research-article

Planar shape interpolation with bounded distortion

Authors:

Mirela Ben-ChenAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 32, Issue 4

Article No.: 108, Pages 1 - 12

https://doi.org/10.1145/2461912.2461983

Published: 21 July 2013 Publication History

Abstract

Planar shape interpolation is widely used in computer graphics applications. Despite a wealth of interpolation methods, there is currently no approach that produces shapes with a bounded amount of distortion with respect to the input. As a result, existing interpolation methods may produce shapes that are significantly different than the input and can suffer from fold-overs and other visual artifacts, making them less useful in many practical scenarios. We introduce a novel shape interpolation scheme designed specifically to produce results with a bounded amount of conformal (angular) distortion. Our method is based on an elegant continuous mathematical formulation and provides several appealing properties such as existence and uniqueness of the solution as well as smoothness in space and time domains. We further present a discretization and an efficient practical algorithm to compute the interpolant and demonstrate its usability and good convergence behavior on a wide variety of input shapes. The method is simple to implement and understand. We compare our method to state-of-the-art interpolation methods and demonstrate its superiority in various cases.

Supplementary Material

ZIP File (a108-chen.zip)

Supplemental material.

Download
31.85 MB

MP4 File (tp044.mp4)

Download
29.86 MB

References

[1]

Alexa, M., Cohen-Or, D., and Levin, D. 2000. As-rigid-as-possible shape interpolation. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, ACM Press/Addison-Wesley Publishing Co., 157--164.

Digital Library

[2]

Alexa, M. 2002. Recent advances in mesh morphing. In Computer graphics forum, vol. 21, Wiley Online Library, 173--198.

[3]

Bao, Y., Guo, X., and Qin, H. 2005. Physically based morphing of point-sampled surfaces. Computer Animation and Virtual Worlds 16, 3--4, 509--518.

Digital Library

[4]

Baxter, W., Barla, P., and Anjyo, K. 2008. Rigid shape interpolation using normal equations. In Proceedings of the 6th international symposium on Non-photorealistic animation and rendering, ACM, 59--64.

Digital Library

[5]

Chao, I., Pinkall, U., Sanan, P., and Schröder, P. 2010. A simple geometric model for elastic deformations. ACM Transactions on Graphics (TOG) 29, 4, 38.

Digital Library

[6]

Choi, J., and Szymczak, A. 2003. On coherent rotation angles for as-rigid-as-possible shape interpolation.

[7]

Crane, K., Pinkall, U., and Schröder, P. 2011. Spin transformations of discrete surfaces. ACM Trans. Graph. 40.

Digital Library

[8]

Fröhlich, S., and Botsch, M. 2011. Example-driven deformations based on discrete shells. In Computer Graphics Forum, vol. 30, Wiley Online Library, 2246--2257.

[9]

Fu, H., Tai, C., and Au, O. 2005. Morphing with laplacian coordinates and spatial-temporal texture. In Proceedings of Pacific Graphics, 100--102.

[10]

Gray, A., Abbena, E., and Salamon, S. 2006. Modern differential geometry of curves and surfaces with Mathematica. Chapman & Hall/CRC.

Digital Library

[11]

Gu, D., Luo, F., and Yau, S. 2010. Fundamentals of computational conformal geometry. Mathematics in Computer Science 4, 4, 389--429.

[12]

Hu, S., Li, C., and Zhang, H. 2004. Actual morphing: a physics-based approach to blending. In Proceedings of the ninth ACM symposium on Solid modeling and applications, Eurographics Association, 309--314.

Digital Library

[13]

Igarashi, T., Moscovich, T., and Hughes, J. 2005. As-rigid-as-possible shape manipulation. In ACM Transactions on Graphics (TOG), vol. 24, ACM, 1134--1141.

Digital Library

[14]

Kilian, M., Mitra, N., and Pottmann, H. 2007. Geometric modeling in shape space. In ACM Transactions on Graphics (TOG), vol. 26, ACM, 64.

Digital Library

[15]

Kircher, S., and Garland, M. 2008. Free-form motion processing. ACM Transactions on Graphics (TOG) 27, 2, 12.

Digital Library

[16]

Klassen, E., Srivastava, A., Mio, M., and Joshi, S. 2004. Analysis of planar shapes using geodesic paths on shape spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 26, 3, 372--383.

Digital Library

[17]

Lipman, Y., Sorkine, O., Levin, D., and Cohen-Or, D. 2005. Linear rotation-invariant coordinates for meshes. ACM Transactions on Graphics (TOG) 24, 3, 479--487.

Digital Library

[18]

Lipman, Y., Kim, V., and Funkhouser, T. 2012. Simple formulas for quasiconformal plane deformations. ACM Transactions on Graphics (TOG) 31, 5, 124.

Digital Library

[19]

Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics (TOG) 31, 4, 108.

Digital Library

[20]

Liu, L., Wang, G., Zhang, B., Guo, B., and Shum, H. 2004. Perceptually based approach for planar shape morphing. In Computer Graphics and Applications, 2004. PG 2004. Proceedings. 12th Pacific Conference on, IEEE, 111--120.

Digital Library

[21]

Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. J. 2008. A local/global approach to mesh parameterization. In Computer Graphics Forum, vol. 27, Wiley Online Library, 1495--1504.

Digital Library

[22]

Sederberg, T., and Greenwood, E. 1992. A physically based approach to 2--d shape blending. In ACM SIGGRAPH Computer Graphics, vol. 26, ACM, 25--34.

Digital Library

[23]

Shapira, M., and Rappoport, A. 1995. Shape blending using the star-skeleton representation. Computer Graphics and Applications, IEEE 15, 2, 44--50.

Digital Library

[24]

Sheffer, A., and Kraevoy, V. 2004. Pyramid coordinates for morphing and deformation. In 3D Data Processing, Visualization and Transmission, 2004. 3DPVT 2004. Proceedings. 2nd International Symposium on, IEEE, 68--75.

Digital Library

[25]

Sorkine, O., and Alexa, M. 2007. As-rigid-as-possible surface modeling. In ACM International Conference Proceeding Series, vol. 257, Citeseer, 109--116.

Digital Library

[26]

Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27, 3 (Aug.), 77:1--77:11.

Digital Library

[27]

Surazhsky, T., and Elber, G. 2002. Metamorphosis of planar parametric curves via curvature interpolation. International Journal of Shape Modeling 8, 02, 201--216.

[28]

Surazhsky, V., and Gotsman, C. 2001. Controllable morphing of compatible planar triangulations. ACM Transactions on Graphics 20, 4, 203--231.

Digital Library

[29]

Surazhsky, V., and Gotsman, C. 2003. Intrinsic morphing of compatible triangulations. International Journal of Shape Modeling 9, 02, 191--201.

[30]

Sykora, D., Dingliana, J., and Collins, S. 2009. As-rigid-as-possible image registration for hand-drawn cartoon animations. In Proceedings of the 7th International Symposium on Non-Photorealistic Animation and Rendering, ACM, 25--33.

Digital Library

[31]

Weber, O., and Gotsman, C. 2010. Controllable conformal maps for shape deformation and interpolation. ACM Transactions on Graphics (TOG) 29, 4, 78.

Digital Library

[32]

Weber, O., Myles, A., and Zorin, D. 2012. Computing extremal quasiconformal maps. In Computer Graphics Forum, vol. 31, Wiley Online Library, 1679--1689.

Digital Library

[33]

Weintraub, S. 1997. Differential forms: a complement to vector calculus. Academic Press, San Diego.

[34]

Whited, B., and Rossignac, J. 2011. Ball-morph: Definition, implementation, and comparative evaluation. Visualization and Computer Graphics, IEEE Transactions on 17, 6, 757--769.

Digital Library

[35]

Winkler, T., Drieseberg, J., Alexa, M., and Hormann, K. 2010. Multi-scale geometry interpolation. In Computer Graphics Forum, vol. 29, Wiley Online Library, 309--318.

[36]

Wolberg, G. 1998. Image morphing: a survey. The visual computer 14, 8, 360--372.

[37]

Xu, D., Zhang, H., Wang, Q., and Bao, H. 2006. Poisson shape interpolation. Graphical models 68, 3, 268--281.

Digital Library

[38]

Zeng, W., Luo, F., Yau, S., and Gu, X. 2009. Surface quasi-conformal mapping by solving beltrami equations. Mathematics of Surfaces XIII, 391--408.

Digital Library

Cited By

Huang ZLiu SFu X(2024)High-order shape interpolationComputer Aided Geometric Design10.1016/j.cagd.2024.102301111(102301)Online publication date: Jun-2024
https://doi.org/10.1016/j.cagd.2024.102301
Huang XRitschel TSeidel HMemari PSingh G(2023)Patternshop: Editing Point Patterns by Image ManipulationACM Transactions on Graphics10.1145/359241842:4(1-14)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592418
Wang YGuo MSolomon J(2023)Variational quasi-harmonic maps for computing diffeomorphismsACM Transactions on Graphics10.1145/359210542:4(1-26)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592105
Show More Cited By

Index Terms

Planar shape interpolation with bounded distortion
1. Computing methodologies
  1. Computer graphics

Recommendations

Bounded distortion harmonic shape interpolation

Planar shape interpolation is a classic problem in computer graphics. We present a novel shape interpolation method that blends C^∞ planar harmonic mappings represented in closed-form. The intermediate mappings in the blending are guaranteed to be ...
Real-Time Nonlinear Shape Interpolation

We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular the fact that the set of all possible interpolated shapes is a low-dimensional object in ...
Bounded distortion mapping spaces for triangular meshes

The problem of mapping triangular meshes into the plane is fundamental in geometric modeling, where planar deformations and surface parameterizations are two prominent examples. Current methods for triangular mesh mappings cannot, in general, control ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 32, Issue 4

July 2013

1215 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/2461912

Issue’s Table of Contents

Copyright © 2013 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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 21 July 2013

Published in TOG Volume 32, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

61
Total Citations
View Citations
860
Total Downloads

Downloads (Last 12 months)40
Downloads (Last 6 weeks)8

Reflects downloads up to 25 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Huang ZLiu SFu X(2024)High-order shape interpolationComputer Aided Geometric Design10.1016/j.cagd.2024.102301111(102301)Online publication date: Jun-2024
https://doi.org/10.1016/j.cagd.2024.102301
Huang XRitschel TSeidel HMemari PSingh G(2023)Patternshop: Editing Point Patterns by Image ManipulationACM Transactions on Graphics10.1145/359241842:4(1-14)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592418
Wang YGuo MSolomon J(2023)Variational quasi-harmonic maps for computing diffeomorphismsACM Transactions on Graphics10.1145/359210542:4(1-26)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592105
Chen YXie TYuksel CKaufman DYang YJiang CLi M(2023)Multi-Layer Thick ShellsACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591489(1-9)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591489
Lin JYuan CCai YLi HRen YZou YHong XZhang F(2023)ImMesh: An Immediate LiDAR Localization and Meshing FrameworkIEEE Transactions on Robotics10.1109/TRO.2023.332122739:6(4312-4331)Online publication date: 1-Dec-2023
https://dl.acm.org/doi/10.1109/TRO.2023.3321227
Fukusato TMaejima AIgarashi TYotsukura T(2023)Exploring inbetween charts with trajectory-guided sliders for cutout animationMultimedia Tools and Applications10.1007/s11042-023-17354-x83:15(44581-44594)Online publication date: 18-Oct-2023
https://doi.org/10.1007/s11042-023-17354-x
Fargion GWeber O(2022)Globally Injective Flattening via a Reduced Harmonic SubspaceACM Transactions on Graphics10.1145/3550454.355544941:6(1-17)Online publication date: 30-Nov-2022
https://dl.acm.org/doi/10.1145/3550454.3555449
Rasheed ARomero VBertails-Descoubes FWuhrer SFranco JLazarus A(2022)A Visual Approach to Measure Cloth-Body and Cloth-Cloth FrictionIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2021.309754744:10_Part_2(6683-6694)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1109/TPAMI.2021.3097547
Zhang JZhu CZheng LXu K(2021)ROSEFusionACM Transactions on Graphics10.1145/3450626.345967640:4(1-17)Online publication date: 19-Jul-2021
https://dl.acm.org/doi/10.1145/3450626.3459676
Ikemakhen AAhanchaou T(2021)Blending of hyperbolic closed curvesComputer Graphics Forum10.1111/cgf.1435840:5(71-79)Online publication date: 23-Aug-2021
https://doi.org/10.1111/cgf.14358
Show More Cited By

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 Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents