local interpolation

[redcoopers]

Hi all,

Well, this may be more directed towards people such as Andrew Mason or people with experience with computational geometry:

I need to find a bivariate interpolation of scattered data (but on a very local nature to each point in question). What I really need is a set of points (r=very close to each point; and theta= 0, 90, 180, 270). Algorithms such as a Monte Carlo method will not work, so I am leaning to a radial weighted method combined with an angular lagrange polynomial.

I've thought of using radial NURBS curves, but this seems computationally expensive since I need only a few radial points for each set of curves. Furthermore, I can't seem to set knots how I would need... For each radial curve I would be interpolating for, I would need the origin point be set absolutely, (ie, delta_ij = kronecker delta), but the end point would not necessarily need to have this condition. As an example for 5 control points of order 3, if I wanted up to C2 continuity at the origin, but didn't care about the rest of my points, why wouldn't a knot vector such as [0,0,0,1/5,2/5,3/5,4/5,1] work? Maybe my recursion algorithm is flawed, but I always get a closed curve, but not an open curve...

In any case, I think I need a least squares fit of order=n>=2 for a set of radial data. Does anyone have experience with weighted least squares methods or NURBS methods of fitting data?

-Jon

[Andrew Mason]

Red

Actually we have been doing a lot of work with Neural Networks for data fitting, if you have access to the Maxsurf Academic site have a look at the paper "Artificial Neural Networks for Hull Resistance Prediction. COMPIT 2004" on the Maxsurf Academic Technical Papers page.

The NN program I recommend is NeuroIntelligence by Alyuda, www.Alyuda.com. They have a 30 day demo version that can handle 1000 data points, full working program is about $400.

Andrew

[redcoopers]

Thanks for the link.

I'm actually looking for a linear method (ie, z2 = K z1) where K is my interpolation matrix.

I think I developed somthing which may work, however. For a univariate case, I would simply use lagrange polynomials of a limited degree centered around each point.

Moving to a bivariate case, now I interpolate in the x and y directions for each point. I use weighted Bernstein polynomials where the weight depends on how close each point is to the axis. I evaluate a few parametric points, and put these into a lagrange polynomial to finally complete the interpolation.

It works better than anything else I've tried and is rather robust. This works only because I do not want a global interpolation - only very locally near each point.

You have some rather good papers on your site. Using genetic algorithms to fit a drawing is impressive, but seems like overkill... Does the algorithm manipulate the control net for a NURBS surface, or does it develop its own surface?

-Jon