How do I get Apple TV mirroring to perform better?

Problem: When I turn on Apple TV mirroring, the image on my little computer screen is MUCH better than on the TV - and it's not because the monitor has better resolution. I've examined this closely, and you can tell that the apparent refresh rate/response time is much slower on the TV, and that it must be using fewer colors (the grays look like big blocks on the TV, but all is smooth on the monitor). The images on the TV look "pixellated" even though the setting is 1080p resolution. Also, the portions of the images that are moving quickly look really poor, but it something stays still for awhile, the image improves.

This seems like a compression problem. I've gone through the Displays settings on the computer, and the Apple TV settings via the TV, and I can't find a way to change the compression settings. The Apple TV is set to 1080p, 60Hz, but the compression quality needs to be fixed.

I've already checked and unchecked the "Overscan compensation" checkbox in the Displays settings, and that didn't seem to have an effect.

We got the Apple TV new in Jan 2013, and I think that makes it an Apple TV 3. I't hooked up to a Mac Mini we got at the same time.

Any insights would be helpful.

Thanks.