By: Bryan Bonvallet Nikolla Griffin Advisor: Dr. Jia Li

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Presentation transcript:

By: Bryan Bonvallet Nikolla Griffin Advisor: Dr. Jia Li 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 Robust for simple and complex applications

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

3D Spherical Harmonics Benefits Process Problems 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

Comparison Method Precision Recall Example 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

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

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

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

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

Future Work Use SVD algorithm for solving Ax=b

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. Problems: Prepare for potential questions Clearly state the importance of research Better explain the problem with radii cuts Picture labels need to be changed.