Some questions about interpolation...

[Paul.Heckbert at HOSTESS.GRAPHICS.CS.CMU.EDU]

Michael Aupetit asked:
> Does it exist "standard" tests to establish that an interpolation 
> method is "good" or "bad"?
>
> Does it exist any interpolation method able to deal with high
> dimension (at least more than 5)

You probably want some metric of smoothness or fairness for a
function.  The most common metric, for univariate functions, is the
integral of the square of the second derivative.  The interpolating
function minimizing this is the "natural cubic spline",
whose solution reduces to a tridiagonal system of equations.  The
metric can be generalized in various ways for multivariate functions.
In general, I don't believe the optimal solution in higher dimensions
has a closed form solution, but you can use numerical variational
techniques, including multigrid or wavelets, to find an approximate
solution efficiently.

For references, search for "variational surface" at
http://www.cs.cmu.edu/~ph/859E/www/project.html
and see the work cited there by Terzopoulos and Kobbelt.

Paul Heckbert, Associate Professor
Computer Science Dept., Carnegie Mellon University
5000 Forbes Ave, Pittsburgh PA 15213-3891, USA

ph at cs.cmu.edu		http://www.cs.cmu.edu/~ph

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