Article

Tightening: curvature-limiting morphological simplification

Authors:

Jason Williams,

Jarek RossignacAuthors Info & Claims

SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modeling

Pages 107 - 112

https://doi.org/10.1145/1060244.1060257

Published: 13 June 2005 Publication History

Abstract

Given a planar set S of arbitrary topology and a radius r, we show how to construct an r-tightening of S, which is a set whose boundary has a radius of curvature everywhere greater than or equal to r and which only differs from S in a morphologically-defined tolerance zone we call the mortar. The mortar consists of the thin or highly curved parts of S, such as corners, gaps, and small connected components, while the boundary of a tightening consists of minimum-length loops through the mortar. Tightenings are defined independently of shape representation, and it may be possible to find them using a variety of algorithms. We describe how to approximately compute tightenings for sets represented as binary images using constrained, level-set curvature flow.

References

[1]

Amenta, N. and Bern, M. 1999. Surface Reconstruction by Voronoi Filtering, Discrete and Computational Geometry 22, 481--504.

[2]

Attali, D. 1997. r-Regular Shape Reconstruction from Unorganized Points, Computational Geometry, 248--253.

Digital Library

[3]

Bajaj, C. and Xu, G. 2003. Anisotropic Diffusion of Surfaces and Functions on Surfaces, ACM Transactions on Graphics, 22, 1, 4--32.

Digital Library

[4]

Celniker, G. and Gossard, D. 1991. Deformable Curve and Surface Finite Elements for Free-Form Shape Design. In Computer Graphics (Proceedings of ACM SIGGRAPH 91), 257--266.

Digital Library

[5]

Cohen, J., Varshney, A., Manocha, D., Turk, G., Weber, H., Agarwal, P., Brooks, F., and Wright, W. 1996. Simplification Envelopes. In Proceedings of ACM SIGGRAPH 96, ACM Press / ACM SIGGRAPH, New York, Computer Graphics Proceedings, Annual Conference Series, ACM, 119--128.

Digital Library

[6]

Desbrun, M., Meyer, M., Schroder, P. and Barr, A. Implicit Fairing of Meshes Using Diffusion and Curvature Flow. 1999. In Proceedings of ACM SIGGRAPH 99, ACM Press / ACM SIGGRAPH, New York, Computer Graphics Proceedings, Annual Conference Series, ACM, 317--324.

Digital Library

[7]

Danielsson, P. 1980. Euclidean Distance Mapping, Computer Graphics and Image Processing 14, 227--248.

[8]

Edelsbrunner, H. and Mucke, E. 1994. Three-Dimensional Alpha Shapes, ACM Transactions on Graphics 13, 1, 43--72.

Digital Library

[9]

El-Sana, J. and Varshney, A. 1998. Topology Simplification for Polygonal Virtual Environments, IEEE Transactions on Visualization and Computer Graphics 4, 2, 133--144.

Digital Library

[10]

Erikson, C. and Manocha, D. 1999. GAPS: General and Automatic Polygonal Simplification. In Proceedings of the 1999 Symposium on Interactive 3D Graphics, 79--88.

Digital Library

[11]

Fleiscman, S., Drori, I., and Cohen-Or, D. 2003. Bilateral Mesh Denoising. ACM Transactions on Graphics, 22, 3, 950--953.

Digital Library

[12]

Grayson, M. 1987. The Heat Equation Shrinks Embedded Plane Curves to Round Points, Journal of Differential Geometry 26, 285--314.

[13]

Hahmann, S. and Konz, S. 1997. Fairing Bi-Cubic B-Spline Surfaces Using Simmulated Annealing. In Curves and Surfaces with Applications in CAGD, Vanderbilt University Press, Nashville, TN, A. Le Mehaute, C. Rabut, and L. Schumaker, Eds., 159--168.

[14]

He, T., Hong, L., Varshney, A., and Wang, S. 1996. Controlled Topology Simplification, IEEE Transactions on Visualization and Computer Graphics 2, 2, 171--183.

Digital Library

[15]

Heijmans, H. 1994. Morphological Image Operators. Academic Press.

[16]

Jones, T. Durand, F., and Desbrun, M. 2003. Non-Iterative, Feature-Preserving Mesh Smoothing. ACM Transactions on Graphics, 22, 3, 943--949.

Digital Library

[17]

Mitchell, J. 2004. Shortest Paths and Networks. In Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, J. Goodman and J. O'Rourke, Eds., 607--641.

Digital Library

[18]

Moreton, H. and Sequin, C. 1992. Functional Optimization for Fair Surface Design. In Computer Graphics (Proceedings of ACM SIGGRAPH 92), 26, ACM, 167--176.

Digital Library

[19]

Nooruddin, F. and Turk, G. 2003. Simplification and Repair of Polygonal Models Using Volumetric Techniques. IEEE Transactions on Visualization and Computer Graphics 9, 2, 191--205.

Digital Library

[20]

Popovic, J. and Hoppe, H. 1997. Progressive Simplicial Complexes. In Proceedings of ACM SIGGRAPH 97, ACM Press / ACM SIGGRAPH, New York, Computer Graphics Proceedings, Annual Conference Series, ACM, 217--224.

Digital Library

[21]

Rossignac, J. 1985. Blending and Offsetting Solid Models. Ph.D. thesis, University of Rochester.

Digital Library

[22]

Rossignac, J. and Requicha, A. 1986. Offsetting Operations in Solid Modeling. Computer Aided Geometric Design 3, 2, 129--148.

Digital Library

[23]

Serra, J. 1982. Image Analysis and Mathematical Morphology. Academic Press.

Digital Library

[24]

Sethian, J. 1999. Level Set Methods and Fast Marching Methods. Cambridge University Press.

[25]

Sklansky, J., Chazin, R., and Hansen, B. 1972. IEEE Transactions on Computers 21, 3, 260--268.

[26]

Szymczak, A. and Vanderhyde, J. 2003. Extraction of Topologically Simple Isosurfaces from Volume Datasets, IEEE Visualization 2003, 67--74.

Digital Library

[27]

Tam, R. and HEIDRICH, W. 2003. Shape Simplification Based on the Medial Axis Transform. IEEE Visualization 2003, 481--488.

Digital Library

[28]

Taubin, G. 1995. Curve and Surface Smoothing without Shrinkage. In Proceedings of the Fifth International Conference on Computer Vision, 852--857.

Digital Library

[29]

Toussaint, G. 1989. Computing Geodesic Properties inside a Simple Polygon. Revue D'Intelligence Artificielle 3, 2, 9--42.

[30]

Westgard, G. and Nowacki, H. 2001. Construction of Fair Surfaces over Irregular Meshes, ACM Symposium on Solid Modeling and Applications 2001, 88--98.

Digital Library

[31]

Williams, J. and Rossignac, J. Mason: Morphological Simplification. To appear in Graphical Models.

Digital Library

[32]

Wood, Z., Hoppe, H., Desbrun, M., and Schroder, P. 2004. Removing Excess Topology from Isosurfaces, ACM Transactions on Graphics 23, 2, 190--208.

Digital Library

[33]

Zhang, H. and Fiume, E. 2003. Butterworth Filtering and Implicit Fairing of Irregular Meshes. In Proceedings of the 11th Pacific Conference on Computer Graphics and Applications, 502--506.

Digital Library

Cited By

de Villiers EOthmer C(2012)Multi-Objective Adjoint Optimization of Intake Port GeometrySAE Technical Paper Series10.4271/2012-01-0905Online publication date: 16-Apr-2012
https://doi.org/10.4271/2012-01-0905
Rossignac J(2011)Ball-based shape processingProceedings of the 16th IAPR international conference on Discrete geometry for computer imagery10.5555/1987119.1987122(13-34)Online publication date: 6-Apr-2011
https://dl.acm.org/doi/10.5555/1987119.1987122
Whited BRossignac JBronsvoort WGonsor DRegli WGrandine TVandenbrande JGravesen JKeyser J(2009)b-morphs between b-compatible curves in the plane2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling10.1145/1629255.1629279(187-198)Online publication date: 5-Oct-2009
https://dl.acm.org/doi/10.1145/1629255.1629279
Show More Cited By

Index Terms

Tightening: curvature-limiting morphological simplification
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Topological simplification of isosurfaces in volumetric data using octrees

Volumetric data, such as output from CT scans or laser range scan processing methods, often have isosurfaces that contain topological noise-small handles and holes that are not present in the original model. Because this noise can significantly degrade ...
Hierarchical topology-preserving simplification of terrains

We present an algorithm for simplifying terrain data that preserves topology. We use a decimation algorithm that simplifies the given data set using hierarchical clustering. Topology constraints, along with local error metrics, are used to ensure ...
Adaptive mesh coarsening for quadrilateral and hexahedral meshes

Mesh adaptation methods can improve the efficiency and accuracy of solutions to computational modeling problems. In many applications involving quadrilateral and hexahedral meshes, local modifications which maintain the original element type are ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modeling

June 2005

287 pages

ISBN:1595930159

DOI:10.1145/1060244

Program Chairs:
Leif Kobbelt
RWTH Aachen University, Germany
,
Vadim Shapiro
University of Wisconsin--Madison, USA

Copyright © 2005 ACM.

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. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 13 June 2005

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Conference

SPM05

Sponsor:

SIGGRAPH

SPM05: 2005 ACM Symposium on Solid and Physical Modeling

June 13 - 15, 2005

Massachusetts, Cambridge

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
258
Total Downloads

Downloads (Last 12 months)1
Downloads (Last 6 weeks)0

Reflects downloads up to 05 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

de Villiers EOthmer C(2012)Multi-Objective Adjoint Optimization of Intake Port GeometrySAE Technical Paper Series10.4271/2012-01-0905Online publication date: 16-Apr-2012
https://doi.org/10.4271/2012-01-0905
Rossignac J(2011)Ball-based shape processingProceedings of the 16th IAPR international conference on Discrete geometry for computer imagery10.5555/1987119.1987122(13-34)Online publication date: 6-Apr-2011
https://dl.acm.org/doi/10.5555/1987119.1987122
Whited BRossignac JBronsvoort WGonsor DRegli WGrandine TVandenbrande JGravesen JKeyser J(2009)b-morphs between b-compatible curves in the plane2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling10.1145/1629255.1629279(187-198)Online publication date: 5-Oct-2009
https://dl.acm.org/doi/10.1145/1629255.1629279
Datar MCates JFletcher PGouttard SGerig GWhitaker R(2009)Particle Based Shape Regression of Open Surfaces with Applications to Developmental NeuroimagingProceedings of the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part II10.1007/978-3-642-04271-3_21(167-174)Online publication date: 1-Oct-2009
https://dl.acm.org/doi/10.1007/978-3-642-04271-3_21
Bronson JRheingans POlano MAnjyo KSillion F(2008)Semi-automatic stencil creation through error minimizationProceedings of the 6th international symposium on Non-photorealistic animation and rendering10.1145/1377980.1377989(31-37)Online publication date: 9-Jun-2008
https://dl.acm.org/doi/10.1145/1377980.1377989
Meyer MWhitaker RKirby RLedergerber CPfister H(2008)Particle-based Sampling and Meshing of Surfaces in Multimaterial VolumesIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2008.15414:6(1539-1546)Online publication date: 1-Nov-2008
https://dl.acm.org/doi/10.1109/TVCG.2008.154
Cates JFletcher PWarnock ZWhitaker R(2008)A shape analysis framework for small animal phenotyping with application to mice with a targeted disruption of Hoxd112008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro10.1109/ISBI.2008.4541045(512-515)Online publication date: May-2008
https://doi.org/10.1109/ISBI.2008.4541045
Cates JFletcher PStyner MHazlett HWhitaker R(2008)Particle-Based Shape Analysis of Multi-object ComplexesProceedings of the 11th international conference on Medical Image Computing and Computer-Assisted Intervention - Part I10.1007/978-3-540-85988-8_57(477-485)Online publication date: 6-Sep-2008
https://dl.acm.org/doi/10.1007/978-3-540-85988-8_57
Rossignac J(2007)Solid and Physical ModelingWiley Encyclopedia of Electrical and Electronics Engineering10.1002/047134608X.W7526.pub2Online publication date: 15-Jun-2007
https://doi.org/10.1002/047134608X.W7526.pub2
Rossignac J(2005)Shape complexityThe Visual Computer10.1007/s00371-005-0362-721:12(985-996)Online publication date: 24-Nov-2005
https://doi.org/10.1007/s00371-005-0362-7

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten