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 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 Fraction of retrieved imag

es 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 CartesianCoordinatesDiscreet

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

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