First of all I realized that audio processing is at a really awesome point right now: you could train a system to capture your dog's voiceprint when it barks! I suspect that would be sufficiently unique to only match your own dog in your neighborhood. But this could be attacked by recording the barks and playing them back, means your dog is making noise every time it wants to be let in, and also depends on the dog learning new tricks ;)

Then I wondered about GPS tracking, then realized that would be stupid since this doesn't require absolute positioning.

So then I thought about Bluetooth LE. I think this is reasonably short-range(?).

Then I thought about those tiny 2.4GHz 2Mbps radio modules you can get on eBay, and fiddling with one of those to set the output power weak enough so it only works when really close. But that might be so close to "not transmitting at all" it might be very flaky.

But the above 3 ideas all require battery replacement in a collar.

Image recognition does sound like the way to go. One thought that does come to mind though - capturing a wide view of the environment and requiring both "dog within small perimeter around dog door" AND "empty environment with no additional objects in it", for additional security (especially if the dog door is positioned such that it's not too hard to reach up through the hole to the doorhandle). Some tweaking would probably be needed to make the system resilient to birds and objects moved around the environment.