An Introduction to Information Sets with an Application to Iris Based Authentication

This paper presents the information set which originates from a fuzzy set on applying the Hanman-Anirban entropy function to represent the uncertainty. Each element of the information set is called the information value which is a product of the information source value and its membership function value. The Hanman filter that modifies the information set is derived by using a fil-tering function. Adaptive Hanman-Anirban entropy is formulated and its properties are given. It paves the way for higher form of information sets called Hanman transforms that evaluate the information source based on the information obtained on it. Based on the information set six features, Effective Gaussian Information source value (EGI), Total Effective Gaussian Information (TEGI), Energy Feature (EF), Sigmoid Feature (SF), Hanman transform (HT) and Hanman Filter (HF) features are derived. The performance of the new features is evaluated on CASIA-IRIS-V3-Lamp database using both Inner Product Classifier (IPC) and Support Vector Machine (SVM). To tackle the problem of partially occluded eyes, majority voting method is applied on the iris strips and this enables better performance than that obtained when only a single iris strip is used.

troduced the concept of specificity as an important measure of uncertainty in a fuzzy set or possibility distributions. As we are aware any crisp set is deemed to have zero fuzziness, finding the difference between the uncertainty and the specificity [3] of a fuzzy subset containing one and only one element is one way of measuring the uncertainty. Representing the uncertainty in the fuzzy sets by the entropy functions is another way.
Most of the entropy functions were defined in the probabilistic domain as an entropy measure gives the degree of uncertainty associated with a probability distribution. The Shannon entropy function [4] defined in the probabilistic domain has the logarithmic gain function which creates problems with zero probability; so it is replaced with the exponential gain in Pal and Pal entropy function [5]. The Hanman-Anirban entropy function [6] contains polynomial exponential gain with free parameters which enable it to become a membership function.

Motivation
The motivation for this work stems from two reasons. 1) To expand the scope of information sets in [6] by defining an adaptive exponential gain function that empowers a membership function to act as an agent, and 2) To develop higher form of information sets such as Hanman Transform that helps evaluate the information source values by way of higher level uncertainty representation and Hanman filter that helps modify the information.
In our previous work [7] we have introduced the information set and also developed some features and inner product classifier (IPC) for the authentication based on ear. In the present work we embark on extending the information sets to represent higher forms of uncertainty in addition to formulating a new classifier. The original information set features were derived from the non-normalized Hanman-Anirban entropy, which is not suitable for representing higher forms of uncertainty because of its constant parameters; hence this entropy needs to be made adaptive by assuming its parameters as variables. The power of the resulting adaptive entropy is immense as it can tackle both time varying and spatially varying situations. Our main consideration is here to see the applicability and suitability of information set based features for the distinct and unique iris textures.
The paper is organized as follows: Section 2 introduces the information set and describes the extraction of features based on this set in Section 3. Segmentation of iris and use of the information set based features for iris authentication are discussed in Section 4. Inner Product classifier (IPC) is described along with the formulation of Hanman Transform classifier in Section 5. The results of application of IPC on the Iris database using the proposed features are given in Section 6 followed by the conclusions in Section 7.  (2) can't model the possibility distribution of ij I due to lack of parameters in them. Unlike probability distribution the possibilistic distribution requires a membership function which in turn needs parameters to model the distribution.

An Introduction to Information Sets
As Hanman-Anirban entropy function being information theoretic entropy function contains parameters in its exponential gain function, which we can use to convert the gain function into a membership function. The non-normalized form of this function is defined as.
This gives more spread than possible with variance. We will now consider a triangular membership function given by Similarly with another choice of parameters, 0 a = , 3) takes another form with gain function becoming Gaussian: It may be noted that in the derivation of (8) and (9) We can also derive the entropy function using the triangular membership function. Assuming 0 a = , In the context of information sets, the role of the membership is enlarged by terming it as an agent, which can be its complement, square or intuitive. The agent can take care of both spatially and time varying information source values.
Definition of Information Set: A set of information source values can be converted into an Information set by representing the uncertainty in their distribution. The basic information set consists of a set of information values with each value being the product of information source value (property/attribute) and its membership value (agent in the general case). It is denoted by Note that the membership function not only represents the distribution of information source values but also acts as an agent that helps generate different information sets such as { }

Hanman Transforms
These transforms are higher form of information sets. Note that information sets are the result of determining the uncertainty in the information source values whereas the transforms will be shown to be the result of determining the uncertainty in the information source values by the information gathered on them. The formulation of transforms is only possible if the parameters in the Hanman-Anirban entropy function are varying though they are assumed to be constant [6]. We now present the adaptive entropy function and its properties.

The Adaptive Hanman-Anirban Entropy Function
The non-normalized Hanman-Antropy function with the varying parameters is called the adaptive entropy function which is relevant to spatially varying and time varying information source values. To this end, we modify this entropy function by taking two parameters a and b as zeros and other two parameters c and d as variables. The resulting adaptive entropy function is therefore: so H is an increasing function of n.
The function is concave if the Hessian matrix is negative definite.
as ij c and ij p are in the range [0, 1].
The Hessian matrix is the second order partial derivative of square matrix having the following form: In that case, results if the parameter is also an agent by itself. We will now derive the transforms based on this concept. (13)

The Adaptive Hanman-Anirban Entropy Function as the Transform
In this, the exponential gain is made as a function of information value ij ij I µ which is already shown to be a measure of the uncertainty. The new gain function termed as an agent is a function of the information value.
Note that the information source value weighted by this new agent in Hanman transform (21) gives a better representation of the uncertainty. The division of ij µ by the maximum gray level in a window, max I is necessitated from the fact that this ratio serves as a better statistic than mere ij µ in (21). Note that if information source values are normalized already, no division is needed.
Proof: The zero order transform can be obtained if we take 0 (13)   2) Obtain the normalized information value by dividing the information value with the maximum gray level in the window.
3) Multiply the normalized information value from Step 2 with the corresponding gray level in Equation (21). 4) Repeat Steps 2 and 3 in a window and sum all the products to get a feature value. 5) Form a feature vector by repeating Steps 1 -4 on all windows of an iris strip.

Hanman Filter
Invariably the information sets derived from the fuzzy sets may not possess desirable characteristics. By modifying the information sets by certain functions or operators it is possible to get better features. The modification of the information is required to meet certain objectives like better classification or a new interpretation.
Let us see how to modify the information This is done by taking the membership function as a function of parameter s.
In type-1 fuzzy sets, the fuzzier is constant as in (24) where the parametric frequency of the cosine function is defined as We can write the r.h.s. of Equation (25)  , which is a product of the information value and the cosine function. This filter is different from Gabor filter which is the convolution of image and the product of the Gaussian and cosine functions. We have no such restriction for We can fix "s" in (25) An algorithm for the extraction of Hanman filter features is as follows: 1)  Table 1. A comparison of recognition rates due to different feature types is shown in Table 2

Divergence
If two memberships in the role of agents evaluate the same information source value, we get the divergent information. Let ij I be the set of information source values and let 1 ij µ and 2 ij µ be the two membership functions that look at ij I differently. Then the divergent information is expressed as We can use this measure in quantifying the quality of evaluation of any information source.

Random Information
By changing the membership function values randomly one can distort the distribution pattern present in the information values. If r is the random number the basic information can be turned into random by using: The corresponding random evaluation is expressed as,

Effective Information Source Value
This feature directly emerges from the definition of the basic information set.
The Effective Information source value from the k th window is computed from: Replacing ij µ with the Gaussian membership function g ij µ in (33) leads to what we term as Effective Gaussian Information (EGI):

Total Effective Gaussian Information (TEGI)
Just as the above, this feature also comes directly from the basic information.
TEGI is defined as the product of Effective Gaussian Information We can also consider e ij µ instead of g ij µ or any arbitrary function but we have adopted only g ij µ in our study.

Energy Features (EF)
From (12) It may be noted that the choice of an appropriate membership function is an important issue that is evaded here by going in for an experimentally proven function.

Sigmoid Features (SF)
where avg I is the average gray level in the k th window.
To extract features an iris strip is divided into windows of size 7 × 7 and the gray levels are normalized. The number of features is equal to the number of non overlapping windows fitted into an iris strip. The classification of features is performed using the Inner Product Classifier (IPC) in [7].

Formulation of Inner Product Classifier (IPC)
This classifier makes use of the error vectors between the training feature vectors of a user and a single test feature vector. As our objective is to get the error vector of the least disorder we generate all possible t-normed error vectors by applying t-norms on any two error vectors of a user at a time. As each normed error vector involves two training feature vectors; these are averaged to get the aggregated training feature vector. The inner product of each t-normed error vector and the corresponding aggregated training feature vector must be the least to represent a user. The infimum of all the least inner products of all users gives the identity of a user. This is the concept behind the design of IPC. Before where F t is the Frank t-norm given by:

Application to Iris Based Authentication
The above information-set based features are now implemented on iris textures to demonstrate their effectiveness in the authentication of users. Many approaches are in vogue in the literature for the iris recognition but they fail to yield good recognition rates on the partially occluded irises. As the texture is a region concept the proposed approach proceeds with the granularization of an image by varying the window size on the iris strip so as to get an appropriate texture representation. Moreover the proposed information set based approach is capable of modifying the information on the texture to facilitate easy classification. No new approach is attempted on segmentation of iris, so we have used the existing methods for segmentation. In this case study our emphasis is mainly on the texture representation and classification using the information set based features.

A Brief Review of Iris as a Biometric
Iris has been a topic of interest for person authentication ever since the pioneering works of Daugman [10] and Wildes [11]. In iris recognition, the onus is on selecting the most suitable features that enable accurate classification. As iris is endowed with a specific texture, it can be used for investigating new texture representations and classifiers.
Gabor filter has played a significant role in characterizing the iris texture by way of iris codes generated using the phase information; hence it is one of the best tools to characterize and classify textures [12]. The advantage of using Gabor filter is its ability to quantify the spatio-temporal component of texture. It may be noted that better recognition of irises can only stem out of better understanding of textures. Even after nearly 20 years of the inception of iris technology, efforts are still on finding better features and classifiers [13] [14].

Literature Survey
The original works of Daugman [10] and Wildes [11] are the harbinger for the iris based personal authentication. Daugman [10] [15] uses Gabor wavelet phase information whereas Wildes uses the Laplacian of the Gaussian filter at multiple scales as features. Some important contributions on iris recognition are now discussed.
Segmentation of iris texture region plays a pivotal role in the iris recognition.
Different approaches like morphological operations [16], thresholding using histogram curve analysis [17] are used for segmentation. Camus and Wildes [18] M.  [19] determines the accuracy of iris recognition for a partial iris image. There are a host of problems such as non-circular shape of iris and pupil and off axis images, which have prompted special consideration [20] [21]. It has been proved that better iris segmentation will help in improving the overall performance of iris recognition [22]. Many new methods on iris segmentation can be found in [23]. Gabor filter features are the most sought after so far as the texture is concerned [24]. Other feature extraction methods like Hilbert transform [25], Wavelet based filters [26] are also extensively used in the literature. About the classification algorithms, mention may be made of the correlation of phase information from windows [27], Support Vector Machine (SVM) [28] apart from simple Euclidean distance classifiers.
Practical implementation of iris based biometrics requires faster and more efficient data storage and a possible solution to this problem is suggested using FPGA [29]. Spoofing of iris from iris codes is a sure bet and to circumvent this, counterfeiting measures are developed in [30]. Factors affecting the quality of iris images captured using visible wavelength are investigated in [31]. Concerns regarding degradation of quality due to compression techniques are dispelled in [32]. The quality of iris images and its effect on the recognition rates are analysed with respect to the visible area of the iris texture region [33]. An attempt is made to enable iris recognition using directional wavelets [34]. New methodology on biometric recognition using periocular region (facial region close to the eye) rather than the texture features from the visible iris in Near Infrared (NIR) lighting conditions are discussed in [35] whereas iris recognition using the score level fusion techniques on video frames is presented in [36].

Segmentation of Iris and Generation of Strips
Segmentation forms a very important part of iris recognition as is evident from its effect on the performance improvement [22]. Though segmentation is not the main concern of this paper we will discuss the segmentation methodology briefly. The iris segmentation is done using the Hough transform based approach [37]. In this, Canny edge detector [38] is applied to get the segmented regions followed by the Hough transform that detects the boundaries of circular regions in the segmented regions. For strip generation polar to rectangular conversion is employed without recourse to the interpolation. A sample image from the database and the corresponding iris are depicted in Figure 1. The iris strips are affected by the occlusion of eyes due to eyelids and eyelashes as evident from Figure 1(b). To rectify this problem the iris strip is juxtaposed with itself and the middle portion of the resulting strip is bereft of occlusion as in Figure 2

Results and Discussion
The extracted features from each iris strip are EGI, TEGI, SF, EF, HF and HT.
The dimensions of all test strips are normalized before matching with the training strips.

The Features Used for Comparison
The performance of the above feature types is evaluated and compared with that of the conventional Gabor filter using SVM in [40]. After numerous trails the parameters of Gabor filter are set as follows: The standard deviations: σ x = 3 and σ y = 3, Phase offset: 0, Aspect ratio: 1, Orientations: θ = π/4, 2 π /4, 3 π /4 and π and Wavelengths: λ = 1, 2, 3.

Performance Evaluation of the Proposed Features
As shown in Table 3 To tackle the problem of partially occluded eyes, we will apply the majority voting on the iris strips which enables better performance than that of the individual iris strips.

Majority Voting
As noted in [41] certain regions of an iris strip like the middle region possess the discriminative texture. It may be noted that significant texture regions are present in iris at different radial distances away from the papillary boundary. This might be attributed to the fact that for some persons, the iris textures are spread over the region between the papillary boundary and the limbic boundary [41] while the majority of people have iris texture features lying closer to the papillary boundary. The aggregation of results from iris strips of different sizes enhances the overall recognition rate. In a few cases, correct classification is obtained with the small sized iris strips; hence the need for considering features from iris strips of different sizes. Based on the above observation, the iris region between the papillary boundary and limbic boundary is divided into three sizes along with full size. The number of features depends upon the window size chosen to partition an iris. In our study, the window size is taken as 7. The feature vectors corresponding to iris strips of 1/4, 1/2, 3/4 and full size are 78, 156, 234 and 273 respectively. The original iris strip size is 48 × 270. The accuracy achieved with IPC on a particular strip size is given in the 3 rd column of  As mentioned above, when the decisions from the individual feature types on strips of different sizes are combined using the majority voting method, the final decision is as shown in the last but one column of Table 4. Further enhancement in the recognition rates is obtained when the results from all iris strips are combined using the classification accuracies due to individual feature types as weights similar to ranks [43] using IPC. Then the combined recognition rate from all the feature types on all four strips attains 100% as shown in the last column of Table 4. By applying the majority voting on the matching results of four iris strips of different sizes the effect of occlusions can be minimized to a great extent.
This type of segmental approach for iris recognition is proposed in [44]. Instead of accept option that we have used in the majority voting method, the reject option can also be used to detect the possibility of erroneous classification in case we are unable to reach a consensus by the accept option.

A Comparison with the Existing Methods
We have also compared the performance of our features as in Table 4 in which the results correspond to 3/4 th size of iris with that of the existing features such as PCA, ICA [45], Local binary patterns (LBP) [46], Gabor [24] and Log Gabor [47] on the same database using k-fold validation in Table 5. The highest performance (99.35%) is obtained with HF, EF, SF and HT using IPC whereas the highest performance of 96.2% is obtained with ICA using SVML.

Verification Evaluation
At the verification level, IPC is compared with Euclidean distance classifier (EC) on the proposed features. The performance of IPC and EC is shown in terms of two separate ROCs on six features denoted by EF, HF, SF, EGI, TEGI, and HT also judged by the recognition rates.
The Euclidean distance based ROC plot in Figure 4(b) shows the maximum GAR of 93.3% at FAR of 0.1% with HF features. A maximum GAR of 99% at FAR of 0.1% is achieved with HT by IPC in ROC of Figure 4(a). The perfromance of IPC is better than that of EC as shown in Figure 4(a) and Figure 4(b).
At the verification level the proposed features are also compared with Gabor filter as it is extensively used for iris. As shown in Figure 4(a) the proposed features perform better than Gabor filter.   One ramification of this work is that we can generate a plethora of features from information sets for tackling different kinds of problems though we have chosen iris to vindicate the effectiveness of our features.