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 replaces the Hilbert transform. The monogenic signal is derived from the combination of a 2-D signal with its Riesz transform, resulting in a sophisticated 2-D analytic signal. The method is derived from irrotational and solenoidal vector fields, preserving the invariance-equivariance property of the 1-D analytic signal, which decomposes a signal into structural and energetic information. The monogenic signal also preserves properties such as symmetry, energy, allpass transfer function, and orthogonality. A geometric phase interpretation is introduced, relating the 1-D analytic signal and the 2-D monogenic signal via the Radon transform. The monogenic signal is used for applications such as edge detection, stereo correspondence, and image denoising. The paper also discusses the relationship between the monogenic signal and the 1-D analytic signal, showing that the monogenic phase corresponds to the complex phase in 1-D. The monogenic signal is isotropic and preserves the split of identity, decomposing the signal into energetic, structural, and geometric information. The paper concludes that the monogenic signal is a generalization of the analytic signal to two dimensions, with applications in image processing and signal analysis.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 replaces the Hilbert transform. The monogenic signal is derived from the combination of a 2-D signal with its Riesz transform, resulting in a sophisticated 2-D analytic signal. The method is derived from irrotational and solenoidal vector fields, preserving the invariance-equivariance property of the 1-D analytic signal, which decomposes a signal into structural and energetic information. The monogenic signal also preserves properties such as symmetry, energy, allpass transfer function, and orthogonality. A geometric phase interpretation is introduced, relating the 1-D analytic signal and the 2-D monogenic signal via the Radon transform. The monogenic signal is used for applications such as edge detection, stereo correspondence, and image denoising. The paper also discusses the relationship between the monogenic signal and the 1-D analytic signal, showing that the monogenic phase corresponds to the complex phase in 1-D. The monogenic signal is isotropic and preserves the split of identity, decomposing the signal into energetic, structural, and geometric information. The paper concludes that the monogenic signal is a generalization of the analytic signal to two dimensions, with applications in image processing and signal analysis.