What is the general technique that one extract an image of a person from various backgrounds.

I am learning optimization theory right now, and we covered how to create different facial expression by finding the rotational and translational vectors of a static background image (basically the L2 norm of a minimization of Rx + r - y), now, although that is interesting, that made be wonder how you would solve the problem with different backgrounds, or with the same person in an image but with different centering.

Specifically, how would I capture the person in the image in the first place, a pixel by pixel Rotational and translational movement would not work here, I was thinking of doing a low rank approximation if the person is at least most part of the image, but what if the person is a small part of the image?

Any advice would be appreciated