simplifying and recognizing patterns in fingerprints

I haven't done any reading of the academic literature, but my guess is a PCA analysis on image scan. Basically, compile a huge collection of finger-print scans and find the 'eigen-features'. That way you can identify a specific scan by the linear combination of eigen-features. You can then measure the distance between two scans as the distance between their factor loadings.

I believe this technique is used in some facial recognition software as well.

If you already know where the fingerprint is, and can be sure that there is no 'extra' data in the fingerprint image:

1. Naive way
Binarize the image and do straight pixel-by-pixel compare against fingerprints in your database. This technique will be VERY sensitive to rotation, translation, and scale (i.e. wont work).

2. Less naive way
Convert the fingerprint in to vectors (a HoG if you're lazy), normalize the entire collection, and then do a straight comparison against your database. This method will generate a lot of useless data, but should (generally) be pretty robust to scale, rotation, and translation. It should be fairley accurate provided your data is clean and consistently sampled.

3. Better way
Use a feature extraction algorithm (like SIFT, SURF, or one designed specifically for fingerprints) to extract features, and then match the feature descriptors against those in a database. You'll store a lot less data this way, and boost your accuracy by having a nice tight description of a fingerprint that is insensitive to scale and rotation.

Long story short: There is no right answer. It really depends on what you're using it for. IIRC there is a 'standard' format for storing fingerprints. That might give you some hints.