This paper introduces a two-dimensional (2-D) generalization of the analytic signal, known as the monogenic signal. The approach is based on the Riesz transform, which is used instead of the Hilbert transform. The combination of a 2-D signal with its Riesz transformed version yields the monogenic signal, which is derived analytically from irrotational and solenoidal vector fields. The local amplitude and local phase of the monogenic signal preserve the split of identity, meaning they are invariant and equivariant with respect to energetic and structural information. The paper also discusses the properties of the analytic signal, such as symmetry, energy, aliased transfer function, and orthogonality, and compares these properties with other 2-D analytic signal approaches. A geometric phase interpretation based on the relationship between the 1-D analytic signal and the 2-D monogenic signal established by the Radon transform is introduced, along with possible applications of this relationship. The monogenic signal is shown to be isotropic, preserving the split of identity, and orthogonally decomposing the signal into energetic, structural, and geometric information. The paper concludes by discussing the potential applications of the monogenic signal and its advantages over previous 2-D analytic signal approaches.This paper introduces a two-dimensional (2-D) generalization of the analytic signal, known as the monogenic signal. The approach is based on the Riesz transform, which is used instead of the Hilbert transform. The combination of a 2-D signal with its Riesz transformed version yields the monogenic signal, which is derived analytically from irrotational and solenoidal vector fields. The local amplitude and local phase of the monogenic signal preserve the split of identity, meaning they are invariant and equivariant with respect to energetic and structural information. The paper also discusses the properties of the analytic signal, such as symmetry, energy, aliased transfer function, and orthogonality, and compares these properties with other 2-D analytic signal approaches. A geometric phase interpretation based on the relationship between the 1-D analytic signal and the 2-D monogenic signal established by the Radon transform is introduced, along with possible applications of this relationship. The monogenic signal is shown to be isotropic, preserving the split of identity, and orthogonally decomposing the signal into energetic, structural, and geometric information. The paper concludes by discussing the potential applications of the monogenic signal and its advantages over previous 2-D analytic signal approaches.