Removing drift from integration of accelerometer data
Double integration amplifies any offsets, non-linearities and noise.  These can't be removed without the use of some type of external reference point measurements (e.g. not from the accelerometer) or other information.  Calibration might be able to remove most of the ones that do not vary with time (temperature, etc.)  

For instance, if the size of the room is known, you can robustly clamp position to be inside the walls and above the floor.