Actually once you know the circle's position (this needs to be accurate) and approximate size you can convert it into log-polar coordinates and then apply phase

Actually once you know the circle's position (this needs to be accurate) and approximate size you can convert it into log-polar coordinates and then apply phase correlation [1] to a known "template" to determine the rotation angle very accurately. I'm not sure what would be the best pattern for this, but I might actually try implementing a prototype. [1] https://en.wikipedia.org/wiki/Phase_correlation https://en.wikipedia.org/wiki/Phase_correlation