Geometric Modeling and Processing - GMP 2006: 4th International Conference, GMP 2006, Pittsburgh, PA, USA, July 26-28, 2006, Proceedings (Lecture Notes in Computer Science, 4077) 354036711X, 9783540367116

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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

4077

Myung-Soo Kim Kenji Shimada (Eds.)

Geometric Modeling and Processing – GMP 2006 4th International Conference Pittsburgh, PA, USA, July 26-28, 2006 Proceedings

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Volume Editors Myung-Soo Kim Seoul National University, School of Computer Science and Engineering Seoul 151-742, Korea E-mail: [email protected] Kenji Shimada Carnegie Mellon University, Mechanical Engineering Pittsburgh, PA 15213, USA E-mail: [email protected]

Library of Congress Control Number: 2006929220 CR Subject Classification (1998): I.3.5, I.3.7, I.4.8, G.1.2, F.2.2, I.5, G.2 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-540-36711-X Springer Berlin Heidelberg New York 978-3-540-36711-6 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 11802914 06/3142 543210

Preface

This book contains the proceedings of Geometric Modeling and Processing 2006, the fourth in a biennial international conference series on geometric modeling, simulation and computing, which was held July 26–28, 2006 in Pittsburgh, USA. The previous conferences were in Hong Kong (2000), Tokyo (2002), and Beijing (2004). The next conference (GMP 2008) will be held in China. GMP 2006 received 84 paper submissions, covering various areas of geometric modeling and processing. Based on the recommendations of 114 reviewers, 36 regular papers were selected for conference presentation, and 21 short papers were accepted for poster presentation. The authors of these proceedings come from Austria, Belgium, Canada, Chile, China, Colombia, Greece, Indonesia, Israel, Japan, Korea, Lebanon, Singapore, the UK, and USA. We are grateful to the authors who submitted to GMP 2006 and to the many dedicated reviewers. Their creativity and hard work substantially contributed to the technical program of the conference. We would also like to thank the members of the Program Committee for their strong support. We also wish to thank David Gossard, GMP2006 Conference Chair, and past GMP Program Co-chairs Shimin Hu, Ralph Martin, Helmut Pottmann, Hiromasa Suzuki, and Wenping Wang, whose current and previous work has helped to establish this conference as a major event in geometric modeling and processing. Further, we wish to thank the members of the Computer Integrated Engineering Laboratory at Carnegie Mellon University, in particular Soji Yamakawa, for their invaluable assistance throughout the conference preparation. We gratefully acknowledge the financial support of Carnegie Mellon University. Finally, we wish to thank all conference participants for making GMP 2006 a success. We hope that the readers will enjoy this book. In our view, it impressively demonstrates the rapid progress in geometric modeling and processing. It shows the importance and range of this field, with its impact in such areas as computer graphics, computer vision, machining, robotics, and scientific visualization. Finally, we hope that the conference and its proceedings will stimulate further exciting research. Myung-Soo Kim Kenji Shimada

Conference Committee

Conference Chair David Gossard (Massachusetts Institute of Technology, USA)

Program Co-chairs Myung-Soo Kim (Seoul National University, Korea) Kenji Shimada (Carnegie Mellon University, USA)

Steering Committee Shimin Hu (Tsinghua University, China) Ralph Martin (Cardiff University, UK) Helmut Pottmann (Institut f¨ ur Geometrie, TU Wien, Austria) Hiromasa Suzuki (University of Tokyo, Japan) Wenping Wang (Hong Kong Univsersity, Hong Kong)

Program Committee Chandrajit Bajaj (University of Texas at Austin, USA) Hujun Bao (Zhejiang University, China) Alexander Belyaev (Max-Planck-Institut f¨ ur Informatik, Germany) Wim Bronsvoort (Delft University of Technology, The Netherlands) Stephen Cameron (Oxford University, UK) Fuhua (Frank) Cheng (University of Kentucky, USA) Falai Chen (University of Science and Technology, China) Eng Wee Chionh (National University of Singapore, Singapore) Jian-Song Deng (University of Science and Technology, China) Gershon Elber (Technion, Israel) Rida Farouki (University of California, Davis, USA) Gerald Farin (Arizona State University, USA) Anath Fischer (Technion, Israel) Michael Floater (SINTEF Applied Mathematics, Norway) Xiao-Shan Gao (Chinese Academy of Sciences, China) Ron Goldman (Rice University, USA) Craig Gotsman (Technion, Israel) Xianfeng Gu (State University of New York, Stony Brook, USA) Baining Guo (Microsoft Research Asia, China) Satyandra K. Gupta (University of Maryland, USA)

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Organization

Soonhung Han (KAIST, Korea) Shimin Hu (Tsinghua University, China) Christoph Hoffmann (Purdue University, USA) Leo Joskowicz (The Hebrew University of Jerusalem, Israel) Tao Ju (Washington University, St. Louis, USA) Bert J¨ uttler (Johannes Kepler Universit¨ at Linz, Austria) Satoshi Kanai (Hokkaido University, Japan) Takashi Kanai (RIKEN, Japan) Deok-Soo Kim (Hanyang University, Korea) Tae-Wan Kim (Seoul National University, Korea) Young J. Kim (Ewha Womans University, Korea) Leif Kobbelt (RWTH Aachen, Germany) Haeyoung Lee (Hongik University, Korea) In-Kwon Lee (Yonsei University, Korea) Seungyong Lee (POSTECH, Korea) Ligang Liu (Zhejiang University, China) Weiyin Ma (City University of Hong Kong, Hong Kong) Takashi Maekawa (Yokohama National University, Japan) Ralph Martin (Cardiff University, UK) Hiroshi Masuda (University of Tokyo, Japan) Kenjiro Miura (Shizuoka University, Japan) Ahmad H. Nasri (American University of Beirut, Lebanon) Ryutarou Ohbuchi (Yamanashi University, Japan) Yutaka Ohtake (RIKEN, Japan) Alexander Pasko (Hosei University, Japan) Martin Peternell (Institut f¨ ur Geometrie, TU Wien, Austria) Helmut Pottmann (Institut f¨ ur Geometrie, TU Wien, Austria) Hartmut Prautzsch (Universitaet Karlsruhe, Germany) Hong Qin (State University of New York, Stony Brook, USA) Stephane Redon (INRIA Rhone-Alpes, France) Maria Cecilia Rivara (Universidad de Chile, Chile) Nicholas Sapidis (University of the Aegean, Greece) Vadim Shapiro (University of Wisconsin-Madison, USA) Hayong Shin (KAIST, Korea) Yoshihisa Shinagawa (University of Illinois at Urbana-Champaign, USA) Yohanes Stefanus (University of Indonesia) Kokichi Sugihara (University of Tokyo, Japan) Hiromasa Suzuki (University of Tokyo, Japan) Chiew-Lan Tai (Hong Kong University of Science and Technology, Hong Kong) Shigeo Takahashi (University of Tokyo, Japan) Kai Tang (Hong Kong University of Science and Technology, Hong Kong) Changhe Tu (Shandong University, China) Tamas V´arady (Geomagic Hungary, Hungary) Johannes Wallner (Institut f¨ ur Geometrie, TU Wien, Austria) Charlie Wang (The Chinese University of Hong Kong)

Organization

Guojin Wang (Zhejiang University, China) Jiaye Wang (Shandong University, China) Michael Wang (The Chinese University of Hong Kong) Wenping Wang (Hong Kong University, Hong Kong) Joe Warren (Rice University, USA) Soji Yamakawa (Carnegie Mellon University, USA) Hong-Bin Zha (Peking University, China) Kun Zhou (Microsoft Research Asia, China)

Additional Reviewers Sigal Ar Oscar Kin-Chung Au Sergei Azernikov Anna Vilanova i Bartroli Silvia Biasotti Jung-Woo Chang Yoo-Joo Choi Hongbo Fu Iddo Hanniel Yaron Holdstein Martin Isenburg David Johnson Sujeong Kim Ji-Yong Kwon Shuhua Lai Yu-Kun Lai Torsten Langer Jae Kyu Lee Jieun Jade Lee

Zhouchen Lin Yang Liu Yong-jin Liu Alex Miropolsky Muthuganapathy Ramanathan Malcolm Sabin Oliver Schall Guy Sela Olga Sorkine Raphael Straub Han-Bing Yan Yong-Liang Yang Xu Yang Min-Joon Yoo Jong-Chul Yoon Seung-Hyun Yoon Weiwei Xu Xinyu Zhang Qian-Yi Zhou

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Table of Contents

Shape Reconstruction Automatic Extraction of Surface Structures in Digital Shape Reconstruction Tamas V´ arady, Michael A. Facello, Zsolt Ter´ek . . . . . . . . . . . . . . . . . . . .

1

Ensembles for Normal and Surface Reconstructions Mincheol Yoon, Yunjin Lee, Seungyong Lee, Ioannis Ivrissimtzis, Hans-Peter Seidel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Adaptive Fourier-Based Surface Reconstruction Oliver Schall, Alexander Belyaev, Hans-Peter Seidel . . . . . . . . . . . . . . . .

34

Curves and Surfaces I Least–Squares Approximation by Pythagorean Hodograph Spline Curves Via an Evolution Process ˇır, Bert J¨ Martin Aigner, Zbynek S´ uttler . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Geometric Accuracy Analysis for Discrete Surface Approximation Junfei Dai, Wei Luo, Shing-Tung Yau, Xianfeng David Gu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Quadric Surface Extraction by Variational Shape Approximation Dong-Ming Yan, Yang Liu, Wenping Wang . . . . . . . . . . . . . . . . . . . . . . .

73

Geometric Processing I Tracking Point-Curve Critical Distances Xianming Chen, Elaine Cohen, Richard F. Riesenfeld . . . . . . . . . . . . . .

87

Theoretically Based Robust Algorithms for Tracking Intersection Curves of Two Deforming Parametric Surfaces Xianming Chen, Richard F. Riesenfeld, Elaine Cohen, James Damon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Subdivision Termination Criteria in Subdivision Multivariate Solvers Iddo Hanniel, Gershon Elber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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Towards Unsupervised Segmentation of Semi-rigid Low-Resolution Molecular Surfaces Yusu Wang, Leonidas J. Guibas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Curves and Surfaces II Piecewise Developable Surface Approximation of General NURBS Surfaces, with Global Error Bounds Jacob Subag, Gershon Elber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Efficient Piecewise Linear Approximation of B´ezier Curves with Improved Sharp Error Bound Weiyin Ma, Renjiang Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Approximate μ-Bases of Rational Curves and Surfaces Liyong Shen, Falai Chen, Bert J¨ uttler, Jiansong Deng . . . . . . . . . . . . . . 175

Shape Deformation Inverse Adaptation of Hex-dominant Mesh for Large Deformation Finite Element Analysis Arbtip Dheeravongkit, Kenji Shimada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Preserving Form-Features in Interactive Mesh Deformation Hiroshi Masuda, Yasuhiro Yoshioka, Yoshiyuki Furukawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Surface Creation and Curve Deformations Between Two Complex Closed Spatial Spline Curves Joel Daniels II, Elaine Cohen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Shape Description Computing a Family of Skeletons of Volumetric Models for Shape Description Tao Ju, Matthew L. Baker, Wah Chiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Representing Topological Structures Using Cell-Chains David E. Cardoze, Gary L. Miller, Todd Phillips . . . . . . . . . . . . . . . . . . . 248 Constructing Regularity Feature Trees for Solid Models M. Li, F.C. Langbein, R.R. Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

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XIII

Insight for Practical Subdivision Modeling with Discrete Gauss-Bonnet Theorem Ergun Akleman, Jianer Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Shape Recognition Shape-Based Retrieval of Articulated 3D Models Using Spectral Embedding Varun Jain, Hao Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Separated Medial Surface Extraction from CT Data of Machine Parts Tomoyuki Fujimori, Yohei Kobayashi, Hiromasa Suzuki . . . . . . . . . . . . . 313 Two-Dimensional Selections for Feature-Based Data Exchange Ari Rappoport, Steven Spitz, Michal Etzion . . . . . . . . . . . . . . . . . . . . . . . 325

Geometric Modeling Geometric Modeling of Nano Structures with Periodic Surfaces Yan Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Minimal Mean-Curvature-Variation Surfaces and Their Applications in Surface Modeling Guoliang Xu, Qin Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Parametric Design Method for Shapes with Aesthetic Free-Form Surfaces Tetsuo Oya, Takenori Mikami, Takanobu Kaneko, Masatake Higashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Curves and Surfaces III Control Point Removal Algorithm for T-Spline Surfaces Yimin Wang, Jianmin Zheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Shape Representations with Blossoms and Buds L. Yohanes Stefanus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Manifold T-Spline Ying He, Kexiang Wang, Hongyu Wang, Xianfeng Gu, Hong Qin . . . . 409

Subdivision Surfaces √ Composite 2 Subdivision Surfaces Guiqing Li, Weiyin Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

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Tuned Ternary Quad Subdivision Tianyun Ni, Ahmad H. Nasri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

Geometric Processing II Simultaneous Precise Solutions to the Visibility Problem of Sculptured Models Joon-Kyung Seong, Gershon Elber, Elaine Cohen . . . . . . . . . . . . . . . . . . 451 Density-Controlled Sampling of Parametric Surfaces Using Adaptive Space-Filling Curves J.A. Quinn, F.C. Langbein, R.R. Martin, G. Elber . . . . . . . . . . . . . . . . . 465

Engineering Applications Verification of Engineering Models Based on Bipartite Graph Matching for Inspection Applications Fabricio Fishkel, Anath Fischer, Sigal Ar . . . . . . . . . . . . . . . . . . . . . . . . . 485 A Step Towards Automated Design of Side Actions in Injection Molding of Complex Parts Ashis Gopal Banerjee, Satyandra K. Gupta . . . . . . . . . . . . . . . . . . . . . . . . 500 Finding All Undercut-Free Parting Directions for Extrusions Xiaorui Chen, Sara McMains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

Short Papers Robust Three-Dimensional Registration of Range Images Using a New Genetic Algorithm John Willian Branch, Flavio Prieto, Pierre Boulanger . . . . . . . . . . . . . . 528 Geometrical Mesh Improvement Properties of Delaunay Terminal Edge Refinement Bruce Simpson, Maria-Cecilia Rivara . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 Matrix Based Subdivision Depth Computation for Extra-Ordinary Catmull-Clark Subdivision Surface Patches Gang Chen, Fuhua (Frank) Cheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Hierarchically Partitioned Implicit Surfaces for Interpolating Large Point Set Models David T. Chen, Bryan S. Morse, Bradley C. Lowekamp, Terry S. Yoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

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XV

A New Class of Non-stationary Interpolatory Subdivision Schemes Based on Exponential Polynomials Yoo-Joo Choi, Yeon-Ju Lee, Jungho Yoon, Byung-Gook Lee, Young J. Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Detection of Closed Sharp Feature Lines in Point Clouds for Reverse Engineering Applications Kris Demarsin, Denis Vanderstraeten, Tim Volodine, Dirk Roose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 Feature Detection Using Curvature Maps and the Min-cut/Max-flow Algorithm Timothy Gatzke, Cindy Grimm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 Computation of Normals for Stationary Subdivision Surfaces Hiroshi Kawaharada, Kokichi Sugihara . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Voxelization of Free-Form Solids Represented by Catmull-Clark Subdivision Surfaces Shuhua Lai, Fuhua (Frank) Cheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Interactive Face-Replacements for Modeling Detailed Shapes Eric Landreneau, Ergun Akleman, John Keyser . . . . . . . . . . . . . . . . . . . . 602 Straightest Paths on Meshes by Cutting Planes Sungyeol Lee, Joonhee Han, Haeyoung Lee . . . . . . . . . . . . . . . . . . . . . . . . 609 3D Facial Image Recognition Using a Nose Volume and Curvature Based Eigenface Yeunghak Lee, Ikdong Kim, Jaechang Shim, David Marshall . . . . . . . . . 616 Surface Reconstruction for Efficient Colon Unfolding Sukhyun Lim, Hye-Jin Lee, Byeong-Seok Shin . . . . . . . . . . . . . . . . . . . . . 623 Spectral Sequencing Based on Graph Distance Rong Liu, Hao Zhang, Oliver van Kaick . . . . . . . . . . . . . . . . . . . . . . . . . . 630 An Efficient Implementation of RBF-Based Progressive Point-Sampled Geometry Yong-Jin Liu, Kai Tang, Joneja Ajay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Segmentation of Scanned Mesh into Analytic Surfaces Based on Robust Curvature Estimation and Region Growing Tomohiro Mizoguchi, Hiroaki Date, Satoshi Kanai, Takeshi Kishinami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

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Finding Mold-Piece Regions Using Computer Graphics Hardware Alok K. Priyadarshi, Satyandra K. Gupta . . . . . . . . . . . . . . . . . . . . . . . . . 655 A Method for FEA-Based Design of Heterogeneous Objects Ki-Hoon Shin, Jin-Koo Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 Time-Varying Volume Geometry Compression with 4D Lifting Wavelet Transform Yan Wang, Heba Hamza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 A Surface Displaced from a Manifold Seung-Hyun Yoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Smoothing of Meshes and Point Clouds Using Weighted Geometry-Aware Bases Tim Volodine, Denis Vanderstraeten, Dirk Roose . . . . . . . . . . . . . . . . . . 687 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Automatic Extraction of Surface Structures in Digital Shape Reconstruction Tamas V´arady1 , Michael A. Facello2 , and Zsolt Ter´ek1 1

2

Geomagic Hungary, Ltd., Budapest, Hungary Geomagic, Inc., Research Triangle Park, North Carolina, USA

Abstract. One of the most challenging goals in digital shape reconstruction is to create a high-quality surface model from measured data with a minimal amount of user assistance. We present techniques to automate this process and create a digital model that meets the requirements in mechanical engineering CAD/CAM/CAE. Such a CAD model is composed of a hierarchy of different types of surfaces, including primary surfaces, connecting features and vertex blends at their junctions, and obey a well-defined topological structure that we would like to reconstruct as faithfully as possible. First, combinatorially robust segmentation techniques, borrowed from Morse theory, are presented. This is followed by an algorithm to create a so-called feature skeleton, which is a curve network on the mesh that represents the region structure of the object. The final surface structure comprises the optimally located boundaries of edge blends and setback vertex blends, which are well aligned with the actual geometry of the object. This makes the surface structure sufficient for an accurate, CAD-like surface approximation including both quadrangular and trimmed surface representations. A few representative industrial objects reconstructed by Geomagic systems illustrate the efficiency and quality of the approach. Keywords: digital shape reconstruction, segmentation, combinatorial Morse theory, curve tracing, vertex blends.

1

Introduction

Digital Shape Reconstruction (formerly reverse engineering) deals with converting physical objects into a computer representation. DSR is a particular chapter within a general discipline called Digital Shape Sampling and Processing (DSSP) that integrates all point cloud related computations emerging in various fields [Marks05, Geom06]. There are well-established techniques to create polygonal meshes from measured data, which need to be further converted to a representation suitable for CAD, CAM, and CAE. The biggest challenge is to automate this conversion process while producing a model that meets the requirements of downstream applications, including good structure and high quality surfaces. In Computer Aided Design and – in particular – in mechanical engineering, the majority of objects are composed of (i) relatively large, primary surfaces M.-S. Kim and K. Shimada (Eds.): GMP 2006, LNCS 4077, pp. 1–16, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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T. V´ arady, M.A. Facello, and Z. Ter´ek

connected by (ii) highly-curved connecting features, such as edge blends or freeform steps, and (iii) vertex blends at their junctions. Segmentation is the process of partitioning the polygonal mesh into an accurate and consistent region structure, where each region is a pre-image of the final CAD model [VarMar02]. The quality of segmentation fundamentally determines the quality of the final surface model [VarFac05]; a faithful and geometrically well-aligned region structure is a necessary condition to accurately approximate regions by standard implicit and parametric surfaces according to the above surface hierarchy. In the last few years several approaches have been reported to segment triangular meshes. One group of methods cover the shape by a collection of quadrangles in a consistent manner, that are created by tiling a strongly decimated triangular mesh, or using Voronoi diagrams [EckHop96, HecGar97, LeSSCD98]. Unfortunately, these surface models – due to their uniformity and four-sidedness – cannot properly reproduce standard CAD objects which require general topology and different types of surfaces. Another group of approaches puts the main emphasis on extracting connected regions whose triangles are likely to belong together based on their geometric characteristics. These approaches – including region growing [SapBes95, LeoJaS97], watershed methods [ManWhi99, RazBae03] – are strong in collecting matching geometric data, but they face difficulties in creating a full, consistent topological structure and representing smooth transitions between primary surfaces. A third group of approaches limits the class of bounding surfaces to simple surfaces only [FitEgF97, BenVar01]. This – of course – helps to automate the segmentation process, however, excludes a large set of objects where conventional prismatic parts are combined with complex free-form surfaces. The majority of the segmentation methods use curvature estimations or other indicators to locally qualify the vertices of a triangular mesh using local point neighborhoods [CsaWal00, BenVar04, HuaMen01]. It is a hard problem to find an appropriate threshold to segment a given object. Difficulties are partly due to differences in dimensions, level of measurement noise and unevenly distributed triangulations. It is also hard to find a single, global threshold due to great variations in curvature – take for example the simultaneous detection of very small and large radius fillets. In this paper we introduce a new segmentation approach which combines results from combinatorial Morse Theory with special geometric modeling algorithms that have been adapted for digital shape reconstruction. We would like to create CAD-like structures that reflect the original design intent and make high-quality surface approximation possible. Our main goals are the following: – separate primary regions and highly curved transition regions – create complete and consistent region structures without topological limitations – avoid threshold setting, but offer alternative segmentations, if necessary – provide an automatic procedure with no or minimal user assistance – develop a computationally efficient and robust procedure, that can be used for large scanned data sets and objects with high complexity.

Automatic Extraction of Surface Structures in DSR

3

The paper is structured as follows. In Section 2 we present an overview of the overall process. In Sections 3, 4 and 5 we discuss three particular topics in more detail – Morse complex-based segmentation, curve tracing for building feature skeletons and using setback type vertex blends to complete face loops in the final surface model. Before the concluding remarks, the results of the algorithm are illustrated using an industrial part in Section 6.

2

Overview of the Process

In this section we summarize our shape reconstruction algorithm. A simple schematic part was chosen to illustrate the consecutive phases, as shown in Figures 1(a) and 1(b). The input is a triangulated mesh that was created previously by approximating the measured data points. There are five phases: 1. 2. 3. 4. 5.

Hierarchical Morse Complex Segmentation Feature Skeleton Construction Computing Region Boundaries Surface Structure Creation Surface Fitting

Phase 1: Hierarchical Morse Complex Segmentation. Morse theory studies smooth functions over manifolds [Miln63]. A Morse complex partitions the manifold into a collection of regions by a network of curves that connect the nondegenerate, critical points (minima, maxima, and saddles) of a given function. In our context, we use curvature indicator values estimated at each vertex of the polygonal mesh, and create a piecewise linear function to highlight the highly curved parts of the shape. Utilizing concepts of persistence and prioritization [EdeLeZ02], a hierarchy of topologically simplified segmentations is obtained. Each segmentation consists of monotonic regions, whose boundaries form a combinatorially correct curve network. These curves are composed of polylines of connected triangle edges and are typically ragged. We also perform an additional mesh operation to thicken these boundaries on the mesh. As a result, strips of triangles – called separator sets – are created; simultaneously, the original monotonic regions shrink (see red triangles in Figure 1(a)). Thus, the result of Phase 1 is a set of relatively flat regions having clearly separated by triangles of highly curved transitions. Further details follow in Section 3. Phase 2: Feature Skeleton Construction. In this phase we construct an intermediate data structure called a feature skeleton, which is a smooth curve network of edges running in the middle of separator sets. Special curve tracing algorithms are applied using an estimated translational vector field (Section 4). As shown in Figure 1(a), a separator set may indicate (i) a smooth connecting feature between two adjacent regions, such as a fillet, (ii) a sharp feature, where there is tangential discontinuity between two surfaces, or (iii) a smooth subdividing curve that is defined, or computed, to cut large regions into smaller ones. The above three types of edges are classified in advance, since they require different treatments. Feature edges are only temporary entities and will later

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(a)

(b) Fig. 1. (a) Separator sets and feature skeleton. (b) Features, vertex blends and the final surface structure.

be replaced by a pair of boundary curves. Sharp edges are precisely extracted from the polygon model, and become boundaries in the final surface structure. Smooth subdividing curves lie in the interior of regions where there is no curva-

Automatic Extraction of Surface Structures in DSR

5

ture variation; and similarly to sharp edges, they are smoothed and preserved. We also classify the vertices of the skeleton. In particular, we identify collinear edge pairs that approach a vertex from opposite directions; as an example take T-nodes that are degree-3 vertices with one pair of collinear edges. Phase 3: Computing Region Boundaries. In this phase we create boundary edges for the individual connecting features and vertex blends, see Figure 1(b). First we replace each feature edge of the skeleton by a pair of longitudinal boundaries, then replace each vertex of the skeleton by a loop of a setback vertex blend [Braid97, VarHof98]. As will be discussed in Section 4, the vertex loop may consist of an alternating sequence of profile curves and spring curves. Profile curves terminate the corresponding connecting features, spring curves connect pairs of adjacent feature boundaries lying on the same primary region. Spring curves must be inserted due to various reasons – see the T-node, or the convexconcave vertex blend in Figure 1(b); they may also degenerate with zero length. Phase 4: Surface Structure Creation. The thickened feature skeleton comprises the previously determined boundaries of the connecting features and the loops of the vertex blends that also determine the loop structure of the primary regions, as shown in Figure 1(b). The connecting features are generally four-sided lying between two primaries and terminated by two vertex blends. The vertex blends terminate the features, but they may also share spring curves with adjacent primary regions. For a given primary region, the thickened edges of the corresponding loop may contain self-intersections, which must be removed for a valid structure. This step may modify the topology of the original thickened feature skeleton, as will be discussed in Section 5. Phase 5: Surface Fitting. The above algorithm leads to a geometrically wellaligned structure whose existence is necessary for high-quality surface fitting. By construction the relatively large primary regions are empty of highly curved features, and the feature regions are free from possible artifacts coming from the primaries. There are two alternative techniques to approximate this surface structure. (i) A collection of smoothly connected quadrangular tiles can be used for rapid surfacing, or (ii) trimmed surfaces and special features can be fitted to obtain conventional CAD models. The latter yields much better surface quality, but requires more extensive computations. Alternative concepts of surface fitting techniques have been analyzed in [VarFac05], however, in this paper our only interest is the extraction of surface structures.

3

Hierarchical Morse Complex Segmentation

Our segmentation utilizes results from Combinatorial Morse Theory, which analyzes functions defined over manifolds [EdeHaZ03]. These manifolds are represented as polygonal meshes, produced from point clouds. In our context a function approximates an unknown, smooth function in a piecewise linear form; often the term indicator function will be used. In the following paragraphs we introduce basic concepts from Morse Theory in a nutshell, and simplification

6

T. V´ arady, M.A. Facello, and Z. Ter´ek

(a)

(b)

Fig. 2. (a) Morse Complex segmentation and related quadrilaterals. (b) Feature-based segmentation using the Morse regions.

algorithms applied to Morse complexes that eventually lead to the feature-based region structures. Morse Theory. Classical Morse theory studies critical points of generic, smooth functions on manifolds [Miln63]. Let M be a 2-manifold and f : M → R a smooth function. At a point x ∈ M , assume a local orthonormal coordinate system; the gradient ∇f (x) is the vector of the local partial derivatives. The point x is critical if its gradient is zero, and regular otherwise. A critical point is non-degenerate if its Hessian is nonsingular, which is a property independent of the coordinate system. There are three types of non-degenerate critical points: minima, saddles, and maxima. It can be shown that the non-degenerate critical points are isolated. Technically f is considered a Morse function if (i) all critical points are non-degenerate; and (ii) the critical points have pair-wise different heights. The gradient forms a smooth vector field on M . An integral line is a curve traced on the surface whose tangent coincides with the local gradient of M for all of its points. It always starts at a critical point and ends at another critical point without containing the endpoints. Because f is smooth, two integral lines are either disjoint or identical. A descending manifold D(x) of a critical point x is the set of points that flow toward x, i.e. the point x and all points of the integral lines ending at x. For a minimum, D(x) is identical to point x; for a saddle, D(x) contains two connecting integral lines; for a maximum, D(x) is an open disk. The Morse Complex is the collection of these disjoint manifolds: the boundary curves connect the saddles and the minima, and thus form a curve network that separates all simple monotonic regions of the surface, each assigned to a maximum. Piecewise linear functions. The main effort in combinatorial topology is to turn the mathematical ideas of Morse theory into algorithms that operate on piecewise linear functions. Our assumption is that we have a triangulation K homeomor-

Automatic Extraction of Surface Structures in DSR

7

phic to a 2-manifold M . Our indicator function f is explicitly defined at the vertices and linearly interpolated over the edges and triangles of K. For a good, global segmentation, we assume that our indicator function distinguishes highly curved and relatively flat parts. The majority of practical indicator functions are related – directly or indirectly – to surface curvature or its reciprocal value, including planarity, mean or Gaussian curvature, dimensionality, slippage, and others, which are estimated using a point neighborhood around a given vertex; for details see [CsaWal00, BenVar04, GelGui04]. Our experience is that the choice of the indicator function is not so crucial, and most of them provide reasonable segmentation when Morse theory is applied. The simplest indicator is planarity, defined by the error term of a local best-fit-plane; alternatively, curvature estimations based on local, low-degree implicit functions were also found reasonably stable and computationally acceptable. Locally estimated mean curvature was used for the examples of this paper; see Figure 8(c). Constructing and simplifying a Morse complex. Our goal is to construct the descending manifolds of the local maxima, and create “rivers” which separate the regions and flow through the local minima. The details of the algorithm can be found in [EdeLeZ02, EdeHaZ03, AEHW04]; here we present a simplified version. The vertices are classified as regular or critical by taking the local star of triangles, and computing the function values at the related vertices. Starting at maximum points and taking the ordered vertices by function values we can merge triangles into descending 2-manifolds. The algorithm creates open disks, and it can be proven that the topology of the regions is the same as for the smooth case. Due to the nature of piecewise linear functions and numerical estimations, the initial Morse complex is likely to consist of too many regions, which needs to be simplified. A possible procedure, which guarantees that the regions always remain simple, merges adjacent regions by removing a pair of critical points from the structure on M . Either a saddle with a maximum is cancelled by erasing two curves from M , or a saddle with a minimum by contracting curves with degree1 endpoints. The sequence of cancellations creates a hierarchy of progressively coarser segmentations. Although there are many possible strategies the mathematically most elegant method to prioritize cancellations is based on the idea of persistence, as introduced in [EdeLeZ02]. Here the minima and maxima are paired with saddles, and the persistence of a pair is the absolute difference in function values between the two critical points. Figure 2(a) shows a Morse Complex segmentation after simplification, the indicator function takes its maxima at the flat parts of the mesh. As can be observed, each monotonic region is bounded by a loop of orange arcs, alternating between minimum and saddle points; there are also black arcs that connect the maximum points to the saddles. In other words – at a saddle always four arcs meet connecting two minima and two maxima. The orange loop represents the final boundary of the Morse regions and serves as the basis of our further computations. This loop will be smoothed, aligned, thickened and transformed into a feature-based region structure (Figure 2(b)), as will be described in the next sections.

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T. V´ arady, M.A. Facello, and Z. Ter´ek

Fig. 3. Morse complex segmentation and separator sets; setting low, medium or high sensitivity defines different surface structures

Separator sets. Morse segmentation provides a clear topological structure; however, it ignores the geometry of the “flat” regions and the highly curved transitions. Based on a local threshold we can identify the triangles that belong to the connecting features. The zigzagged polylines will be thickened and we obtain triangle strips that likely belong to transitional regions, see red triangles in Figure 3. At the same time, the original Morse regions shrink and the remaining triangles now represent the primary regions in a “feature-free” form. The three pictures in Figure 3 illustrate the previously mentioned concept of hierarchical segmentation. Based on different sensitivity values three different region structures have been created; less sensitive segmentations can always be embedded into more sensitive ones.

4

Feature Based Curve Extraction

In the previous phase, a topologically valid and consistent structure has been created, where the triangles of the mesh have been labeled to belong to one of the primary regions or the separator set. In the second and third phases we focus on creating a topologically identical, but geometrically correct curve network. The ragged region boundaries will be replaced by smoothed polylines crossing through the triangles. The feature skeleton consists of mid-curves running in the middle of the feature surfaces, while the thickened feature skeleton is composed of pairs of feature boundary curves. Translational vector field. An important indicator that can be assigned to the vertices of the mesh is the translational vector, which characterizes the strength of the local extrusion in a given point neighborhood. Take vertices Pi that surround a given vertex P , and estimate the related normal vectors Ni . Mapping these normals to the Gaussian sphere, fit a plane which goes through the origin and approximates endpoints Ni . The normal vector of the fitted plane defines the

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Fig. 4. Three points with different translational and similarity indicators

direction of the translation; the error of the least-squares fit characterizes its strength. Clearly, for the points of the connecting features translation will be strong (see Figure 4), while for the points of primary regions and vertex blends it will become weak or vanish. Similarity filters. For the computation of mid-curves and feature boundaries another indicator called similarity proved to be useful, see [BenVar04]. Assume that f denotes an indicator function. Let us take f (P ) and n indicator values in the neighborhood, and measure the sum of differences expressed as s(P ) =

 1 |f (P ) − f (Pi )| . n |f (P )| i

(1)

If the indicator values are similar to that of the central point, s(P ) will be very close to zero. In Figure 4 three points are marked. At points A and C the similarity will be strong, being in the middle of a primary or a feature region. At point A, translation is weak, while at point C strong translation is indicated. The strongest similarity value will indicate the most likely location of the midcurve. At point B, similarity vanishes and the translational strength is divided by the two halves of the neighborhood. These criteria help to determine likely locations of feature boundaries. Curve tracing. In computer aided geometric design there are well-established techniques to trace curves including the computation of intersection curves or boundaries of various features, such as edge blends. In these cases, the surfaces and the procedures are fully defined, and it is possible to trace exact points on the curve using derivatives of the surfaces. In digital shape reconstruction the situation is different since the surfaces have not yet been created. Fortunately, the underlying mesh helps, and using the local indicators, we can compute the related feature curves.

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T. V´ arady, M.A. Facello, and Z. Ter´ek

(a)

(b)

Fig. 5. (a) Curve tracing in the middle of a feature. (b) Guided curve tracing within a vertex blend area.

It is worth distinguishing three types of curve tracing. 1. Feature tracing is applied when the translational vector field is strong. The translational vectors are naturally defined in the interior of triangles as well by weighting the three related vectors at the vertices. Tracing trajectories of the vector field is analogous to solving an ordinary differential equation on the mesh using a Runge–Kutta-like numerical integration. An example is shown in Figure 5(a), where feature tracing started at the middle point of a separator set. This point was determined longitudinally by halving the arc-length of the related polyline. To locate its cross-sectional position we searched for the extreme value of the similarity indicator; see Point C in Figure 4. After defining the middle point we move towards the two vertex blends on the separator set. The translational vectors become weaker and feature tracing is terminated at the estimated setback position of the vertex blends; this will be explained in the next section. Figure 5(a) shows the original polyline (blue) coming from the Morse segmentation, the estimated cross section (yellow) at the middle, and the traced mid-curve (green). Note that a mid-curve – by construction – always remains within the separator set. It is terminated when it (i) leaves the separator set, (ii) gets closed into itself, or (iii) reaches the boundary of the mesh. 2. Guided tracing is used when the translational vector field becomes weak or “ill-defined”, but we have a rough, guiding polyline and constraints to satisfy at the endpoints. This situation occurs along smooth subdividing edges that may partition a large smooth region, or in vertex areas where the direction of the translational vector field cannot be robustly estimated. Tracing is performed in a step-wise manner extending the current tracing direction, enforcing a smoothing term and satisfying constraints from the given polyline. As an example, take Figure 5(b), where feature tracing stopped at the setbacks of the vertex blend. The initial tangents of the connecting curves are thus given, and the curves (in green) must smoothly run into the vertex where the three blue polylines meet. In fact, in a later phase of feature skeleton construction, the vertex positions of the original polylines are also relocated.

Automatic Extraction of Surface Structures in DSR

11

3. For many parts “vanishing” features also occur, where the strength of the translational field gradually disappears. For example, in many car body panels there are several features which gradually and smoothly flow into the larger primary surfaces. In these cases the so-called balanced tracing is used, which combines the benefits of feature and guided tracing by taking an affine combination of the translational direction and a direction estimated at a point of the guiding polygon.

5

Vertex Blends with Setback

In the previous section we dealt with the generation of longitudinal boundary curves of connecting features that are necessary for the surface structure. In this section we focus on the junctions where they run together or interfere with each other. Generating vertex blends is a complex issue from both the topological and geometrical points of view. This topic was in the focus of geometric modeling research a decade ago, but now we revisit and adopt these techniques in digital shape reconstruction. Setback type vertex blends. The well-known “suitcase corner” connects three edge blends with the same radii, see Figure 6(a). This vertex blend is a 3-sided patch, which can be represented by an octant of a sphere. The naive approach to create a vertex blend is to intersect the boundaries (trimlines) of two edge blends meeting on the same primary face and use these points as corner points for this 3-sided patch. The general situation, however, is much more complicated and may require forming complex blends where an arbitrary number of edges meet. These edges can be locally convex or concave, the angles between them are not necessarily close to 90 degrees, and the edge blends may vary from high to low cross-sectional curvature. We need to handle tangential and cuspate edges, as well, and allow keeping an edge sharp without replacing it by a blend. To deal with these complex cases, the concept of setbacks was introduced by [Braid97, VarHof98], in which the boundaries of the edge blends are terminated before they reach the intersection points and a larger surface piece is inserted as shown in Figure 6(b). Setbacks help to avoid difficult shape configurations: compare Figures 6(c) and (d) with the aesthetically pleasing setback vertex blends in Figures 6(e) and (f). A setback type vertex blend has maximum 2n-sides, where in the most general case n spring curves and n profile curves alternate. Once again, profile curves terminate the edge blends and spring curves connect two corner points lying on the same primary surfaces. Depending on the geometric configuration, the length of any of these curves can be chosen to be zero, and the resulting vertex blend may be treated as a degenerate form of the 2n-sided blend [VarHof98]. As illustrated in Figure 6(g), a single spring curve and two zero-length spring curves can make this vertex blend four-sided. Figure 6(h) shows another example where a profile curve has zero length, since the corresponding edge is sharp and is not replaced by a blend.

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T. V´ arady, M.A. Facello, and Z. Ter´ek

Fig. 6. Different vertex blend configurations

Setbacks. The setback concept is heavily used when we intend to determine the best configuration of unknown vertex blends. Let sbi denote the distance between the original unblended vertex and the cross-sectional termination of an edge blend, and ri−1 and ri+1 denote the range constraints computed from the widths of the previous and next blended edges. To push setbacks further off either for aesthetic reasons or handling degenerate situations we introduce a correction term si . Compare Figure 6(g) and 6(e): in the latter case si has a positive value, and a six-sided vertex blend is created instead of a four-sided one. The final setback value can be expressed by sbi = si + max(ri−1 , ri+1 ) .

(2)

As explained earlier, in the feature skeleton building phase we determine only approximate values for the setbacks by detecting that the translational strength falls under a certain level; see Figure 5(b). The exact setback values are computed after tracing the exact feature boundary curves, which determine the above range values. Spring curves. Special care is needed to handle vertex blends at T-nodes, or when convex and concave edge blends meet, see examples in Figure 1(b). The feature skeleton in these cases will connect two mid-curve pairs with collinear tangents. This situation can be detected by matching the ingoing and outgoing curve trajectories within the separator set of a vertex blend. Such an example is shown in Figure 5(b), where two close trajectories and two nearly equal estimated radii on the left and the right sides confirm the hypothesis that we are dealing with a T-node. Once collinear edge pairs are detected the insertion of a spring curve is compulsory; in the remaining cases we compute the corners by intersecting the related two features boundaries running on the mesh. At the end, the number of sides of the vertex blend will be equal to the number of profile curves plus the number of spring curves.

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Fig. 7. Shrinking primary edge loops

Complex vertex blends. Without going into a detailed analysis we remark that building consistent loops for the primary faces may often require further computations. The basic problem is that the independently generated feature boundaries may interfere with each other, or become degenerate. A simple example shows how a primary region shrinks in Figure 7. On the left side, boundary b1 is intersected with boundary b2, which is intersected with b3, i.e. the internal region loop inherits the loop structure of the feature skeleton, see e1 − e2 − e3 and b1 − b2 − b3. In the other case on the right side, the thickened feature boundary b2 vanishes due to width of the transitions and the originally disjoint vertex blends at points P and Q merge. As a result, an “artificial” spring curve sP Q is inserted to connect the shrunk boundaries b1 and b3 to complete the loop.

6

An Example

An industrial object using real measured data has been chosen to illustrate the proposed process of creating surface structures. Figure 8(a) shows the automatically generated primary regions and the separator sets that correspond to a particular Morse segmentation. Figure 8(b) shows the extracted feature skeleton structure that runs in the middle of the separator sets and has already been smoothed. The thickened feature skeleton, including edge and vertex blend boundaries, is shown in Figure 8(c). As can be seen the initial estimation are well-aligned with the numerical curvature map computed using the polygonal mesh. In the last Figure 8(d) the created primaries (red) and the connecting feature regions (grey) can be seen. Note that there is no connecting feature between the top face and the adjacent cylinder, since the common edge was classified as sharp and it is computed by surface-surface intersection. This model contains 253030 triangles. Using a Pentium 2 GHz processor and the Geomagic Studio shape reconstruction system, the following computation times were measured.

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T. V´ arady, M.A. Facello, and Z. Ter´ek

(a) Separator sets

(b) Feature skeleton

(c) Thickened feature skeleton, aligned with curvatures

(d) Primary regions

Fig. 8. An example

Creating the polygonal mesh 12 sec Computing separator sets 7 sec Generating a feature skeleton 3 sec Computing the thickened surface structure 3 sec Time measurements for the final surface fitting have not been included into the table, since it strongly depends on whether rapid surfacing or trimmed surface fitting is applied. In this example there was no need for user intervention, however, for complex parts or noisy data sets the user may want to enhance the results of the automatic algorithms.

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15

Conclusion

An automatic process to create a CAD-like surface structure over a polygonal mesh was presented. The consistent topology of the structure is assured by applying results from combinatorial Morse theory, while the correct geometric location of the segmenting curve network is the result of tracing methods that utilize local indicators estimated at the vertices of the mesh. The final loop structures were created by applying setback type vertex blends. This process ends with a well-aligned, feature-based structure; however, further steps are necessary to create a complete CAD model. Different issues emerge when quadrilateral or trimmed surface models are fitted. To enhance the quality of the reconstructed models further research and development efforts are needed; these include exact feature boundary relocation, stitching issues, surface fairing with dependencies and enforcing various engineering constraints.

Acknowledgements This algorithm has been developed and implemented by Geomagic’s international engineering team residing in North Carolina and Hungary. The authors would like to acknowledge the important contribution of Herbert Edelsbrunner concerning the fundamentals of Morse Complex segmentation, and that of Tobias Gloth and Dmitry Nekhayev, who implemented the initial modules of the above algorithm. This research has been supported by two NSF–SBIR grants, namely Award #0450230, “Creating functionally decomposed surface models from measured data” and Award #0521838, “Applications of Morse theory in reverse engineering.”

References [AEHW04]

[BenVar01] [BenVar04]

[Braid97] [CsaWal00]

[EckHop96]

[EdeLeZ02]

P. Agarwal, H. Edelsbrunner, J. Harer, Y. Wang: Extreme elevation on a 2-manifold, Proc. 20th Ann. Symp. on Computational Geometry, 2004, 357–365 P. Benko, T. Varady: Algorithms for reverse engineering boundary representation models, Computer-Aided Design, 33 (11), 2001, 839–851 P. Benko, T. Varady: Segmentation methods for smooth point regions of conventional engineering objects, Computer-Aided Design, 36, 2004, 511–523 I. Braid: Non-local blending of boundary models, Computer-Aided Design, 29 (2), 1997, 89–100 P. Csakany, A. M. Wallace: Computation of local differential parameters on irregular meshes, In: The Mathematics of Surfaces IX, Eds.: R. Cipolla, R. R. Martin, Springer-Verlag, 2000, 19–33 M. Eck, H. Hoppe: Automatic reconstruction of B-spline surfaces of arbitrary topological type, Computer Graphics, SIGGRAPH’96, 1996, 325– 334 H. Edelsbrunner, D. Letscher, A. Zomorodian: Topological persistence and simplification, Discrete Computational Geometry, 28, 2002, 511–533

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[EdeHaZ03] H. Edelsbrunner, J. Harer, A. Zomorodian: Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds, Discrete Computational Geometry, 30, 2003, 87–107 [FitEgF97] A. W. Fitzgibbon, D. W. Eggert, R.B. Fisher: High-level CAD model acquisition from range images, Computer-Aided Design, 29 (4), 1997, 321–330 [GelGui04] N. Gelfand, L. J. Guibas: Shape Segmentation Using Local Slippage Analysis, Eurographics Symposium on Geometry Processing, 2004, 214– 223 [Geom06] http://www.geomagic.com/en/dssp resources/ [HecGar97] P. S. Heckbert, M. Garland: Survey of Polygonal Surface Simplification Algorithms, Multiresolution Surface Modeling Course, SIGGRAPH’97, 1997 [HuaMen01] J. Huang, C.H. Menq: Automatic data segmentation for geometric feature extraction from unorganized 3D coordinate points, IEEE Trans. Rob. Aut., 17 (3), 2001, 268–279 [LeoJaS97] A. Leonardis, A. Jaklic, F. Solina, Superquadrics for segmenting and modeling range data, IEEE PAMI, 19 (11), 1997, 1298–1295 [LeSSCD98] A. Lee, W. Sweldens, P. Schroeder, L. Cowsar, D. Dobkin: M