By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site (Wayback Machine):

I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:

Is it possible to find and draw corresponding conformal maps?

Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.

The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.