How can I convert a doubly periodic image into a regular one?

I'm trying to create a Droste effect animation of a single image inspired by M.C. Escher's print gallery version from a Leiden University study. Here is a sketch I made :

Thanks to Ultra Fractal and a Droste plugin from Jos Leys, I managed to create the desired effect :

As you can see I have a resolution problem on this last sketch due to the very high magnification (about 256 times).

I was able to create a doubly periodic image of the sketch that I could use to make my detailed drawing in a variable resolution :

My problem is that I don't find a way to convert back the image into a spiraling one. How could I do?

Thanks in advance for reading me and eventual help.

Fourier transforms can easily convert doubly periodic images to regular ones. A Fourier transform decomposes the periodic image into discrete frequency components, which can be recombined into a non-periodic image. "De-periodizing" the image involves this.

Thanks for the answer bertramegon1!

I'm glad to learn the name of the function to "de-periodize" my image but as I'm quite bad at maths, I don't know how I could apply it. Is it possible with Ultra Fractal?

In the meantime I actually found a way to do this with the help of 3D programs (Cinema 4D and Blender), making distorted UV maps. It's not very precise though and I have a bit of jagged edges when rendering.

I guess I could stick with my UV method, especially with the abilities to render animations with such programs (and even remaking the drawing in 3D) - but if by chance you could tell me how I could "Fourier transform" my drawing, that'd be nice!

Thanks again!