How exactly is x''' determined? The thin lens equation only tells you z_o, but I'm not sure how the diagram gives the location x'''.

I think x''' is selected based on what you want to focus on in the scene. It should be where the focal point of the simulated camera is.

x''' is the point found by passing a ray directly from x' through the center of the lens, at a distance zo from the lens.

How exactly is x''' determined? The thin lens equation only tells you z_o, but I'm not sure how the diagram gives the location x'''.

I think x''' is selected based on what you want to focus on in the scene. It should be where the focal point of the simulated camera is.

x''' is the point found by passing a ray directly from x' through the center of the lens, at a distance zo from the lens.