A versatile framework for shape description

We present a shape description framework that generates a multitude of shape descriptors through a variety of design and continuum of parameter choices. Our parameter is a surface mesh, referred to as the template, which is supplied at run time, and allows generating different shape descriptors for the same model. Our framework extracts a numerical shape descriptor by computing a selected function on the model mesh, mapping (transferring) it to the template, expanding the mapped function in terms of a basis on the template, and collecting the expansion coefficients into a vector. We investigate possible design choices for the steps in the framework, and introduce novel approaches that provide further freedom in generating a multitude of previously unknown descriptors. We show that our approach is a generalization of the way some of the existing numerical descriptors are defined, and that for appropriate template choices one is able to reproduce some of the well-known descriptors. Finally, we show empirically that design and parameter choices have non-trivial effects on the descriptor’s performance, and that better retrieval results can be obtained by combining descriptors obtained via different templates.

[1]  Martin Reimers,et al.  Mean value coordinates in 3D , 2005, Comput. Aided Geom. Des..

[2]  P. Daras,et al.  Using Ellipsoidal Harmonics for 3D Shape Representation and Retrieval , 2008 .

[3]  Karthik Ramani,et al.  Three-dimensional shape searching: state-of-the-art review and future trends , 2005, Comput. Aided Des..

[4]  Petros Daras,et al.  3D Object Retrieval based on Resulting Fields , 2008 .

[5]  Arthur M. Rosenberg,et al.  Proceedings of the 1968 23rd ACM national conference , 1968 .

[6]  W. J. Gordon,et al.  Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation , 1978 .

[7]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[8]  Facundo Mémoli,et al.  Eurographics Symposium on Point-based Graphics (2007) on the Use of Gromov-hausdorff Distances for Shape Comparison , 2022 .

[9]  Guillermo Sapiro,et al.  Comparing point clouds , 2004, SGP '04.

[10]  Daniel A. Keim,et al.  A method for similarity search of 3D objects , 2001 .

[11]  Dietmar Saupe,et al.  3D Model Retrieval , 2001 .

[12]  Hao Zhang,et al.  Robust 3D Shape Correspondence in the Spectral Domain , 2006, IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06).

[13]  Marc Rioux,et al.  Nefertiti: a query by content system for three-dimensional model and image databases management , 1999, Image Vis. Comput..

[14]  Szymon Rusinkiewicz,et al.  Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors , 2003, Symposium on Geometry Processing.

[15]  Daniela Giorgi,et al.  3D Shape Description and Matching Based on Properties of Real Functions , 2007, Eurographics.

[16]  Raif M. Rustamov,et al.  On Mesh Editing, Manifold Learning, and Diffusion Wavelets , 2009, IMA Conference on the Mathematics of Surfaces.

[17]  Ming Ouhyoung,et al.  On Visual Similarity Based 3D Model Retrieval , 2003, Comput. Graph. Forum.

[18]  Bernd Radig,et al.  Proceedings of the 23rd DAGM-Symposium on Pattern Recognition , 2001 .

[19]  Bernard Chazelle,et al.  Shape distributions , 2002, TOGS.

[20]  Ralph R. Martin,et al.  Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII , 2005 .

[21]  Guillermo Sapiro,et al.  A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data , 2005, Found. Comput. Math..

[22]  J. Joseph,et al.  Fourier Series , 2018, Series and Products in the Development of Mathematics.

[23]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Masayuki Nakajima,et al.  Spherical Wavelet Descriptors for Content-based 3D Model Retrieval , 2006, IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06).

[25]  G. Tolstov Fourier Series , 1962 .

[26]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[27]  Wenyu Liu,et al.  Force Field Based Expression for 3D Shape Retrieval , 2007, HCI.

[28]  Daniela Giorgi,et al.  Describing shapes by geometrical-topological properties of real functions , 2008, CSUR.

[29]  Michael G. Strintzis,et al.  Ellipsoidal Harmonics for 3-D Shape Description and Retrieval , 2009, IEEE Transactions on Multimedia.

[30]  Kunihiro Chihara,et al.  Discriminative Spherical Wavelet Features for Content-Based 3D Model Retrieval , 2007, Int. J. Shape Model..

[31]  J. Gibbs Fourier's Series , 1898, Nature.

[32]  Tao Ju,et al.  Mean value coordinates for closed triangular meshes , 2005, ACM Trans. Graph..

[33]  R. Coifman,et al.  Diffusion Wavelets , 2004 .

[34]  Mikhail Belkin,et al.  Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples , 2006, J. Mach. Learn. Res..

[35]  Leonidas J. Guibas,et al.  Gromov‐Hausdorff Stable Signatures for Shapes using Persistence , 2009, Comput. Graph. Forum.

[36]  Arthur D. Szlam,et al.  Diffusion wavelet packets , 2006 .

[37]  Raif M. Rustamov,et al.  Template Based Shape Descriptor , 2009, 3DOR@Eurographics.

[38]  Daniel A. Keim,et al.  Methods for similarity search on 3D databases , 2002 .

[39]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark , 2004, Proceedings Shape Modeling Applications, 2004..

[40]  Remco C. Veltkamp,et al.  A survey of content based 3D shape retrieval methods , 2004, Proceedings Shape Modeling Applications, 2004..

[41]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[42]  D. Shepard A two-dimensional interpolation function for irregularly-spaced data , 1968, ACM National Conference.

[43]  Michael Garland,et al.  Surface simplification using quadric error metrics , 1997, SIGGRAPH.

[44]  Claudio Gutierrez,et al.  Survey of graph database models , 2008, CSUR.

[45]  Guillermo Sapiro,et al.  A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching , 2010, International Journal of Computer Vision.

[46]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[47]  BENJAMIN BUSTOS,et al.  Feature-based similarity search in 3D object databases , 2005, CSUR.

[48]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[49]  Dietmar Saupe,et al.  3D Model Retrieval with Spherical Harmonics and Moments , 2001, DAGM-Symposium.