jumping on late here, but what about a simple limit switch at either end of the garage door 'swing'. if the arduino starts up from BOR or POR and does not see either 'all the way open' or 'all the way closed', it will monitor or ignore sensor input until it gets the shut/open signal. then it can start taking reliable readings. You would need two digital pins dedicated to 'open' and 'shut'.
love the idea of the tape measure! I will use that in future!