1. 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

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 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

5. 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

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.  March 2, 1999.
T. Funkhouser, et al. A search engine for 3D models. In ACM Transactions on Graphics, pages , 2003.
X. Gu and S. J. Gortler, and H. Hoppe. Geometry images. In Proceedings of SIGGRAPH, pages , 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 , 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 ,1995.
A. V. Meremianin. Multipole expansions in four-dimensional hyperspherical harmonics.  Journal of Physics A: Mathematical and General.  Issue 39, pages   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 , 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 , 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.