What is the ruling on photographing the stages of pregnancy and a woman asking her obstetrician-gynecologist to photograph and video-record the birth as a memento?

If the doctor deems it necessary to scan the pregnancy, then there is no objection. However, scanning the fetus at different stages of pregnancy without necessity is impermissible, as it involves the woman uncovering her body without need.

As for a woman requesting the doctor to film the birth process for a keepsake, this is even more forbidden and a greater abomination, because it entails filming severe awrat (private parts), which is prohibited and a heinous act.