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The specific approach taken by the WAX software is to assign apparitions to small lanes (spatial bins corresponding to a range of declinations and right ascensions), and then determining, given a lane , which other lanes contain apparitions that potentially belong to match sets of apparitions in . For any single apparition, only a tiny area of the sky is examined to find a match set. This is the key to obtaining good performance in the swiss cheese algorithm.
Specifically, each band is split into lanes where , and , . Each lane covers a range of right ascensions :
Without loss of generality, let (the angle between unit vectors is invariant under rotation). Furthermore, define . This minimum distance occurs between the positions and , expressed in polar coordinates (unless , in which case this minimum distance is either undefined or zero).
When expressed as vectors in , and . Using the relation , the following formula is obtained :
To find the number of lanes in a band the actual lane width for each possible value of is computed. The number of lanes is determined from the largest resulting in a lane width greater than the requested minimum, or if this is impossible. For example, bands containing a pole have a lane width of zero regardless of and will therefore contain just a single lane. Since this directly effects the efficiency of the match set finding algorithm, the pole bands should be as skinny as possible.
The WAX application uses a minimum lane width of arcsecond when the -o option is specified, and arcseconds otherwise.
Once all apparitions in a band have been retrieved and assigned to the appropriate lanes, the apparitions in each lane are sorted in order of increasing declination.
Using the relations derived above, the lane window is the smallest integer satisfying
or if the relation cannot be satisfied.
Furthermore, if the position has a declination , then apparitions in the match set will have declinations in the range . For each being searched, the apparitions in this range can be found using a simple binary search (since apparitions within a lane are declination sorted). Each such apparition is then tested against , and included in if and only if
Number the apparitions based on their enclosing lane and position within that lane as follows :
where has declination . Since the apparitions in a lane are declination sorted,
By "walking" up the apparitions within a single lane , finding match sets for , then , etc... the match set finding process can be made even more efficient :
This leads to the lane walking algorithm described below.
Consider the first apparition . Find the smallest non-negative integer such that has declination (using a binary or linear search).
Now, let . While :
At this point, all apparitions in belonging to the match set for have been found, so the next apparition in , is considered. To find the first apparition in possibly belonging to the match set for , increment until . Then repeat the steps above, finding all apparitions in belonging to the match set for .
Continue in similar fashion until all apparitions in have been considered.
By walking each lane in this manner, match sets for every apparition in are found.