research-article

Shape comparison through mutual distances of real functions

Author:

Silvia BiasottiAuthors Info & Claims

3DOR '10: Proceedings of the ACM workshop on 3D object retrieval

Pages 33 - 38

https://doi.org/10.1145/1877808.1877816

Published: 25 October 2010 Publication History

Abstract

Spectral analysis provides a library of shape description elements intrinsically defined by the shape itself.

Among all, the eigenfunctions of the Laplace-Beltrami operator can be thought as a set of real valued functions that implicitly abstract and code the shape. In this scenario, this paper introduces a new shape signature derived from the mutual distances between couples of Laplace-Beltrami eigenfunctions. This signature can be seen as a feature vector that acts as an intrinsic shape pattern. Experiments show that it can be effectively used for shape retrieval and its robustness with respect to changes in topology, model resampling, small perturbations and pose variations.

References

[1]

M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computations, 15(6):1373--1396, 2003.

Digital Library

[2]

M. Belkin, J. Sun, and Y. Wang. Discrete Laplace operator for meshed surfaces. In Symposium on Computational Geometry, pages 278--287. ACM, 2008.

Digital Library

[3]

M. Belkin, J. Sun, and Y. Wang. Constructing Laplace operator from point clouds in $\mathbbR^d$. In Symposium on Discrete Algorithms, pages 1031--1040, 2009.

Digital Library

[4]

S. Biasotti and M. Attene. Shape retrieval contest 2008: Stability of watertight models. Shape Modeling and Applications, 0:217--218, 2008.

[5]

S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, and M. Spagnuolo. Describing shapes by geometrical-topological properties of real functions. ACM Comput. Surv., 40(4):1--87, 2008.

Digital Library

[6]

S. Biasotti, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, G. Patanè, and M. Spagnuolo. 3D shape description and matching based on properties of real functions. In EG 2007 Tutorial, pages 949--998, 2007.

[7]

S. Biasotti, G. Patanè, M. Spagnuolo, and B. Falcidieno. Analysis and comparison of real functions on triangulated surfaces. Curve and Surface Fitting, Modern Methods in Mathematics:41--50, 2007.

[8]

A. M. Bronstein, M. M. Bronstein, and R. Kimmel. Efficient computation of isometry-invariant distances between surfaces. SIAM J. Sci. Comput., 28(5):1812--1836, 2006.

Digital Library

[9]

B. Bustos, D. A. Keim, D. Saupe, T. Schreck, and D. V. Vranic. Feature-based similarity search in 3D object databases. ACM Comput. Surv., 37(4):345--387, 2005.

Digital Library

[10]

M. Desbrun, M. Meyer, P. Schröder, and A. Barr. Implicit fairing of irregular meshes using diffusion and curvature flow. In ACM Siggraph, pages 317--324, 1999.

Digital Library

[11]

T. K. Dey, P. Ranjan, and Y. Wang. Convergence, Stability, and Discrete Approximation of Laplace Spectra. In ACM/SIAM Sympos. Discrete Algorithms (SODA 2010), pages 650--663. ACM Press, 2010.

Digital Library

[12]

M. M. Deza and E. Deza. Encyclopedia of Distances. Springer Berlin Heidelberg, 2009.

[13]

G. Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations, volume 1357/LNM, pages 142--155. Springer Berlin / Heidelberg, 1988.

[14]

H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Local and global comparison of continuous functions. In IEEE Visualization, pages 275--280, 2004.

Digital Library

[15]

F. Escolano, P. Suau, and B. Bonev. Information Theory in Computer Vision and Pattern Recognition. Springer, New York, 2009.

Digital Library

[16]

K. Gebal, J. A. Bærentzen, H. Aanæs, and R. Larsen. Shape analysis using the auto diffusion function. Comput. Graph. Forum, 28(5):1405--1413, 2009.

Digital Library

[17]

D. Giorgi, S. Biasotti, and L. Paraboschi. Watertight models track. Technical Report no. 09, CNR - IMATI, Genova, Italy, 2007.

[18]

V. Jain and H. Zhang. Robust 3D shape correspondence in the spectral domain. In Proc. Shape Modeling International 2006, pages 118--129, 2006.

Digital Library

[19]

C. Luo, I. Safa, and Y. Wang. Approximating gradients for meshes and point clouds via diffusion metric. Comput. Graph. Forum, 28(5):1497--1508, 2009.

Digital Library

[20]

M. Mahmoudi and G. Sapiro. Three-dimensional point cloud recognition via distributions of geometric distances. Graph. Models, 71(1):22--31, 2009.

Digital Library

[21]

S. Marini, G. Patané, M. Spagnuolo, and B. Falcidieno. Feature selection for enhanced spectral shape comparison. In Workshop on 3D Object Retrieval, pages 31--38, 2010.

Digital Library

[22]

M. Ovsjanikov, A. M. Bronstein, M. M. Bronstein, and L. J. Guibas. Shape google: a computer vision approach to invariant shape retrieval. In Proc. NORDIA, pages 320--327, 2009.

[23]

L. Paraboschi, S. Biasotti, and B. Falcidieno. Comparing sets of 3D digital shapes through topological structures. In GbRPR, volume 4538 of LNCS, pages 114--125. Springer, 2007.

Digital Library

[24]

O. Pele and M. Werman. A linear time histogram metric for improved sift matching. In ECCV, 2008.

Digital Library

[25]

O. Pele and M. Werman. Fast and robust earth mover's distances. In ICCV, 2009.

[26]

U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics, 1(2), 1993.

[27]

M. Reuter, S. Biasotti, D. Giorgi, G. Patanè, and M. Spagnuolo. Discrete Laplace-Beltrami operators for shape analysis and segmentation. Comput. Graph., 33(3):381--390, 2009.

Digital Library

[28]

M. Reuter, F.-E. Wolter, and N. Peinecke. Laplace-spectra as fingerprints for shape matching. In Proc. of the ACM Symp. on Solid and Physical Modeling, pages 101--106, 2005.

Digital Library

[29]

M. Reuter, F.-E. Wolter, and N. Peinecke. Laplace-Beltrami spectra as Shape-DNA of surfaces and solids. CAD, 38(4):342--366, 2006.

Digital Library

[30]

M. Reuter, F.-E. Wolter, M. Shenton, and M. Niethammer. Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis. Computer Aided Design, 2009.

Digital Library

[31]

S. Rosenberg. The Laplacian on a Riemannian Manifold. Cambridge University Press, 1997.

[32]

Y. Rubner, C. Tomasi, and L. J. Guibas. The earth mover's distance as a metric for image retrieval. Int. J. of Computer Vision, 40:99--121, 2000.

Digital Library

[33]

R. Rustamov. Robust volumetric shape descriptor. In Workshop on 3D Object Retrieval, pages 1--5, 2010.

Digital Library

[34]

R. M. Rustamov. Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In Symp. on Geometry Processing, pages 225--233, 2007.

Digital Library

[35]

O. Schall, R. Zayer, and H.-P. Seidel. Controlled field generation for quad-remeshing. In SPM '08: Proceedings of the 2008 ACM symposium on Solid and physical modeling, pages 295--300, New York, NY, USA, 2008. ACM.

Digital Library

[36]

T. Skopal and B. Bustos. On nonmetric similarity search problems in complex domains. ACM Comput. Surv., 2010. to appear.

Digital Library

[37]

J. Sun, M. Ovsjanikov, and L. Guibas. A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum, 28(5):1383--1392, 2009.

[38]

B. Vallet and B. Levy. Spectral geometry processing with manifold harmonics. Comput. Graph. Forum, 27(2), 2008.

[39]

M. Wardetzky, S. Mathur, F. Kalberer, and E. Grinspun. Discrete Laplace operators: No free lunch. In Proc. Symp. Geom. Proc. 2007, 2007.

Digital Library

[40]

G. Xu. Discrete Laplace-Beltrami operators and their convergence. Comput. Aided Geom. Des., 21(8):767--784, 2004.

Digital Library

[41]

H. Zhang, O. van Kaick, and R. Dyer. Spectral methods for mesh processing and analysis. In Eurographics STAR Report, pages 1--22, 2007.

Cited By

Biasotti SCerri ABronstein ABronstein M(2016)Recent Trends, Applications, and Perspectives in 3D Shape Similarity AssessmentComputer Graphics Forum10.1111/cgf.1273435:6(87-119)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1111/cgf.12734
Kuang ZLi ZLv QWeiwei TLiu Y(2015)Modal function transformation for isometric 3D shape representationComputers and Graphics10.1016/j.cag.2014.09.03346:C(209-220)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1016/j.cag.2014.09.033
Berger BVais AWolter F(2015)Subimage sensitive eigenvalue spectra for image comparisonThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-014-1038-y31:2(205-221)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s00371-014-1038-y
Show More Cited By

Index Terms

Shape comparison through mutual distances of real functions
1. Computing methodologies
  1. Computer graphics
2. Information systems
  1. Information retrieval

Recommendations

Interpolated eigenfunctions for volumetric shape processing
Special Issue on 3DOR 2010

This paper introduces a set of volumetric functions suitable for geometric processing of volumes. We start with Laplace–Beltrami eigenfunctions on the bounding surface and interpolate them into the interior using barycentric coordinates. The ...
Computing a family of skeletons of volumetric models for shape description

Skeletons are important shape descriptors in object representation and recognition. Typically, skeletons of volumetric models are computed using iterative thinning. However, traditional thinning methods often generate skeletons with complex structures ...
Ellipsoidal harmonics for 3-D shape description and retrieval

In this paper, a novel approach for 3-D Shape description and retrieval based on the theory of ellipsoidal harmonics is presented. Four novel descriptors are introduced: the surface ellipsoidal harmonics descriptor, which concerns 3-D objects that are ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

3DOR '10: Proceedings of the ACM workshop on 3D object retrieval

October 2010

96 pages

ISBN:9781450301602

DOI:10.1145/1877808

Program Chairs:
Mohamed Doudi
TELECOM Lille1/LIFL, France
,
Michela Spagnuolo
CNR-IMATI, Italy
,
Remco Veltkamp
Utrecht University, The Netherlands

Copyright © 2010 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

SIGMM: ACM Special Interest Group on Multimedia

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 25 October 2010

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

MM '10

Sponsor:

SIGMM

MM '10: ACM Multimedia Conference

October 25, 2010

Firenze, Italy

Upcoming Conference

MM '24

Sponsor:
sigmm

The 32nd ACM International Conference on Multimedia

October 28 - November 1, 2024

Melbourne , VIC , Australia

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
231
Total Downloads

Downloads (Last 12 months)3
Downloads (Last 6 weeks)0

Reflects downloads up to 15 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Biasotti SCerri ABronstein ABronstein M(2016)Recent Trends, Applications, and Perspectives in 3D Shape Similarity AssessmentComputer Graphics Forum10.1111/cgf.1273435:6(87-119)Online publication date: 1-Sep-2016
https://dl.acm.org/doi/10.1111/cgf.12734
Kuang ZLi ZLv QWeiwei TLiu Y(2015)Modal function transformation for isometric 3D shape representationComputers and Graphics10.1016/j.cag.2014.09.03346:C(209-220)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1016/j.cag.2014.09.033
Berger BVais AWolter F(2015)Subimage sensitive eigenvalue spectra for image comparisonThe Visual Computer: International Journal of Computer Graphics10.1007/s00371-014-1038-y31:2(205-221)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1007/s00371-014-1038-y
Biasotti SFalcidieno BGiorgi DSpagnuolo M(2014)Mathematical Tools for Shape Analysis and DescriptionSynthesis Lectures on Computer Graphics and Animation10.2200/S00588ED1V01Y201407CGR0166:2(1-138)Online publication date: 7-Sep-2014
https://doi.org/10.2200/S00588ED1V01Y201407CGR016
Biasotti SCerri AGiorgi DSpagnuolo MFalcidieno BSpagnuolo MLipman YZhang H(2013)PHOGProceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing10.1111/cgf.12168(13-22)Online publication date: 3-Jul-2013
https://dl.acm.org/doi/10.1111/cgf.12168
Biasotti SSpagnuolo MFalcidieno B(2013)SMI 2013Computers and Graphics10.1016/j.cag.2013.05.00737:6(608-619)Online publication date: 1-Oct-2013
https://dl.acm.org/doi/10.1016/j.cag.2013.05.007

View Options

Get Access

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.

Media

Figures

Other

Tables

View Table of Contents