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Building Interpolating and Approximating Implicit Surfaces Using Moving Least Squares PDF

104 Pages·2007·7.59 MB·English
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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). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. 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 iii

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This dissertation addresses the problems of building interpolating or .. This capacity allows us to generate a family of increasingly Implicit methods provide mathematical tractability and are becoming extremely useful for modeling
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