The monogenic signal

This paper introduces a two-dimensional (2-D) generalization of the analyti c signal. This novel approach is based on the Riesz transform, which is use d instead of the Hilbert transform. The combination of a 2-D signal with th e Riesz transformed one yields a sophisticated 2-D analytic signal: the mon ogenic signal. The approach is derived analytically from irrotational and s olenoidal vector fields. Based on local amplitude and local phase, an appro priate local signal representation that preserves the split of identity, i. e., the invariance-equivariance property of signal decomposition, is presen ted. This is one of the central properties of the one-dimensional (1-D) ana lytic signal that decomposes a signal into structural and energetic informa tion. We show that further properties of the analytic signal concerning sym metry, energy, allpass transfer function, and orthogonality are also preser ved, and we compare this with the behavior of other approaches for a 2-D an alytic signal. As a central topic of this paper, a geometric phase interpre tation that is based on the relation between the 1-D analytic signal and th e 2-D monogenic signal established by the Radon transform is introduced. Po ssible applications of this relationship are sketched, and references to ot her applications of the monogenic signal are given.