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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

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

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  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.

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