I need to stretch oval shapes on cartographic bitmaps like this one -

in order to avoid distortions while wrapping images on sphere.
Shapes should be stretched horizontally (suppose according to log function):
maximum distortion on the top and the bottom,
minimum in the middle, like on this example:

or on this one:

Three frames of the same sky map area - shapes on the middle one
should be distorted like these on the right and the left image.
Is there any way to do this job with help of IM?
I guess 'blur' would be suitable tool but of course I'm not sure about that
and I don't know how to tell IM what I need to have been made.
I will be grateful for any help or references.
Forgive me my rough English please.
Best -
yarr