1 / 11

3D Shape Descriptors: 4D Hyperspherical Harmonics “ An Exploration into the Fourth Dimension ”

3D Shape Descriptors: 4D Hyperspherical Harmonics “ An Exploration into the Fourth Dimension ”. By: Bryan Bonvallet Nikolla Griffin Advisor: Dr. Jia Li. Introduction: The Problem. Increased availability of 3D shapes Text based searches are not effective

prentice
Download Presentation

3D Shape Descriptors: 4D Hyperspherical Harmonics “ An Exploration into the Fourth Dimension ”

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3D Shape Descriptors: 4D Hyperspherical Harmonics“An Exploration into the Fourth Dimension” By: Bryan Bonvallet Nikolla Griffin Advisor: Dr. Jia Li

  2. Introduction: The Problem • Increased availability of 3D shapes • Text based searches are not effective • Robust for simple and complex applications

  3. Shape Descriptors • Definition: Computational 3D shape representation • Characteristics • Easy comparison • Independent of original representation • Concise to store • Insensitive to noise • Challenges • Rotation • Translation • Scale

  4. 3D Spherical Harmonics • Benefits • Invariant to scale and rotation • Relatively invertible • High precision/ recall • Process • Voxelize • Cut along radius • Analyze harmonics • Problems • 3D storage • Error due to radii cuts • Harmonic truncation

  5. Comparison Method • Precision • Fraction of retrieved images which are relevant • Recall • Fraction of relevant images which are retrieved • Example • 20 cows total • 30 results • 10 results are cows • Precision = 1/3 • Recall = 1/2

  6. 4D Hyperspherical Harmonics • Theory Basis • Want harmonics over entire shape • No slicing across radii • n-sphere harmonics • 2D plane to 3D sphere mapping

  7. 4D Hyperspherical Harmonics • Theory • 3D volume to 4D hypersphere mapping • Hyperspheric harmonic analysis • No radii cuts

  8. 4D Spherical Harmonics Voxelization Cartesian Coordinates Discreet Cartesian Continuous: 4D Unit Sphere Hyperspherical Coordinate continuous 4D Harmonic Coefficients

  9. Conclusion • Inconclusive • we are using a square matrix for solving coefficients (LU decomposition algorithm for solving Ax=b) • we can only sample a fixed number of points • we cannot use the entire sample set of points

  10. Future Work • Use SVD algorithm for solving Ax=b

  11. References • J. Avery. Hyperspherical Harmonics and Generalized Sturmians. Dordrecht: Kluwer Academic Publishers, 2000. • N. D. Cornea, et al. 3d object retrieval using many-to-many matching of curve skeletons. In Shape Modeling and Applications, 2005. • D. Eberly. Spherical Harmonics.  http://www.geometrictools.com.  March 2, 1999. • T. Funkhouser, et al. A search engine for 3D models. In ACM Transactions on Graphics, pages 83-105, 2003. • X. Gu and S. J. Gortler, and H. Hoppe. Geometry images. In Proceedings of SIGGRAPH, pages 355-361, 2002. • M. Kazhdan. Shape Representations And Algorithms For 3D Model Retrieval. PhD thesis, Princeton University, 2004. • M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz. Rotation invariant spherical harmonic representation of 3D shape descriptors. In Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (2003) pages 156-164, 2003. • A. Matheny, and D. B. Goldgof. The Use of Three- and Four-Dimensional Surface Harmonics for Rigid and Nonrigid Shape Recovery and Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, volume 17, pages 967-981,1995. • A. V. Meremianin. Multipole expansions in four-dimensional hyperspherical harmonics.  Journal of Physics A: Mathematical and General.  Issue 39, pages 3099-3112.  March 8, 2006. • C. Misner. Spherical Harmonic Decomposition on a Cubic Grid.  Classical and Quantum Gravity, 2004. • M. Murata, and S. Hashimoto. Interactive Environment for Intuitive Understanding of 4D Object and Space. In Proceedings of International Conference on Multimedia Modeling, pages 383-401, 2000. • W. Press, S. Teukolsky, W. Vetterling, B. Flannery. Numerical Recipes in C: The Art of Scientific Computing (Second Edition).  Cambridge University Press, 1992. • J. Tangelder, and R. Veltkamp. A survey of content based 3d shape retrieval methods. In Shape Modeling International, pages 145-156, 2004.

More Related