SOHO: Orthogonal and Symmetric Haar Wavelets on the
Sphere
















Figure
1: Our
derivation yields two bases that are orthogonal and symmetric. The two
sets of basis functions for a partition at level 0 are shown above. 
Abstract:
We propose the
SOHO wavelet basis. To our knowledge this is the first spherical Haar
wavelet basis that is both orthogonal and symmetric, making it
particularly well suited for the approximation and processing of
allfrequency signals on the sphere. The key to the derivation of the
basis is a novel spherical subdivision scheme that defines a partition
acting as domain of the basis functions. The construction of the SOHO
wavelets refutes earlier claims doubting the existence of such a basis.
We also investigate how signals represented in our new basis can be
rotated. Experimental results for the representation of spherical
signals verify that the superior theoretical properties of the SOHO
wavelet basis are also relevant in practice.
Publications
Christian Lessig and Eugene Fiume, SOHO: Orthogonal
and Symmetric Haar Wavelets on the Sphere, ACM Transactions of Graphics, March 2008 (Presented at SIGGRAPH 2008). (preprint, bibtex)
Christian Lessig, Orthogonal
and Symmetric Haar Wavelets on the Sphere, Master of Science
thesis, Department of Computer Science, University of Toronto, April 2007. (pdf, bibtex)
Miscellaneous
Matlab implementation of the SOHO wavelets.
Presentation
slides from SIGGRAPH 2008.
Comparison of Spherical Harmonics and the SOHO wavelets.
Video detailing the construction of the SOHO wavelets.
Presentation
slides from my MSc. defense.
Mathematica notebook detailing the construction of the basis functions of the SOHO wavelets.
Mathematica notebook detailing the construction of the subdivision scheme for the partition of the SOHO wavelets.
A short writeup on second generation wavelets (used as handout for a blackboardbased talk).
