Orthogonal and Symmetric Haar

Orthogonal and Symmetric Haar Wavelets on the Sphere Christian Lessig Master of Science Graduate Department of Computer Science University of Toronto 2007 The efficient representation of signals defined over spherical domains has many applications. We derive a new spherical Haar wavelet basis (SOHO) that is both orthogonal and symmetric, rebutting previous work that presumed the nonexistence of such a basis. The key to obtaining the basis is a novel spherical subdivision scheme that defines a partition acting as the domain of the basis functions. We also derive basis transformation matrices that permit the rotation of a signal represented in our new basis. The elements of these matrices can be computed analytically, in contrast to previous work that required numerical computations. Experimental results for the approximation and rotation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis also have practical benefits over previous representations.