Statistical behavior of fingerprint matches

Dr. Manfred Bromba
http://www.bromba.com/contacte.htm
2001-05-27
The match scores of fingerprints show a statistical behavior which shall be analyzed in this paper. Especially questions about independence, stationarity etc. are under consideration. 

The matching process

To match two fingerprints means to compare two fingerprints (one request fingerprint and one reference fingerprint) and calculate their similarity. The measure for the similarity is the match score which generally assumes values between 0 (no similarity) and 100 (complete identity). (Sometimes, not the similarity is calculated but some kind of distance between request and reference fingerprint. The greater the similarity, the smaller the distance. All subsequent considerations remain applicable also for such inverse similarity measures.) 

Of course, if exactly the same two sets of digital request data are compared with the reference fingerprint, always the same match score will result. In practice, if the request fingerprint changes, the score values also will also change: 

  • If the request fingerprints come from the same finger but from different acquisitions, the score will show a statistical behavior
  • If the request fingerprints come from different fingers (and different acquisitions), the score shows a statistical behavior with different mean values, depending on the finger

Reasons for match score deviations between different fingerprints

If request and/or reference fingers change, the match score value will also change because the degree of similarity changes. This is very natural and does not explain the statistical behavior. However, if the request fingerprint originates from the same finger but from different acquisitions, changes of the score value are not obvious. Indeed, there are a lot of reasons for such changes: 
  • It is not possible to hit always the same position on the sensor, especially if the sensor only acquires a small window of the whole fingerprint
  • Fingerprint images may be disturbed by image deterioration which may lead to errors in the match score result
Image deterioration itself has a lot of reasons: 
  • Finger contamination
  • Finger dryness or wetness
  • Sensor problems (humidity and moisture, e.g.), leading to sensor surface contamination
  • Finger pressure effects like minutia type inversion and warping
  • Sensor noise due to digitization
  • etc.
In the following, we will discuss the effect of position errors and image deterioration. 

Position errors

Position errors come from different placements of the finger on the sensor. Since the placement is statistical, the score value will change statistically. To minimize this error, a well formed finger guide is necessary, depending on the size of the sensor relative to the finger size. However, since the finger guide never allows an exact reproduction of the placement (due to the elastic state of a finger), a small error will remain. This error should become larger as the time interval between two acquisitions rises. It should be zero if the finger is not lifted between two acquisitions and is maximal if the user has completely forgotten his last finger position.

If request and reference prints come from the same finger, position errors relative to the reference template always should decrease the match score. If they belong to different fingers, the match score may decrease or increase.

Image deterioration

Finger contamination

Depending on the consistency of the contamination, the match score may change for a constant offset value for succeeding acquisitions (stable contamination) or show a non stationary behavior (if contamination disappears for subsequent prints or is transferred to the sensor surface).

If request and reference prints come from the same finger, finger contamination always should decrease the match score. If they belong to different fingers, the match score may decrease or increase.

Finger dryness or wetness

If the finger is too dry or too wet, image deterioration may lead to erroneous match scores. For dry fingers the continual contact with the impermeable sensor surface will increase moisture with time. This will increase the score value for subsequent acquisitions in the case of same request and reference finger. For wet fingers, the opposite becomes true: The match score will decrease.

Sensor problems (humidity and moisture, e.g.), leading to sensor surface contamination

Surface contamination will decrease the performance of the system over time, if the time interval is short. If time between subsequent acquisition is long enough, moisture contamination will evaporate and performance will re-improve. 

If the sensor reacts on humidity (weather), the match score will also depend on climate and hence will show a slowly varying non stationary behavior. 

Finger pressure effects like minutia type inversion and warping

Assuming the same finger for request and reference, the match score will always decrease, if there is a difference between enrollment and verification. This decrease will vary in the same statistical manner as position errors. 
 

Sensor noise due to digitization

The statistical behavior of the match score will strongly depend on the properties of the digitizing noise. If it is white (uncorrelated) within the time interval for fingerprint acquisition, the score "noise" should also be white. 

Conclusions

Assuming the same (genuine) request finger but different acquisitions, the match score depends on: 
  • the last acquisition, leading to statistical dependence
  • slowly varying surrounding conditions, leading to non stationary noise
Assuming acquisitions of different fingers (impostors) without acquisition errors (as described for different acquisitions of the same finger), the match score depends only on the specific finger. However, since each finger is assumed to be unique, the match score varies statistically from finger to finger. 

Modeling a series of match score measurements, it must be stated that each value comes with its own probability function which depends on the probability function of the latest score values and on surrounding conditions.