Combination of multiple fingers in fingerprint verification

Heidi Wesemann-Schlegel - Manfred Bromba
http://www.bromba.com/contacte.htm
Permanent address for citation: urn:nbn:de:0125-2008032549
2004-09-25 (First version 2001-11-23, first release 2004-09-19)

Introduction

Decreasing the area of an fingerprint sensor significantly decreases the performance in terms of FAR and FRR if the absolute size falls below a certain value. To compensate for this effect, the idea is to use more than one finger for verification. The verification result for each finger then has to be combined to prepare the final decision whether the user is to be authenticated or not. In practice, this is to be achieved by combining the resulting score values for each finger using an appropriate mathematical procedure, for example using the smallest or the largest of the two values to pass a certain threshold. This document describes the results for several cases when taking into account realistic impostor behavior.

Target

The following 5 cases have partly been examined:
  • Verification with 2 different fingers
    • Both fingers are to be accepted ("AND")
    • At least 1 of 2 fingers is to be accepted ("OR")
  • Verification with 2 equal fingers
    • The finger must be accepted twice ("AND")
    • The finger must be accepted at least once in two efforts ("OR")
    • The mean value of two subsequent prints of a finger has to be accepted ("MEAN")

Theory

Estimation of error rates

There are three kinds of different estimation levels for the desired results (methods):

Real-life test

Performing a real-life test with acquiring a series of fingerprints with two of them being directly consecutive and then determining the error rate by off-line testing

Real-life approximation

Take the series of score values from the ID Mouse Field Test calculation (accepting that their acquisition time distance is at least one minute)

Mathematical approximation

Mathematical analysis based on the FAR and FRR results from the ID Mouse Field Test
Only the second and third method are used in this investigation. If the fingerprint match score values are statistically independent and stationary, it is to be expected that the same result should be obtained in all cases.

Two different fingers OR

Real-life approximation

The maximum of two consecutive score values has to exceed the threshold. This is equivalent to the requirement that the score value of the first fingerprint OR the score value of the second fingerprint exceeds the threshold. This calculation is done directly on the score series from the ID Mouse Field Test. The calculation of the FAR/FRR curves is done straight forward.

Mathematical approximation

The FAR and FRR curves for 2 fingers (index 2) are calculated directly from the curves for 1 finger (index 1) assuming that all fingers have the same FAR and FRR, respectively, using the formulae:
 
FAR2(th) = 1 - (1 - FAR1(th))2
FRR2(th) = (FRR1(th))2

Two different fingers AND

Real-life approximation

The minimum of two consecutive score values has to exceed the threshold. This is equivalent to the requirement that the score value of the first fingerprint AND the score value of the second fingerprint exceed the threshold. This calculation is done directly on the score series from the ID Mouse Field Test. The calculation of the FAR/FRR curves is done straight forward.

Mathematical approximation

The FAR and FRR curves for 2 fingers (index 2) are calculated directly from the curves for 1 finger (index 1) assuming that all fingers have the same FAR and FRR, respectively, using the formulae:
 
FAR2(th) = FAR1(th)
FRR2(th) = 1 - (1 - FRR1(th))2

For the FAR we assume that it is the best strategy for an impostor to use the same finger instead of different fingers. Thus the FAR should remain the same as in the single finger case instead of being squared.

Two equal fingers OR

Real-life approximation

The maximum of two consecutive score values has to exceed the threshold. This is equivalent to the requirement that the score value of the first fingerprint OR the score value of the second fingerprint exceeds the threshold. This calculation is done directly on the score series from the ID Mouse Field Test. The calculation of the FAR/FRR curves is done straight forward.

Mathematical approximation

The FAR and FRR curves for 2 fingers are calculated directly from the curves for 1 finger assuming that all fingers have the same FAR and FRR, respectively. For the FAR we assume that it is the best strategy for an impostor to use different fingers instead of the same. Thus the FAR should increase by a factor of nearly 2:
 
FAR2(th) = 1 - (1 - FAR1(th))2
FRR2(th) = (FRR1(th))2

Two equal fingers AND

Real-life approximation

The minimum of two consecutive score values has to exceed the threshold. This is equivalent to the requirement that the score value of the first fingerprint AND the score value of the second fingerprint exceeds the threshold. This calculation is done directly on the score series from the ID Mouse 4.0 test. The calculation of the FAR/FRR curves is done straight forward.

Mathematical approximation

The FAR and FRR curves for 2 fingers are calculated directly from the curves for 1 finger assuming that all fingers have the same FAR and FRR, respectively. Using the same finger should not greatly increase the chance for false acceptance. This is also the best strategy for an impostor. For that reason it is supposed that the FAR does not change:
 
FAR2(th) =  FAR1(th)
FRR2(th) = 1 - (1 - FRR1(th))2

Two equal fingers MEAN

Real-life approximation

The mean value of two consecutive score values has to exceed the threshold. This calculation is done directly on the score series from the ID Mouse Field Test. The calculation of the FAR/FRR curves is done straight forward.

Mathematical approximation

The FAR and FRR curves for 2 fingers are calculated directly from the curves for 1 finger assuming that all fingers have the same FAR and FRR, respectively. The calculation is based on the fact that the probability density function of two added random variables is equal to the convolution of the probability density functions of each variable. 

Using the same finger should not greatly increase the chance for false acceptance. For that reason it is supposed that the FAR does not change.

Results

Basis of the following evaluations is the (internal) performance evaluation of the Siemens ID Mouse 4.0 with MCM (Minutia Correlation Matcher) with quality rejection switched on or off. This performance evaluation is based on the genuine fingerprint collection of the (internal) ID Mouse Field Test.
Note: The following two diagrams have been calculated with different parameters of the algorithm. As a result, a direct comparison between the diagrams is not possible. Only a comparison within one diagram makes sense!

Real-life approximation

  • Only OR and MEAN cases have been investigated
  • The investigation is restricted to equal fingers
  • MEAN (blue curves) and OR (green curves) combinations of equal fingers improve the performance of the system for the real-life approximation scenario (solid lines)
  • An OR (green curves) combination seems to deliver slightly better results than the MEAN (blue curves) combination in the real-life approximation scenario 
  • The real-life approximation method (solid lines) delivers significantly less improvements than the mathematical approximation method ("est.", dashed lines)

Mathematical approximation

  • Note that equal and different fingers yield the same result as shown above!
  • An AND combination (red curves) degrades the performance of this biometric system in any case
  • An OR combination (green curves) improves the performance of the system in any case
  • There is no significant difference between OR and MEAN (blue curves) combinations
  • As to be expected, the FMR/FNMR behavior is slightly better if the quality rejection is switched on (default, off: dashed lines with label woQR). The FAR/FRR behavior will mainly be determined by the quality rejection which is 3% for the 1-finger case.