One hint is to look into the literature. It seems that the 2D extension of the HT is not uniquely defined:

The Hilbert Transform (HT) and the analytic signal (AS) are widely used in their one-dimensional version for various applications. However, in the bi-dimensional (2D) case as occur for images, the definition of the 2D-HT is not unique and several approaches to it have been developed

The cited paper is freely available and comparatively practical and points out the difficulties. As far as I can see from a glance, equation (5) is what you need to apply after calculating the Fourier transform. The definitions for sgn are explicitly given under the Materials section.

An Approach to the 2D Hilbert Transform for Image Processing Applications