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 C n 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 • overlapping patches take care of regions where POU is small

Interpolation using POU FFT and division data ∗ POU partition of unity original 1D and 2D by POU (regions unevenly spaced where POU ¿ 1 data are discarded)

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 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

Theoretical Results

Continuation Non-periodic Data Direct FFT: Given data Gibbs phenomenon Continuation Double-period continuation: and smoothing! oscillations

Convergence of Continuation Method Generalizes to any number of dimensions!

Comparison with other methods Gegenbauer-polynomial approach • Does not use information about discontinuity location (+) • Requires much finer discretizations for given Gottlieb and Shu [1992]-… error tolerance (-) • Only applies to square domains (-) • Requires use of data at (generally unavailable) data points (-) Function Error

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 Function domains, non-uniformly- Fourier sum 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)

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!)

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

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!

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 (N 3 ) 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