Typically, vendors of biometric systems as well as the organizations which use iris recogni-tion on a large scale, desire to have a unified quality score as a measure of overall suitability for authenticating an individual. Further, recent “quality-enhanced” multi-modal biometric fusion algorithms [70,75,106] tend to use a single quality estimate for each biometric modal-ity included in the fusion framework. To generate an overall qualmodal-ity of iris images based on the estimated individual factors, we adopt an approach based on Dempster-Shafer (DS) theory [96]. This approach was proposed as a solution to a number of problems in the fields of artificial intelligence, software engineering, and pattern classification. We utilize DS the-ory because of its low complexity (compared to say Bayesian thethe-ory), explicit treatment of statistical dependence between dimensions (compared to, say, Fuzzy logic approaches) and its demonstration of good performance in many applications such as robotics [74], software fault prediction [41] and biometrics [107]. We evaluated other approaches to fuse iris quality estimates (namely, Bayesian networks and Fuzzy logic), but due to limited space, we discuss only the DS approach. It is worth mentioning that we found the DS to be the most straight-forward, in terms of low performance overhead and an intuitive description, and that in our experiments other fusion algorithms did not offer any performance advantages.
DS theory relies on concepts of beliefs, propositions, and evidence. The belief for propo-sitions (similar to events in Bayesian theory) start at 0, with uncertainty equal to 1. Based on incoming evidence, belief assignments are updated. This results in decreasing the uncer-tainty. In DS theory, belief models are built on a finite boolean algebra of mutually exclusive propositions known as the frame of discernment, denote it by Θ. The belief in a proposition
Bel(A) is a measure of certainty that evidence A is true. Shafer [96] gives the following expressions for assigning and measuring beliefs.
If Θ is a frame of discernment, then a function m : 2Θ → [0, 1] is called a basic probability assignment when:
1. m(∅) = 0 2. ∑
A⊂Θm(A) = 1.
To measure the belief of a proposition A, one must add up the belief in all subsets B belonging to A:
Bel(A) = ∑
B⊂A
m(B). (3.7)
Dempster’s rule of combination is used to combine beliefs over the same frame of discernment that are from distinct sources of evidence. This is measured by computing the orthogonal sum of all belief functions m which results in a new belief function based on the combined evidence:
where m(C) is the new belief resulting from the combination of the beliefs for evidence’s A and B. Note that Dempster’s rule assumes that the evidence are independent. The problem with this assumption in our application lies in the fact that we do not have a good understanding of the dependencies between the quality factors and to assume independence between them is unreasonable (since our evidence is from the same source). In light of this, Murphy [73,74] modified Dempster’s rule such that it is suitable to use information from the
same source as seen in equations (3.9) and (3.10):
m(C) =
∑
Ai∩ Bj=C
f (m1(Ai)m2(Bj))
1− ∑
Ai∩ Bj=∅
f (m1(Ai)m2(Bj)), (3.9)
where f (·) is given by:
f (m1(Ai)m2(Bj)) = [m1(Ai)m2(Bj)]n, n∈ [0, 1], (3.10)
in Murphy’s rule. Murphy characterizes n as a method to weight evidence. Choosing n >
0.5 gives more weight when combining new evidence, while choosing n < 0.5 gives less weight when combining new evidence [74]. Other proponents of Murphy’s rule characterize n as governing correlation between evidence [71]. It is explained in [71], that choosing n > 0.5 assumes more independence between the evidence while choosing n < 0.5 assumes correlation. In light of both views, choosing n = 0.5 is considered neutral and equal weight is applied to all evidence during integration.
3.4.1 Quality Score Normalization
Prior to fusing the factors, they need to be normalized between [0, 1] [71,73]. Occlusion, specular, and pixel counts are ratios which are already between the desired score range.
Defocus and motion blur are normalized based on a modified form of min-max normalization:
Qnew= Qold− Qmin
Qmax− Qmin
. (3.11)
With respect to defocus, Qold represents the raw power scores obtained from the bandpass kernal. In the case of motion, Qold is the raw power score corresponding to the localized peak which is extracted from directional filter that is perpendicular to the estimated angle.
Power constants Qmax and Qmin represent the maximum and minimum values for which the quality scores are normalized between. These values were learned based on a subset of images from ICE 1.0. They were not changed when evaluating other data sets. Recall that lighting variation is just the variance of the means extracted from un-occluded iris regions.
However, inaccurate occlusion estimation can cause outliers in the estimate resulting in a score greater than one. In this case, the score is simply set to one.
3.4.2 Dempster Shafer Theory Applied to Quality Assessment
We adopt a frame of discernment containing two propositions which represent opposite be-liefs:
• A - Image quality is bad (our belief that quality is bad).
• B - Image quality is good (our belief that quality is good).
The normalized values for each quality factor are assigned as beliefs to proposition A. Since these propositions represent opposite beliefs, the assigned belief to B is essentially the com-plement of the assigned belief to A. We adopt Murphy’s rule to combine beliefs with param-eter n = 0.5 for all evidences. A generalized expression for combining beliefs from k quality
factors m1 to mk is given by:
ˆ
mi(A) = (mi−1(A)· mi(A))n
((mi−1(A)· mi(A))n+ (mi−1(B)· mi(B))n, i = 2, .., k,
(3.12)
where mi(B) = 1− mi(A), since our propositions are complements of each other. Murphy has shown that different orderings result in different values for combined beliefs [74]. Since we have seven quality factors, that will result in 7! combinations. Our goal is to attain the orderings that result in the minimum and maximum values. These values are important because we can consider them as the worst case and best case quality for a specific image.
Mladenovski [71] has proved that a maximum value can be attained by first sorting the beliefs in ascending order with n = 0.5. Similarly, if sorted in descending order a minimum value can be obtained. The following section illustrates application of the fusion rule on samples from WVU, CASIA 3.0 and ICE 1.0 data sets.
3.4.3 Evidence Fusion Examples Based on Murphy’s Rule
The sample iris images in Fig. 3.13 are from WVU, ICE 1.0, CASIA 3.0 data sets. Image (a) represents a good quality image from WVU and (c) represents a good quality image from ICE 1.0 (based on visual evaluation). Images (e) and (f) represent good and poor quality images from CASIA 3.0 respectively. Images (b) and (d) represent degraded quality images which are effected by motion (b) and occlusion, defocus, and lighting (d). Image (f) scores low because the iris is flat with almost no visible texture. The estimated direction of motion blur for Fig. 3.13 (b) is 85◦ counter-clockwise with respect to the pupil center being at 0◦.
(a) WVU (b) WVU (c) ICE 1.0 (d) ICE 1.0
(e) CASIA 3.0 (f) CASIA 3.0
Figure 3.13: Sample Images from WVU, ICE 1.0, and CASIA 3.0 data sets
Table 3.2 lists the estimated factors (factors are normalized to take values between 0 and 1, with 1 implying heavy degradation) for these images and the combined quality for them.
The quality column (last column in the table) represents the lower value (minimum value attained from fusion of all factors which we loosely consider as the worst case quality) on image quality. We use this value as our global quality score.
Image Factor Defocus Motion Occlusion Specular Lighting Pixel Count Quality
(a) 0.01 0.01 0.01 0.00 0.01 0.01 0.97
(b) 0.27 0.66 0.04 0.00 0.21 0.05 0.63
(c) 0.00 0.00 0.08 0.00 0.03 0.08 0.95
(d) 0.83 0.17 0.55 0.00 0.99 0.54 0.08
(e) 0.09 0.10 0.06 0.03 0.01 0.08 0.91
(f) 0.90 0.13 0.20 0.03 0.06 0.2 0.19
Table 3.2: Estimated Factors for images in Fig. 3.13