High-order Surface Representation Oscar P. Bruno and Matthew M. - - PowerPoint PPT Presentation

High-order Surface Representation

Oscar P. Bruno and Matthew M. Pohlman, California Institute of Technology

Background

• Standard Methods

– Piecewise interpolation methods are fast, but only Cn for "small" n, n ∈ {0, 1} for surfaces – Efficient trigonometric interpolation methods (using the FFT) are C∞ but require evenly spaced data

• Current Approach

– Use Fourier Series for C∞ representation of surface patches – Take advantage of an Unevenly Spaced Fast Fourier Transform (USFFT) to obtain a Fourier coefficients from irregular data with accuracy ε in O(N log N + N log 1/ε) time [Dutt & Rokhlin, 1993] – Continue discrete data to smooth periodic functions in order to preserve spectral convergence of Fourier methods (Continuation Method)

Unevenly Spaced Fast Fourier Transform

• From unevenly spaced data f, sample the convolution f∗g, at

evenly spaced locations in O(MN) time

– the convolution filter g is known analytically – for error tolerance ε, the necessary sample rate of g for discrete convolution step is M=O(log 1/ε) → M≈32 when ε≈1e-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
• verlapping patches take care of regions where POU is small

Interpolation using POU

partition of unity data∗POU FFT and division by POU (regions where POU ¿ 1 are discarded)

• riginal 1D and 2D

unevenly spaced data

Interpolation of Non-periodic Data

• patches can’t overlap an edge or corner smoothly, so POU

can’t 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

• riginal 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

Theoretical Results

Direct FFT: Gibbs phenomenon Double-period continuation:

• scillations

Continuation and smoothing! Given data

Continuation Non-periodic Data

Convergence of Continuation Method

Generalizes to any number of dimensions!

Gegenbauer-polynomial approach

Comparison with other methods

• 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]-… Error Function

Comparison w/ other methods (contd.):

Singular Padè-Fourier approach

Function Fourier sum

“The proposed approach exhibits the significant advantage of being able to deal with arbitrary data sets (non-square domains, non-uniformly- spaced data, arbitrary dimensionality), and yet, it yields more accurate results than other available methods”

Padè-Fourier sum Singular Padè-Fourier sum

Driscoll and Fornberg [2001]

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)

Wing represented by eleven

• verlapping patches, each patch

given explicitly by three coordinate functions (which are Fourier Series!)

Refined mesh shown here generated via surface interpolation

Surface Interpolation of Wing Patch

Wing Edges

A change of variables in parameter space gives an unevenly sampled surface for accurate resolution of edge- scattering

It is easy to compute surface normals and curvatures by differentiation of Fourier Series representation! Fine array of surface normals plotted on interpolated wing surface

Wing Normals

A Few Other Patches

Canopy Nose Top

Conclusions

• interpolation using USFFT together with POU or Periodic Continuation is spectrally accurate

– USFFT is O(N log N) for fixed accuracy (ε≈1e-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