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Warping Functions Warping Functions

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Submitted by bpittman. on 2005-12-22 16:29.
It would be nice to have a family of simple high-order multi-dimensional warping functions that were easily invertible -- ideally, the family would contain its own inverse functions.

Unfortunately, this goal doesn't appear possible. However, I was thinking about the following mathematical factoid:
  y(x) = x − a⋅sin(x)
has the solution (for |a|<1, which is the well-posed case)
  x(y) = y + ∑n=1 (2/n)⋅Jn(a⋅n)⋅sin(ny)
(This type of series of Bessel functions is known as a Kapteyn series; also see Chapter 17 of GN Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition.) Furthermore, Jn(a⋅n) decreases exponentially fast as n→∞. This means that for warps represented as low-order finite Fourier series, the inverse functions, although requiring an infinite number of terms to be exact, are pretty well represented themselves by low-order finite Fourier series.

In higher dimensions, the obvious generalization is a tensor-product Fourier series; this would be reasonable over a rectangular domain. Slightly less obvious, over the interior of a spherical domain, a better thing to do would be spherical harmonic series based on Ynm(θ,φ)jnpr). On the surface, drop the half-order Bessel functions. Must investigate numerics.

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