Meshfree Approximation Methods with MATLAB (Interdisciplinary Mathematical Sciences)

Category: Mathematics
Author: Gregory F. Fasshauer
5.0

Comments

by anonymous   2018-08-06

I am interested in parameterizing a surface without a mesh. One technique used in the field of optics is to use Radial Basis Functions (e.g. Gaussians).

From a naive point of view, the decomposition of a 2D scalar field into Gaussian functions centered at various spatial locations doesn't sound that much different than wavelet transforms.

Being a novice at both wavelets and RBF's, both appear to be decomposition into a series of functions with finite extent as opposed to Fourier decomposition, Bessel function decomposition, Legendre polynomials, ... which tend to be distributed over the entire area of the 2D field.

  1. For each, don't you need to exercise subjective judgement in choosing the spatial scale (smallest size) of the RBF or wavelets?
  2. Is there a fundamental difference that helps one choose which approach to use? (RBF vs. wavelet)
  3. Is one more computationally efficient than the other?
  4. What would be a metric for determining which is best?

Refs

  • http://math.iit.edu/~fass/#Book
  • https://www.elsevier.com/books/a-wavelet-tour-of-signal-processing/mallat/978-0-12-374370-1
  • https://www.amazon.com/Meshfree-Approximation-Interdisciplinary-Mathematical-Sciences/dp/9812706348