of the z surface

A = reshape(g.coefs,2,2); %clear visualization of coefs

So just a simple square, with the borders unite by lines.

My question is: How do I obtain this coefficients? What is the mathematical system to obtain them? I need to obtain them by hand in order to fully understand everything

Any pointer to an specific bibliography that can help would be appreciated as well. Thank you so much !

Answer 1

[Avadhoot]

Hi Jaime,

I understand that you have already calculated the coefficients for a bivariate second order spline using MATLAB but want to understand the mathematics behind the calculations. In multivariate splines, you must calculate a spline basis function in each dimension first and then use the tensor product to form a grid which approximates the surface of a higher dimensional function.

The process to calculate the coefficients for multivariate splines can be described in 3 stages as follows:

1. Choose a knot sequence for the spline.
2. Calculate the B-spline basis functions based on the knots sequence.
3. Find the tensor product of the basis functions to get the function for the surface of the bivariate spline.

The formula for the tensor product is as follows:

Here and are the univariate basis functions and are the coefficients. To obtain the coefficients, you must solve a system of linear equations. This system comes from the conditions that the spline must satisfy at the given data points. For your example where ( z = xx + yy ), the coefficients must be chosen so that the spline interpolates the values of z at the points of the grid. After creating the constraints, a system of linear equations is formed which can then be solved to obtain the coefficients.

I hope it helps.