Re: Reconstruction of a continuous function starting from a multidimensional data set

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 02/24/05 

Date: 24 Feb 2005 22:49:46 GMT

figueroa wrote:
> The region is connected and I do have a data set of points which lie
> outside the region.

Hmm, so we're trying to approximate a nonconvex region in possibly
high dimensions where all we know is that the region is connected, and
a sample of points together with whether or not they are internal or
external. Ouch.

I can think of ways to try to tackle it -- for example, clustering
nearby internal points into a hypersphere until such a hypersphere
must contain an external point. Then trying to find an overlapping
hypersphere containing internal points not already covered.

However, eventually this is going to run into hypersphere clusters
that can not overlap. The question then is how to connect them.
There are many choices, for example it should always be possible to
find a hyperellipsoid with one focus in each of the clusters.

What you would end up with is a set of quadratic inequalities, with a
point being inside the region if any of them are satisfied.

There are undoubtably better ways, but that's the one that came to
mind.

- Tim

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