Building Interpolating and Approximating Implicit Surfaces Using Moving Least Squares

104 Pages·2007·7.59 MB·English

Building Interpolating and Approximating Implicit
Surfaces Using Moving Least Squares
Chen Shen
Electrical Engineering and Computer Sciences
University of California at Berkeley
Technical Report No. UCB/EECS-2007-14
http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-14.html
January 12, 2007
Copyright © 2007, by the author(s).
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Building Interpolating and Approximating Implicit Surfaces Using Moving
Least Squares
by
Chen Shen
M.S. (Chinese Academy of Science, China) 2000
A dissertation submitted in partial satisfaction
of the requirements for the degree of
Doctor of Philosophy
in
Computer Sciences
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge:
Professor James F. O’Brien, Chair
Professor Jonathan Shewchuk
Professor Carlo H. Squin
Professor Sara McMains
Fall 2006
The dissertation of Chen Shen is approved.
Chair Date
Date
Date
Date
University of California, Berkeley
Fall 2006
Building Interpolating and Approximating Implicit Surfaces Using Moving Least
Squares
Copyright (cid:13)c 2006
by
Chen Shen
Abstract
Building Interpolating and Approximating Implicit Surfaces Using Moving Least
Squares
by
Chen Shen
Doctor of Philosophy in Computer Sciences
University of California, Berkeley
Professor James F. O’Brien, Chair
This dissertation addresses the problems of building interpolating or approximating im-
plicit surfaces from a heterogeneous collection of geometric primitives like points, polygons,
and spline/subdivision surface patches. The user can choose to generate a surface that
exactly interpolates the geometric elements, or a surface that approximates the input by
smoothing away features smaller than some user-specified size. The implicit functions are
represented using a scattered data interpolation formulation known as moving least-squares
withconstraintsatinputpointsorintegratedovertheparametricdomainofeachpolygonor
surface patch. This dissertation also proposes an improved technique for enforcing normal
constraints that overcomes undesirable oscillatory behavior produced by previous methods.
Multiple points, polygons and surface patches can be blended together by a single implicit
function whose isosurface is a manifold envelope that either interpolates or approximates
the original input, even when self-intersections, holes, or other defects are present. With
an iterative procedure for ensuring that the implicit surface tightly encloses the input ele-
ments, the resulting clean, manifold surface can then be used for generating volume meshes
for finite elements, manufacturing rapid prototyping models, and other applications that
require manifold surfaces.
1
Professor James F. O’Brien
Dissertation Committee Chair
2
To my wife, Kun,
for your love and support.
i
Contents
Contents ii
List of Figures iv
List of Tables vi
Acknowledgments vii
1 Introduction 1
2 Background 6
2.1 Implicit Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Algebraic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Blobby Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Functional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Scattered Data Interpolation and Approximation . . . . . . . . . . . . . . . 9
2.2.1 Shepard’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.3 Thin Plate Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.5 Meshless Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Related Work on Implicit Surfaces . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Explicit Mesh Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Moving Least-Squares Basics 14
3.1 Standard Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Moving Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
ii
3.3 Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 General Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Implicit Moving Least-Squares 24
4.1 2D Implicit Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Pseudo-Normal Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 True-Normal Constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 Application: Surface Reconstruction from Range Scan Data . . . . . . . . . 32
5 Integrating Moving Least-Squares 34
5.1 Problems of Finite Point Constraints . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Achieving Infinite Point Constraints . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Integration Over Line Segments . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.1 Function Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3.2 Function Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Integration Over Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4.1 Function Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4.2 Function Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4.4 Integration Over Parametric Patches . . . . . . . . . . . . . . . . . . 61
6 Implementation Details 67
6.1 Interpolation and Approximation . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Fast Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1 Details of Hierarchical Fast Evaluation over Triangles . . . . . . . . 72
6.3 PreProcessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7 Results 77
7.1 Polygon Soup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Preprocessor for Rapid Prototyping Machines . . . . . . . . . . . . . . . . . 80
7.3 Simulation Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.4 Parametric Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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