Approximation Properties of Subdivision Surfaces

153 Pages·2012·1.17 MB·English

Approximation Properties of
Subdivision Surfaces
Greg Arden
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
University of Washington
2001
Program Authorized to Offer Degree: Mathematics

University of Washington
Graduate School
This is to certify that I have examined this copy of a doctoral dissertation by
Greg Arden
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
Chair of Supervisory Committee:
Tom Duchamp
Reading Committee:
Tom Duchamp
Ken Bube
Randy LeVeque
Date:

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University of Washington
Abstract
Approximation Properties of
Subdivision Surfaces
by Greg Arden
Chair of Supervisory Committee:
Professor Tom Duchamp
Department of Mathematics
SplinesarepiecewisepolynomialfunctionsdefinedonadomaininEuclideanspace. Because
they are easily computed and have high-order approximation power, they are useful for
modeling surfaces. Modeling a complex surface with splines typically requires a number of
spline patches, which must be smoothly joined, making splines cumbersome to use.
Subdivision schemes generalize splines to domains of arbitrary topology. Thus, subdi-
vision functions can be used to model complex surfaces without the need to join patches.
Like splines, subdivision schemes have a multiresolution structure (i.e, a nested sequence of
function spaces) associated to subdivisions of the domain. This thesis shows that a particu-
lar class of subdivision functions also have high-order approximation power. Although only
one subdivision scheme, Loop’s, is analyzed, the approach appears to be more general.
The main result is an approximation theorem in Sobolev spaces Hs of functions with
square integrable derivatives up to order s. It is shown that each function f in Hr, r 3
≤
can be approximated in Hs, s < r, with error on the order of λ(r s (cid:178))k, where k is the
− −
multiresolution level of the approximation, λ is a number between 1/2 and 5/8, associated
to the valences of the vertices of the domain complex, and (cid:178) > 0 is arbitrary.
This approximation theorem provides a theoretical foundation for various applications
of subdivision schemes, such as the solution of thin shell problems in elasticity.

TABLE OF CONTENTS
List of Figures iv
Glossary v
Chapter 1: Introduction 1
1.1 Subdivision Properties of Splines . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Approximation Results for Meshes and Splines . . . . . . . . . . . . . . . . . 5
1.3 Loop’s Subdivision Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2: Fundamentals of Subdivision Surfaces 11
2.1 Stationary Subdivision on Simplicial Complexes . . . . . . . . . . . . . . . . . 11
2.1.1 Simplicial Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Subdivision Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Subdivision Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 Examples of Subdivision Schemes . . . . . . . . . . . . . . . . . . . . . 18
2.2 Affine Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 The n Regular Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
−
2.4 The Subdivision Map and the Characteristic Map. . . . . . . . . . . . . . . . 23
2.4.1 Characteristic Maps on K . . . . . . . . . . . . . . . . . . . . . . . 27
n
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2.4.2 Equivariant Characteristic Maps . . . . . . . . . . . . . . . . . . . . . 30
2.5 Subdivision Smooth Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6 A Little More Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Some Geometric Properties of Neighborhoods . . . . . . . . . . . . . . . . . . 40
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Chapter 3: Global Approximation in Sobolev Spaces 48
3.1 Sobolev Spaces in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Sobolev Spaces on Simplicial Surfaces . . . . . . . . . . . . . . . . . . . . . . 54
3.3 A Global Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 4: Approximation Theory on the n–Regular Complex 62
4.1 Approximation by Quasi-Interpolants . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Background on Averaged Taylor Polynomials . . . . . . . . . . . . . . . . . . 66
4.3 Proof of the Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 Estimation of the First Term . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 Estimation of the Second Term . . . . . . . . . . . . . . . . . . . . . . 70
4.4 An Alternative Characterization of a Quasi-Interpolant . . . . . . . . . . . . 77
Chapter 5: Constructing Quasi-Interpolants 81
5.1 Box Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Boundedness of Loop’s Subdivision and Restriction . . . . . . . . . . . . . . . 86
5.3 A Quasi-Interpolant on K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
n
5.4 Solving the Polynomial Reproducing System . . . . . . . . . . . . . . . . . . . 103
5.5 Class I: Away From Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6 Class II: Near an edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.7 Class III: Near a Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.7.1 Two Symmetric Box Spline Decompositions . . . . . . . . . . . . . . . 113
5.7.2 Corner Case A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.7.3 Corner Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter 6: Applications and Future Directions 125
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1.1 Smooth Structures on Simplicial Surfaces . . . . . . . . . . . . . . . . 125
6.1.2 Approximation in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . 127
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