Geometric Hashing Technique

Geometric Hashing is a technique for matching a target molecule (or a collection of such) against a set of one or more models(e.g. database). The models are assumed to be known in advance, and thus allow preprocessing, as follows:

For each model object (first molecule in our case or each molecule in database) do:

1.

Pick a reference frame.

2.

Compute the 3D orthonormal basis associated with this reference frame and its shape signature (e.g. triangle sides length).

3.

Compute the coordinates of all the other points (in a pre-specified neighborhood) in this reference frame.

4.

Use each coordinate as an address to the hash (look-up) table. Store the entry [protein id, ref. frame, shape sign., point] at the hash table address.

5.

Repeat above steps for each model reference frame (noncollinear triplet of model points).

Recognition stage of the algorithm uses the hash table, prepared in the preprocessing step. The matching of a target object is accomplished as follows:

1.

For each reference frame of the target:

2.

Compute the 3D orthonormal basis and the shape signatureassociated with it.

3.

Compute the coordinates of all other points in the current reference frame.

4.

Use each coordinate to access the hash-table and retrieve all the records [protein id, ref. frame, shape sign.,point].

5.

For records with matching shape signature ``vote'' for the pair [protein, ref. frame].

6.

Compute the transformations of the ``high scoring'' hypotheses. For each hypothesis one can also register the pairs of matching points. This match list along with the transformation comprise a seed match.

Now we can construct an alignment algorithm that uses Geometric Hashing as follows: