Highorder Surface Representation

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High-order Surface Representation
Oscar P. Bruno and Matthew M. Pohlman, California
Institute of Technology
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Background

• Standard Methods
• Piecewise interpolation methods are fast, but
  only Cn for "small" n,n 2 0, 1 for surfaces
• Efficient trigonometric interpolation methods
  (using the FFT) are C1 but require evenly spaced
  data
• Current Approach
• Use Fourier Series for C1 representation of
  surface patches
• Take advantage of an Unevenly Spaced Fast Fourier
  Transform (USFFT) to obtain a Fourier
  coefficients from irregular data with accuracy ?
  in O(N log N N log 1/?) time Dutt Rokhlin,
  1993
• Continue discrete data to smooth periodic
  functions in order to preserve spectral
  convergence of Fourier methods (Continuation
  Method)

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Unevenly Spaced Fast Fourier Transform

• From unevenly spaced data f, sample the
  convolution fg, at evenly spaced locations in
  O(MN) time
• the convolution filter g is known analytically
• for error tolerance ?, the necessary sample rate
  of g for discrete convolution step is MO(log
  1/?) ! M¼32 when ?¼1e-16
• Use standard FFT to move from spatial domain to
  frequency domain, O(N log N) time
• Divide by the Fourier coefficients of filter g to
  obtain Fourier coefficients of f, O(N) time

Patches and Partitions of Unity

• Use partitions of unity (POU) to divide surface
  into patches, just like in surface scattering
  Bruno, et. al.
• unknowns become smooth periodic functions on each
  patch
• overlapping patches take care of regions where
  POU is small

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Interpolation using POU
FFT and division by POU (regions where POU 1
are discarded)
partition of unity
dataPOU
original 1D and 2D unevenly spaced data
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Interpolation of Non-periodic Data

• patches cant overlap an edge or corner smoothly,
  so POU cant solve all issues for a complicated
  geometry!
• need a spectrally accurate interpolation method
  for smooth non-periodic data as well
• IDEA
• find smooth periodic function (over a larger
  period) that coincides with original data
• finding such a periodic function is easy using a
  least-squares fit of truncated Fourier series
• need to also ensure that the result is smooth
• Fourier coefficients of smooth periodic functions
  decay exponentially, so make sure the
  coefficients of the continuation decay
  exponentially too!
• this can be accomplished during the least-squares
  step by multiplying Fourier coefficients by
  appropriate factors and setting equal to zero

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Theoretical Results
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Continuation Non-periodic Data
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Convergence of Continuation Method
Generalizes to any number of dimensions!
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Comparison with other methods
Gegenbauer-polynomial approach

• Does not use information about discontinuity
  location ()
• Requires much finer discretizations for given
  error tolerance (-)
• Only applies to square domains (-)
• Requires use of data at (generally unavailable)
  data points (-)

Gottlieb and Shu 1992-
Function
Error
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Comparison w/ other methods (contd.)
Singular Padè-Fourier approach
The proposed approach exhibits the significant
advantage of being able to deal with arbitrary
data sets (non-square domains, non-uniformly-space
d data, arbitrary dimensionality), and yet, it
yields more accurate results than other available
methods
Fourier sum
Function
Padè-Fourier sum
Singular Padè-Fourier sum
Driscoll and Fornberg 2001
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Surface Parameterization

• Parameterization of each patch is done with the
  Intrinsic Parameterization initially developed in
  the CG community to minimize texture map
  deformation Desbrun, Meyer and Alliez, 2002
• Human intervention is currently needed to rescale
  singularities in the parameterization (which
  occur where the geometry has large curvature)

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Surface Interpolation of Wing Patch
Refined mesh shown here generated via surface
interpolation
Wing represented by eleven overlapping patches,
each patch given explicitly by three coordinate
functions (which are Fourier Series!)
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Wing Edges
A change of variables in parameter space gives an
unevenly sampled surface for accurate resolution
of edge-scattering
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Wing Normals
Fine array of surface normals plotted on
interpolated wing surface
It is easy to compute surface normals and
curvatures by differentiation of Fourier Series
representation!
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A Few Other Patches
Canopy
Nose
Top
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Conclusions

• interpolation using USFFT together with POU or
  Periodic Continuation is spectrally accurate
• USFFT is O(N log N) for fixed accuracy (?¼1e-16)
• Continuation is O(N3) due to SVD but necessary
  only for small patches of a geometry
• evaluation of Fourier Series is O(N log N) in
  either case
• can accurately evaluate the surface derivatives
  needed by scattering codes for what began as a
  triangle mesh

Future Work

• Self-tuning continuation methods for more general
  interpolation applications
• Automatic parameterization of much more
  complicated surface patches to improve
  computational efficiency in scattering codes

References

• Dutt and Rokhlin, Fast Fourier Transforms for
  Nonequispaced Data, 1993
• Duijndam and Schonewille, Nonuniform fast Fourier
  transform, 1999
• Desbrun, Meyer and Alliez, Intrinsic
  Parameterizations of Surface Meshes, 2002
• Bruno and Pohlman, Smooth Interpolation of
  Unevenly Spaced Data, in preparation