Analytic image using 2-D Hilbert transform

Question

I am working on a problem of phase (/phase derivative) retrieval from closed fringe patterns. The paper https://opg.optica.org/ao/fulltext.cfm?uri=ao-58-16-4420&id=413113 which uses a pseudo-Wigner Ville distribution (PSWVD) to directly estimate the unwrapped phase derivative is what I want to implement.
The PSWVD method requires a complex signal (here, complex/analytic image). The authors have applied the method to holographic interferometry where one gets the complex signal by default. But I am interested in applying the method to photoelastic fringe patterns(another interferometric technique) where one gets only closed fringe patterns (i.e., real part of the complex signal). So, I need to perform real to analytic image conversion before implementing PSWVD.

For experimentation, a Gaussian phase is simulated (as shown below)

The real part of the signal exp(iphi) based on the simulated phase will be cos(phi). This is the closed fringe pattern, which is given below:

The aim is to obtain an analytic image, i.e, get the sin(phi) image from cos(phi) image using Hilbert transform to define the analytic image of the form exp(iphi).

After applying the 2-D Hilbert transform (available in the scipy module) on the cos(phi) image, the result obtained is:

This is clearly not the sin(phi) image. The true sin(phi) image is given below:

Later, I found that this is reported in literature as half plane Hilbert complex image and this real to analytic image conversion is quite involved than a simple function call.

I found that introducing spatial carrier to the real fringe pattern and then applying Hilbert transform is a possible way out. But I am not quite sure how this works as I am not from a signal processing background. I would highly appreciate it if someone can point me to the right resources concerning this (mainly on spatial carriers or single image real to analytic coversion)?

PS:

-> I have been told that phase shifting method can also be used for constructing the analytic image, but this requires multiple images to be captured experimentally. I am interested in single image real to analytic conversion for which spatial carrier fringe method was recommended.
-> I am aware of the closed fringe techniques that are based on iterative optimization, but since experimentally I can easily introduce a spatial carrier, I preferred this method over the others.