Hilbert and Riesz Transforms Using Atomic Function for Quaternionic Phase Computation

Abstract

Complex and hyper-complex valued filtering play a substantial role in signal processing, especially to obtain local features in the frequency and phase domain. In the case of 1D signals, the analytic signal is typically computed using the Hilbert transform. Such complex representation allows us to compute the phase and magnitude of the signal. For high-dimension signals, the Riesz transform and partial Hilbert transforms have been used as extensions of the analytic signal to compute the local phase and local orientation. A major goal of this work is to highlight the role of an atomic function (AF) up(x) as a kernel for the Hilbert and Riesz transforms. It is well known that the use of phase information has a big potential because it is invariant to illumination and rotations. In this regard, we show the advantages to carry out computations of local Riesz phase using the atomic function instead of the classical global-phase approach. In addition, we explain how the atomic function up(x), formulated in the quaternionic algebra framework, can be used as a building block to perform multiple analytical operations commonly used in image processing, such as low-pass filter, derivatives, local phase processing, steering quaternionic filters, multi-resolution analysis and symmetries detection.