Fast, Smooth Interpolation of Unevenly Spaced Data Oscar P' Bruno and Matthew M' Pohlman California

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Fast, Smooth Interpolation of Unevenly Spaced
DataOscar P. Bruno and Matthew M.
PohlmanCalifornia Institute of Technology
in collaboration with Raul A. Radovitzky,
Massachusetts Institute of Technology
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Standard Methods

• Piecewise interpolation methods are Cn for
  "small" n, typically n0, 1 or 2
• Trigonometric interpolation methods require
  evenly spaced data for efficiency

Present Approach

• Use Fourier method to obtain a C interpolation
  of unevenly spaced data in O(N log N N log 1/e)
  where e measures desired accuracy

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Unevenly Spaced FFT (USFFT)

• Algorithm developed by Dutt and Rokhlin (1993) to
  compute for unevenly spaced xj
• Convolve unevenly spaced data f with smooth
  filter h to obtain evenly spaced values of fh
• Use FFT to compute and then
  divide to obtain

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USFFT Algorithm, continued

• To find the Fourier coefficients of a function,
  we must invert the linear transformation in the
  direct USFFT shown above
• This can be done efficiently with the Conjugate
  Gradient method since the resulting matrix is
  both Hermitian and Toeplitz

• The overdetermined linear system is solved using
  the normal equation formulation of least-squares

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1-Dimensional Example

• The periodic function exp(sin(2px)) is
  interpolated using Fourier modes obtained from
  the USFFT algorithm
• Data points are from randomly spaced values on
  curve

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1-Dimensional Example using Partition of Unity
(POU)
USFFT used to obtain Fourier coefficients
for interpolation
Unevenly spaced data of top half of a sphere
is multiplied by POU to enforce periodicity
Result divided by POU and pared to
avoid catastrophic division

• We see spectral accuracy in the interpolant once
  the POU can be resolved

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Interpolation of a sphere

• Sphere is divided into 6 patches, one for each
  axis
• Each patch is multiplied by a POU to enforce
  periodicity
• Once the POU can be resolved by the Fourier
  modes, the interpolation has spectral accuracy

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Interpolation of a bean
Data is from a triangulation of a bean
shape (Radovitzky)
Patch1
Bean is divided into 6 patches, which are fed
into Fourier interpolation method
Patch2
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Accuracy of bean interpolation
Patch3
Patch4
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Bunny
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What's the use?

• Fast high-order scattering method (Bruno,
  Kunyansky, Hyde, Paffenroth) uses several
  continuous derivatives to achieve spectral
  accuracy

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Conclusions

• C interpolation of unevenly spaced data
• Interpolation of surfaces obtained using
  partitions of unity and patches of surface grid
• O(N log N N log 1/e) computational complexity
• Potential improvement using numerically-tuned
  convolution filters (Duijndam and Schonewille,
  1999)