A new approach to dimensionality reduction: theory and algorithms
SIAM Journal on Applied Mathematics
Shape matching using edit-distance: an implementation
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
ACM Transactions on Graphics (TOG)
Extracting Skeletal Curves from 3D Scattered Data
SMI '99 Proceedings of the International Conference on Shape Modeling and Applications
SMI '04 Proceedings of the Shape Modeling International 2004
Fair morse functions for extracting the topological structure of a surface mesh
ACM SIGGRAPH 2004 Papers
The Whitney Reduction Network: A Method for Computing Autoassociative Graphs
Neural Computation
Robust 3D Shape Correspondence in the Spectral Domain
SMI '06 Proceedings of the IEEE International Conference on Shape Modeling and Applications 2006
Efficient Computation of Isometry-Invariant Distances Between Surfaces
SIAM Journal on Scientific Computing
ACM SIGGRAPH 2002 conference abstracts and applications
A survey of content based 3D shape retrieval methods
Multimedia Tools and Applications
Numerical Geometry of Non-Rigid Shapes
Numerical Geometry of Non-Rigid Shapes
Meaningful 3D shape partitioning using Morse functions
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
IPMI'05 Proceedings of the 19th international conference on Information Processing in Medical Imaging
Retrieving articulated 3-d models using medial surfaces and their graph spectra
EMMCVPR'05 Proceedings of the 5th international conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
A generalized divergence measure for robust image registration
IEEE Transactions on Signal Processing
Calculus of Nonrigid Surfaces for Geometry and Texture Manipulation
IEEE Transactions on Visualization and Computer Graphics
Geodesic matching of triangulated surfaces
IEEE Transactions on Image Processing
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We propose to superpose global topological and local geometric 3-D shape descriptors in order to define one compact and discriminative representation for a 3-D object. While a number of available 3-D shape modeling techniques yield satisfactory object classification rates, there is still a need for a refined and efficient identification/recognition of objects among the same class. In this paper, we use Morse theory in a two-phase approach. To ensure the invariance of the final representation to isometric transforms, we choose the Morse function to be a simple and intrinsic global geodesic function defined on the surface of a 3-D object. The first phase is a coarse representation through a reduced topological Reeb graph. We use it for a meaningful decomposition of shapes into primitives. During the second phase, we add detailed geometric information by tracking the evolution of Morse function's level curves along each primitive. We then embed the manifold of these curves into R3, and obtain a single curve. By combining phase one and two, we build new graphs rich in topological and geometric information that we refer to as squigraphs. Our experiments show that squigraphs are more general than existing techniques. They achieve similar classification rates to those achieved by classical shape descriptors. Their performance, however, becomes clearly superior when finer classification and identification operations are targeted. Indeed, while other techniques see their performances dropping, squigraphs maintain a performance rate of the order of 97%.