How do you decide how many points to sample on the aperture plane?
To ray trace a lens in a pinhole camera, we would just send a ray through the pinhole. But a lens has finite area (an aperture). We want to generate multiple points on the lens and sum them up. We generate some point on the lens, and given the lens position and film plane, we can know what point in object space will correspond to that image plane and be in focus. All rays leaving x' will map to x.