The Monogenic Riesz-Laplace Wavelet Transform
Michael Unser, BIG

Test Run • 22 August 2008 • BM 4.235

Abstract
We introduce a family of real and complex wavelet bases of L_2(R^2) that are directly linked to the Laplace and Riesz operators. The crucial point is that the family is closed with respect to the Riesz transform which maps a real basis into a complex one. We propose to use such a Riesz pair of wavelet transforms to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We derive a corresponding wavelet-domain method for estimating the underlying instantaneous frequency of the signal. We also provide a simple mechanism for improving the shift and rotation-invariance of the wavelet decomposition. We conclude the paper by presenting a concrete analysis example.