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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

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

By: Bryan Bonvallet

Nikolla Griffin

Advisor: Dr. Jia Li

introduction the problem
Introduction: The Problem
  • Increased availability of 3D shapes
  • Text based searches are not effective
  • Robust for simple and complex applications
shape descriptors
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
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
comparison method
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
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 harmonics7
4D Hyperspherical Harmonics
  • Theory
    • 3D volume to 4D hypersphere mapping
    • Hyperspheric harmonic analysis
    • No radii cuts
4d spherical harmonics
4D Spherical Harmonics

Voxelization

Cartesian

Coordinates

Discreet

Cartesian Continuous:

4D Unit Sphere

Hyperspherical

Coordinate

continuous

4D Harmonic

Coefficients

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