non-rigid transformation for data correction

Question

I am seeking a mathematical method to perform non-rigid transformations of quadrilaterals as demonstrated below.

Here you can imagine the four coordinates of a rectangular figure where the data interior to the coordinates has been extracted but is known to be incorrect due to the distortion in the original figure. The goal is to determine the transformation which corrects the figure coordinates to known points and to apply that transformation to the figure data; thereby reducing data extraction errors.

One could always use image processing tools to alter the image perspective prior to data extraction, but I would like to learn a clean way to do this using math. Any suggestions?

Answer 1

This can be done with projective transformations. Many thanks to user619894 for pointing me in the right direction.