The monogenic signal

This paper introduces a two-dimensional (2-D) generalization of the analytic signal. This novel approach is based on the Riesz transform, which is used instead of the Hilbert transform. The combination of a 2-D signal with the Riesz transformed one yields a sophisticated 2-D analytic signal: the monogenic signal. The approach is derived analytically from irrotational and solenoidal vector fields. Based on local amplitude and local phase, an appropriate local signal representation that preserves the split of identity, i.e., the invariance-equivariance property of signal decomposition, is presented. This is one of the central properties of the one-dimensional (1-D) analytic signal that decomposes a signal into structural and energetic information. We show that further properties of the analytic signal concerning symmetry, energy, allpass transfer function, and orthogonality are also preserved, and we compare this with the behavior of other approaches for a 2-D analytic signal. As a central topic of this paper, a geometric phase interpretation that is based on the relation between the 1-D analytic signal and the 2-D monogenic signal established by the Radon transform is introduced. Possible applications of this relationship are sketched, and references to other applications of the monogenic signal are given.