Keenan Crane
Fernando de Goes
ABSTRACT
An introduction to geometry processing using discrete exterior calculus (DEC), which provides a simple, flexible, and efficient framework for building a unified geometry-processing platform. The course provides essential mathematical background as well as a large array of real-world examples. It also provides a short survey of the most relevant recent developments in digital geometry processing and discrete differential geometry. Compared to previous SIGGRAPH courses, this course focuses heavily on practical aspects of DEC, with an emphasis on implementation and applications.
The course begins with the core ideas from exterior calculus, in both the smooth and discrete setting. Then it shows how a large number of fundamental geometry-processing tools (smoothing, parameterization, geodesics, mesh optimization, etc.) can be implemented quickly, robustly, and efficiently within this single common framework. It concludes with a discussion of recent extensions of DEC that improve efficiency, accuracy, and versatility.
The course notes grew out of the discrete differential geometry course taught over the past five years at the California Institute of Technology, for undergraduates and beginning graduate students in computer science, applied mathematics, and associated fields. The notes also provide guided exercises (both written and coding) that attendees can later use to deepen their understanding of the material.
Supplemental Material
- {ACOL00} Marc Alexa, Daniel Cohen-Or, and David Levin. As-Rigid-as-Possible Shape Interpolation. In Proc. ACM SIGGRAPH, pages 157--164, 2000. Google Scholar
Digital Library
- {CDS10} Keenan Crane, Mathieu Desbrun, and Peter Schröder. Trivial Connections on Discrete Surfaces. Comp. Graph. Forum, 29(5):1525--1533, 2010.Google ScholarCross Ref
- {CWW13} Keenan Crane, Clarisse Weischedel, and Max Wardetzky. Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. ACM Trans. Graph., 2013.Google Scholar
- {dGC10} Fernando de Goes and Keenan Crane. Trivial Connections Revisited: A Simplified Algorithm for Simply-Connected Surfaces, 2010.Google Scholar
- {DHLM05} Mathieu Desbrun, Anil Hirani, Melvin Leok, and Jerrold Marsden. Discrete Exterior Calculus. ArXiv e-prints, 2005.Google Scholar
- {DKT08} Mathieu Desbrun, Eva Kanso, and Yiying Tong. Discrete Differential Forms for Computational Modeling. In Alexander I. Bobenko, Peter Schröder, John M. Sullivan, and Günther M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 287--324. Birkhäuser Verlag, 2008.Google Scholar
- {DMSB99} Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan Barr. Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow. In Proc. ACM SIGGRAPH, pages 317--324, 1999. Google ScholarDigital Library
- {Epp03} David Eppstein. Dynamic Generators of Topologically Embedded Graphs. In Proc. ACM-SIAM Symp. Disc. Alg. (SODA), 2003. Google ScholarDigital Library
- {EW05} Jeff Erickson and Kim Whittlesey. Greedy Optimal Homotopy and Homology Generators. In Proc. ACM-SIAM Symp. Disc. Alg. (SODA), 2005. Google ScholarDigital Library
- {Hir03} Anil Hirani. Discrete Exterior Calculus. PhD thesis, Pasadena, CA, USA, 2003.Google Scholar
- {KCPS13} Felix Knöppel, Keenan Crane, Ulrich Pinkall, and Peter Schröder. Globally Optimal Direction Fields. In Proc. ACM SIGGRAPH, 2013.Google Scholar
- {Mac49} Richard MacNeal. The Solution of Partial Differential Equations by means of Electrical Networks. PhD thesis, Caltech, 1949.Google Scholar
- {MMdGD11} Patrick Mullen, Pooran Memari, Fernando de Goes, and Mathieu Desbrun. HOT: Hodge-optimized triangulations. In Proc. ACM SIGGRAPH, 2011. Google ScholarDigital Library
- {MTAD08} Patrick Mullen, Yiying Tong, Pierre Alliez, and Mathieu Desbrun. Spectral Conformal Parameterization. Comp. Graph. Forum, 27(5):1487--1494, 2008. Google ScholarDigital Library
- {TACSD06} Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun. Designing Quadrangulations with Discrete Harmonic Forms. In Proc. Symp. Geom. Proc., 2006. Google ScholarDigital Library
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