How to progressively map a plane to a semisphere?

[PhotoComix]

Thank for the quick replies

RobA, Ravell
yes i want something as in your examples, i was thinking from a POV
from the top as in RobA example

About how to treat the center both RobA and Ravell guessed correctly what i wish:

Only a mild distortion on the center, much more severe to the periphery ,and ,if possible confined very closed to the perimeter

this seems fine

EDIT

Seems fine at thumb size, at original size much less but i supposed was intended to be seen at thumb size.
anyway a very little bit more distortion on the center, a bit more smooth transition with more severe distortion to the periphery



BUT the real problem is how to make a smooth progression from the image to the left to the image to the right ?


About the suggestion to use blender or other 3D app:
May be very good, but i have no any previous experience of 3D app,
more i do not use so muck hot key, while the constant use of hot key seems strictly needed in blender.

Do you think blender may solve the problem of the transition?
(NOT by fading! i need a "progressive mapping" )

PS OOOOPSS

I do not know why RobA example come out so big, at this size seems quite wrong but looked fine before at little size