Impact of trivial quantisation on discrimination power in biometric systems Cai Li and Jiankun Hu✉ Trivial quantisation is widely used in cancellable biometrics and biocryptosystems for error tolerance. This method segments the feature domain into several non-overlapping intervals of equal size, whereas all features in the same interval will be considered matched. Although it is intuitive that trivial quantisation brings about the boundary issue, similar features near interval boundaries may be mapped into different intervals and lead to matching failure. Previous works do not provide specific theoretical analysis on how trivial quantisation impacts discrimination power (DP), an important performance index in biometric systems. Assuming genuine features and imposter features follow Gaussian distributions, the DP provided by the conventional matching method and trivial quantisation-based matching method from a theoretical perspective is discussed and compared. Also, formulas are developed for calculating the error tolerance parameter that maximises the DP in each method and the discrimination loss caused by trivial quantisation.
Introduction: Compared with traditional authentication techniques such as passwords and token cards, biometric-based techniques offer a nonrepudiable, more universal and reliable option for individuals’ authentication. As biometric features tend to suffer from intra-class variability and inter-class similarity, a well-designed matching algorithm should provide high discrimination power (DP) between true matches and false matches, i.e. a genuine acceptance rate (GAR)–false acceptance rate (FAR) difference. In a conventional matching method, a query feature fq ∈ F will be regarded genuine if and only if dis( fq, ft) ≤ t, where ft ∈ F is a template feature, t is a predefined threshold for error tolerance and dis is a distance function/similarity measure defined in F. However, this well-established similarity measure is not directly applicable to some particular applications that do not store raw features explicitly, such as cancellable biometrics or bio-cryptosystems. In fact, many of them [1–7] adopt trivial quantisation to deal with biometric noise – the feature domain F is segmented into several non-overlapping intervals of equal size s while all features in the same interval will be considered matched. Although it is well known that similar features near interval boundaries may be mapped into different intervals and lead to matching failure, to the best of our knowledge, previous works do not provide specific theoretic analysis on how trivial quantisation impacts DP in biometric systems. In this Letter, we discuss and compare the DPs provided by the conventional matching method and trivial quantisation-based matching method from a theoretical perspective.
whereas Di can be written as Di = N ( fu , s2i ), where fu is the mean of the distribution Di, and σg and σi are standard deviations. In the case of the conventional matching method, for a template feature ft ∈ F, a query feature fq ∈ F satisfying dis( fq, ft) ≤ t is considered to be a genuine feature. Therefore, the GAR can be calculated √ f +t 2 2 by GARCft = ftt −t (e−(x− ft ) /2sg /sg 2p) dx, whereas the FAR can be √ f +t 2 2 calculated by FARCft = ftt −t (e−(x− fu ) /2si /si 2p) dx (see Fig. 1). Here superscript C means conventional. As ft is distributed over F as a global feature, the expectation of GAR and FAR can then be computed by (1) and (2) as follows: √ 2 2 GARC = e−(y− fu ) /2si /si 2p GARCy dy F
√ y+t −(x−y)2 /2s2 √ 2 2 g /s = e−(y− fu ) /2si /si 2p e g 2p dxdy F
y−t
√ 2 2 e−(y− fu ) /2si /si 2p F(t/sg ) − F( − t sg /) dy = F
√ 2 2 e−(y− fu ) /2si /si 2p FARCy dy) FARC =
(1)
F
√ y+t −(x− f )2 /2s2 √ 2 2 u i /s = e−(y− fu ) /2si /si 2p e i 2p dxdy y−t
F
√ 2 2 e−(y− fu ) /2si /si 2p (F((y + t − fu )/si ) = F
−F((y − t − fu )/si )) dy (2) where Φ denotes the cumulative distribution function of the standard normal distribution. When it comes to the trivial quantisation-based matching method, we assume the first interval ( fu − s/2, fu + s/2] is constructed with the centre at fu, the second interval is set to ( fu − 3s/2, fu − s/2] and the others are set successively as shown in Fig. 2. For a template feature ft ∈ F, the GAR and FAR can then be expressed by √ f +s/2−⌊( f +s/2− f )/s⌋s 2 2 GARQft = fu −s/2−⌊( fu +s/2− ft )/s⌋s (e−(x− ft ) /2sg /sg 2p) dx and FARQft = u u t fu +s/2−⌊( fu +s/2− ft )/s⌋s −(x− f )2 /2s2 √ u i /s (e i 2p) dx, respectively, where Q fu −s/2−⌊( fu +s/2− ft )/s⌋s means quantisation. Correspondingly, the expectation of GAR and FAR can be represented by (3) and (4) √ 2 2 GARQ = e−(y− fu ) /2si /si 2p GARQ y dy F
Conventional matching method and trivial quantisation-based matching method: The biometric identity of a user is often represented by multiple features. We assume that the features are independent. For simplicity, in this Letter, the analysis of DP is for one feature only. The extension to higher dimensions is straightforward.
√ fu +s/2−⌊( fu +s/2−y)/s⌋s 2 2 e−(y− fu ) /2si /si 2p = F fu −s/2−⌊( fu +s/2−y)/s⌋s √ 2 2 × e−(x−y) /2sg /sg 2p dxdy √ 2 2 e−(y− fu ) /2si /si 2p (F( fu + s/2 =
(3)
F
ft Dg
probability
Dg
ft –t
ft + t
− ( fu + s/2 − y)/s s − y)/sg ) − F(( fu − s/2 − ( fu + s/2 − y)/s s − y)/sg ) dy √ 2 2 FARQ = e−(y− fu ) /2si /si 2p FARQ y dy F
ft + t
ft – t Di
fuGARf
Di ft – t
ft + t
t
FARf
t
feature domain F
Fig. 1 GAR and FAR for conventional matching method
√ fu +s/2−⌊( fu +s/2−y)/s⌋s 2 2 e−(y− fu ) /2si /si 2p = F fu −s/2−⌊( fu +s/2−y)/s⌋s √ 2 2 × e−(x− fu ) /2si /si 2p dxdy √ 2 2 e−(y− fu ) /2si /si 2p (F(s/2 − ( fu + s/2 − y)/s s)/si ) = F − F(( − s/2 − ( fu + s/2 − y)/s s)/si )) dy (4)
For a template feature ft ∈ F, let Dg denote the probability distribution of genuine features and Di denote the probability distribution of imposter features/global features. We use the Gaussian distribution for both Dg and Di because it represents a common model for real-world raw data [8]. In this case, Dg can be written as Dg = N ( ft , s2g ),
Discrimination power comparison: It is well known that the DP equals the GAR–FAR difference, i.e. DP = GAR-FAR. Obviously, in both of the above methods, DP varies with the error tolerance parameters, t and s. For fair comparison, a common way is to compare the highest DP provided by these two methods.
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Subsequently, we can compute the discrimination loss (DS) due to trivial quantisation by DS = max (DPC ) − max (DPQ ).
ft
Dg fu + s/2
fu – s/2
Dg probability
fu Di fu – s/2
fu + s/2
Dg
fu GARf
Di
probability
t
FARf
t
1
2
3 5
4 fu – 5s/2
fu – 3s/2
fu – s/2 fu + s/2 feature domain F
fu + 3s/2
fu + 5s/2
GARfu – FARfu
fu–sisg
2(lns i – lns g) s i2 – s g2
fu + sisg
2(lns i – lns g) s i2 – s g2
Di
Fig. 2 GAR and FAR for trivial quantisation-based matching method
As the probability density of Di reaches the peak when ft = fu, the conventional matching method provides the highest DPC when is maximised. Similarly, the trivial GARCfu − FARCfu quantisation-based method provides the highest DPQ when GARQfu − FARQfu is maximised. GARCfu − FARCfu and GARQfu − FARQfu can be expressed by (5) and (6) as follows: fu +t √ 2 2 e−(x− fu ) /2sg /sg 2p GARCfu − FARCfu = fu −t
− e−(x− fu )
2
/2s2i
√ /si 2p dx
(5)
GARQfu − FARQfu fu +s/2 √ √ 2 2 2 2 = e−(x− fu ) /2sg /sg 2p − e−(x− fu ) /2si /si 2p dx
(6)
fu −s/2
The maximum values of GARCfu − FARCfu and GARQfu − FARQfu can be found by setting the derivative of them with respect to t and s to zero, respectively, (s = 2t = 2si sg 2(ln si − ln sg )/(s2i − s2g )). This is also
reflected
GARQfu
in
Fig.
3,
in
which
GARCfu − FARCfu
and
FARQfu ,
− the cross-hatched area, reach the maximum when fu ± t(fu ± s/2) equal the x coordinates of the intersections of the two prob√ 2 2 and ability density functions: y = e−(x− fu ) /2sg /sg 2p √ 2 2 C Q −(x− fu ) /2si y=e /si 2p. In this case, the highest DP and DP can be calculated as (7) and (8) max(DPC ) = max (GARC − FARC ) √ 2 2 = max (e−(y− fu ) /2si /si 2p (GARCy − FARCy ) dy) F
√ 2 2 e−(y− fu ) /2si /si 2p (F(t/sg ) − F( − t/sg ) = F
− F((y + t − fu )/si )
+ F((y − t − fu )/si ) )dy t=si sg √ 2(ln si −ln sg )/(s2 −s2 ) i
g
(7) max(DPQ ) = max (GARQ − FARQ ) √ 2 2 Q (e−(y− fu ) /2si /si 2p (GARQ = max y − FARy ) dy) F
−(y− fu )2 /2s2i
=
(e
√ /si 2p)×(F( fu + t− ( fu + t − y)/2t 2t − y)/sg )
F
− F(( fu − t − ( fu + t − y)/2t 2t − y)/sg ) − F((t − ( fu + t − y)/2t 2t)/si )
+ F(( − t − ( fu + t − y)/2t 2t)/si ) dy t=si sg √ 2(ln si −ln sg )/(s2 −s2 ) i
g
(8)
s feature domain F
Fig. 3 GAR–FAR difference
Conclusion and future work: We have discussed and compared the DPs provided by the conventional matching method and trivial quantisationbased matching method assuming genuine features and imposter features follow Gaussian distributions. We have also developed formulas for computing the error tolerance parameter that maximises DP in each method and the DS caused by trivial quantisation. In the future, we will extend our work to higher dimensions and investigate the impact of trivial quantisation on DP in publicly available biometric databases. Acknowledgement: This work was supported by the Australian Research Council (ARC) Linkage Project LP120100595. This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0/) Submitted: 19 April 2015 E-first: 23 July 2015 doi: 10.1049/el.2015.1349 One or more of the Figures in this Letter are available in colour online. Cai Li and Jiankun Hu (School of Engineering and Information Technology, University of New South Wales, Canberra 2600, Australia) ✉ E-mail:
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ELECTRONICS LETTERS 6th August 2015 Vol. 51 No. 16 pp. 1247–1249
