8th International Conference on
Computational Harmonic Analysis

September 12-16, 2022

Ingolstadt, Germany

"Quaternionic wavelets for colour images"

Hogan, Jeffrey

We present a construction of non-separable, smooth, orthonormal, compactly supported wavelets on ${\mathbb R}^n$ associated with translations along ${\mathbb Z}^n$ and isotropic dilations by powers of $2$. The problem is reduced to that of finding an ensemble of $2^n\times 2^n$ matrices satisfying certain constraints designed to ensure or promote consistency, regularity, symmetry etc. The feasibility problem is solved by employing iterated projection or Douglas-Rachford algorithms and their variants. With a view to building wavelets appropriate for the treatment of multi-dimensional multi-channel signals such as colour images, we adapt the construction to build quaternion-valued wavelets on the plane. This requires the solution of some interesting problems in quaternionic linear algebra. We show that these new bases provide improved performance in some image processing tasks including colour image compression.

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