Comparison of Properties of Analytic, Quaternionic and Monogenic 2-D Signals STEFAN L. HAHN, KAJETANA M. SNOPEK Faculty of Electronics and Information Technology Warsaw University of Technology Nowowiejska 15/19,00-665 Warsaw POLAND Abstract:- The paper compares the features of three extensions of the notion of the Gabor’s analytic signal for 2-D signals: the analytic signals with single-quadrant spectra (AS), the quaternionic signals with quaternionic single-quadrant spectra (QS) and the monogenic signals (MS). A good platform of comparison are the polar representations of the AS and the QS and the spherical coordinate representation of the MS. Key-Words:- analytic, quaternionic, monogenic signals 1 Introduction 2 The notion of the Gabor’s [1] analytic signal is widely used in the theory of 1-D signals and systems. For a real time signal u(t), the analytic signal has the form () () () t ut jv t ψ = + z t , where is the Hilbert transform of u. Of course, many other linear operators could be used instead of H. However, Vakman [2] has shown, that only the Hilbert operator satisfies three very simple conditions. Let us recall, that the analytic signal has a one-sided Fourier spectrum at positive frequencies (the conjugate one at negative frequencies). The adjective “analytic” indicates that, for an analytic function Ψ(z), () [ ( )] vt Hut = j τ = + , i.e., () (,) z ut (,) jv t Ψ τ (, z ut τ 0) (, 0) jv t = + () , the analytic signal represents a boundary distribution of the analytic function Ψ τ τ + + + = = = along the 0 + side of the time axis t [3]. In the past, several extensions of the notion of the Gabor’s analytic signal for 2-D signals have been proposed. In this paper we compare three selected methods which, as we believe, are the most important: 1. Analytic signals (AS) with single quadrant spectra first described in [4] and presented in [5], 2. Quaternionic signals (QS) with single quadrant quaternionic spectrum first described in [6] and also in a regular paper [7], 3. Monogenic signals (MS) first described in a regular paper in [8]. The Fourier spectral analysis of 2-D signals The Fourier transforms of 2-D signals are well known. However, notations and interpretations are important for our presentation. Therefore, let us describe shortly the basic relations. Consider a 2-D signal u(x 1 , x 2 ) defined in the Cartesian coordinates x = (x 1 , x 2 ). The signal u can be decomposed in a union of four terms [5] ( ) ( ) ( ) ( ) ( 1 2 1 2 1 2 1 2 1 2 , , , , ee oe eo oo ux x u x x u x x u x x u x x = + + + ) , (1) where the subscripts denote eveness or oddness w.r.t. the variables x 1 and x 2 . This decomposition plays an important role in next considerations. The 2-D Fourier transform (2-D FT) of u is defined by the integral ( ) ( ) 1 2 1 2 1 2 1 2 , , j j U f f e ux x e dx dx α α − − = ∫∫ (2) where 1 2 1 1 f x α π = and 2 2 2 2 f x α π = . We used the order of functions in the integrand similar, as used in the definition of the quaternionic Fourier transform (see the Eq. (9)). The insertion of (1) in (2) yields ( ) ( 1 2 , ee oo oe eo U ) f f U U j U U = − − + (3) where