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

1 / 11

# 3D Shape Descriptors: 4D Hyperspherical Harmonics “ An Exploration into the Fourth Dimension ” - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about '3D Shape Descriptors: 4D Hyperspherical Harmonics “ An Exploration into the Fourth Dimension ”' - prentice

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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

By: Bryan Bonvallet

Nikolla Griffin

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
• Invariant to scale and rotation
• Relatively invertible
• High precision/ recall
• Process
• Voxelize
• Analyze harmonics
• Problems
• 3D storage
• Error due to radii cuts
• Harmonic truncation
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
4D Hyperspherical Harmonics
• Theory Basis
• Want harmonics over entire shape
• n-sphere harmonics
• 2D plane to 3D sphere mapping
4D Hyperspherical Harmonics
• Theory
• 3D volume to 4D hypersphere mapping
• Hyperspheric harmonic analysis
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.