Skewed 2D Hilbert transforms and computed AM-FM models

Computed AM-FM models represent images in terms of instantaneous amplitude and frequency modulations. However, the instantaneous amplitude and frequency of a real valued image are ambiguous. We apply the directional 2D Hilbert transform to compute a complex extension for a real image. This extension, called the analytic image, admits most of the attractive properties of the 1D analytic signal. However, the analytic image is not unique: for a given real image, taking the Hilbert transform in the horizontal and vertical directions yields different complex extensions and differing computed AM-FM models. We show that these two differing models are essentially equivalent and develop explicit formulations relating them.