Auto distorted image distortion method

OK Say I have 2 images. The first is a set of lines that form a grid with a circle. The second is the same grid and the DISTORTED circle. The second image is a distorted result of the first image when it is stretched over a three-dimensional solid-state element.

What I want to do is morph the circle in the second distorted image, where it will be in the first image before the distortion. In other words, I want to predict what the shape of the circle should be before distortion, so that when it is distorted, it ends up looking like a perfect circle in the first image.

The grid is only there, as an aid, I am really interested in the circle. A circle is simply an idealized image. In real life, the "circle" is a graphic. These two images will be valid scans from the sheet before and after the distortion. Distortion is also just idealized. In real life, this is more complicated than that.

I am open to commercial software or using Matlab or other software for writing something. I just don't know where to start. Any ideas are welcome.

BTW ... here is a message that is somewhat similar to nature.

Original - unconfirmed image: Original - Undistorted

After distortion: Distorted

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