How to warp/stretch a sector/"pie slice" into a whole circle

Onnij,

What you wish is the opposite of what is usual/easy/possible to do, a bit like the UnBlur filter that is joked about in Photoshop.

This could be called an UnArch effect.

Especially the pie slice with its single point at the bottom, which is supposed to be stretched to the same length as the quarter circle at the top, is a tough task.

I believe a better approach would be to perceive which horizontal (set(s) of) shape(s) can be made to look right when stretched and bent to fill a full (circular) ring; and forget about anything extending to the centre, or even near it.

Depending on the actual artwork, maybe working in different ways at different distances from the centre, as in separate concentric rings with different (moderately varying) levels of stretching and bending, going from more stretching and less bending in the outermost ring to the opposite in the innermost one.

The artbrush that pixxxelschubser mentioned could give you this. But the starter image would no be a pie, has to be a rectangle.

It might be possible - with an artbrush ...

Not as such. Can you show the actual image though? It may be possible to achieve what you want via other means.