I want to interpolate the following 3 scattered data points: (80.9,58.5,48.0),(35.0,89.6,82.3),(74.7,17.4,85.9) by an uniform cubic B spline surface on the following control lattice, $ \phi $:
In this figure the data points (blue) have been normalized to [0,1]:
$ x_n = (x-xmin)/(xmax-xmin) $
The B spline interpolator becomes:
$ z(x,y) = \Sigma_{i=-1}^2\Sigma_{j=-1}^2B_{i+1}(x_n)B_{j+1}(y_n)\phi_{i,j} $
where
$ B_0 (t)=(1-t^3)/6 $
$ B_1 (t)=(3t^3-6t^2+4)/6 $
$ B_2 (t)=(-3t^3+3t^2+3t+1)/6 $
$ B_3 (t)=t^3/6 $
How do I calculate the values of the control points $ \phi_{i,j} $?
Introducing:
$ \vec{z} = [z_{0,0},z_{0,1},…,z_{3,3}]^T $,
$ A_i = [B_0(x_i)B_0(y_i), B_0(x_i)B_1(y_i), …, B_3(x_i)B_3(y_i)] $,
$ A=[A_1^T, A_2^T, A_3^T]^T $,
$ \vec{\phi} = [\phi_{0,0},\phi_{0,1},…,\phi_{3,3}]^T $
The interpolation can be written as a matrix equation:
$ \vec{z} = A\vec{\phi} $
I can then find a Least Square Error solution to this equation.
Doing this in Matlab and sampling the resulting interpolator gives the following result:
The interpolator goes exactly trough the data points (black) but the extrapolation in the lower left corner seems bad.
What is causing this problem?
I would guess the matrix equation is underdetermined (3 equations and 16 unknowns).
Is there some easy way I can regularize it so that the interpolator gives me a more planar solution?
Best Answer
The way you set up and solved the problem looks basically correct. But, as you suspected, your particular example problem is highly undetermined. Even the simplest cubic b-spline surface (with a single patch) has 16 degrees of freedom, so it needs 16 data points to fully specify it, and you have only 3.
Least-squares fitting is only appropriate if the number of data points exceeds the number of degrees of freedom in the surface (i.e. the opposite of your situation).
I know that interpolation of scattered points doesn't always work. In certain cases the system of linear equations has no unique solution. I'm not sure if this same problem arises with least-squares fitting, but I'd guess that it does.
Interploation and LSQ fitting are both much easier if you have a gridded rectangular array of data points. If you really have to handle scattered points, this makes the problem somewhat harder.