IRIS PERTURBATION METHODS FOR IMPROVED RECOGNITION

A Thesis

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Computer Science and Engineering

by

Joseph W.Thompson, III

Patrick J. Flynn, Director

Graduate Program in Computer Science and Engineering

Notre Dame, Indiana

March 2012

IRIS PERTURBATION METHODS FOR IMPROVED RECOGNITION

Abstract

by

Joseph W.Thompson, III

Perturbations of software provided circular segmentations are used to generate

multiple probe templates from a single image. The multiple probes are matched

against a gallery in order to improve the verification rate of the original algorithm.

Using too many of the generated probes will result in greatly increased running

times. As a result, various methods of finding and selecting perturbations to be

used are empirically examined. Using ROC analysis of the true accept rate (TAR)

at a given false accept rate of 0.001, many methods are discovered that perform

much better than the original algorithm, approximate the performance of using

every perturbation to less than 0.2%, and require between 500 to 1000 times less

time to verify a single image.

CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Description of Biometric Systems . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Verification vs. Recognition . . . . . . . . . . . . . . . . . . . 21.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Using Iris Texture for Identification . . . . . . . . . . . . . . . 41.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER 2: RELATED WORK . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Segmentation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Perturbation of Segmentation/Normalization Parameters . . . . . . . 8

2.2.1 Facial Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Iris Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER 3: BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 ROC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Significance Testing . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 Calculating the Correlation Coefficient of Two Estimates . . . 183.1.4 ROC Use in Experiments . . . . . . . . . . . . . . . . . . . . . 18

CHAPTER 4: PERTURBATION EXPERIMENTS . . . . . . . . . . . . . . 204.1 Overview of the IrisBEE Recognition System . . . . . . . . . . . . . . 204.2 Experiment Context . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Baseline Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Introduction to Perturbations . . . . . . . . . . . . . . . . . . . . . . 22

4.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

ii

4.6 Revised Experiment Flow . . . . . . . . . . . . . . . . . . . . . . . . 254.7 Determining the Best Perturbation . . . . . . . . . . . . . . . . . . . 27

4.7.1 Minimum Score Method . . . . . . . . . . . . . . . . . . . . . 274.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.7.3 Runtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.8 Segmentation Perturbation Space Search Methods . . . . . . . . . . . 304.8.1 Description of Search Method . . . . . . . . . . . . . . . . . . 314.8.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.8.3 Defining Search Methods . . . . . . . . . . . . . . . . . . . . . 334.8.4 Naming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.8.5 Runtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.8.6 Determining the Best Method . . . . . . . . . . . . . . . . . . 41

4.9 Performance Measurement . . . . . . . . . . . . . . . . . . . . . . . . 424.9.1 Quality of Approximation Metrics . . . . . . . . . . . . . . . . 424.9.2 ROC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.10 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

CHAPTER 5: CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . 575.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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FIGURES

1.1 The verification scenario for a biometric identification system . . . . . 2

1.2 The recognition system for a biometric identification system . . . . . 3

1.3 A diagram of the parts of the human eye pertaining to iris recognitionalong with image artifacts . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 A sample score histogram a) detailing the generation of a single pointon the ROC curve and the associated ROC curve b) with importantpoints marked . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 The original algorithm’s match and nonmatch histograms a) and theassociated ROC curve b). Error bars represent a 2σ (approximately95%) confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 A diagram showing the segmentation parameters that determine theboundary circles of the pupil and limbus . . . . . . . . . . . . . . . . 24

4.3 An example perturbed segmentation. The original segmentation withassociated normalized image is shown to compare to the optimumsegmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 A flowchart of the minimum score method. All possible perturbationswithin the defined space are created and matched against the gallery.Then the minimum score is selected as the true match score. . . . . . 28

4.5 The score distributions a) and resulting ROC b) for the minimumscore method. The distributions and ROC for the original algorithm.Error bars represent a 2σ (approximately 95%) confidence interval . . 29

4.6 A flowchart of the subspace search methods. The search for the bestperturbation is restricted to a prune and search of a subspace of theoriginal perturbation space. Results from this search are used to alterrefine the subspace as all parameters are searched. . . . . . . . . . . . 32

4.7 Prune and search methods on the subspaces in one a), two b), andthree c) dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

iv

4.8 The approximation quality distributions for the 6D search method a)and the 2D2D(xp, xl)(yp)(yl, rl)(rp) method b). Note that the distri-bution of the better performer, 6D, has a mean closer to 1.0 . . . . . 43

4.9 The ROC curves a) of the 180 2D1D search methods compared to theoriginal method. There is very little overlap between the error barsof the ROC for the original algorithm and any of the perturbationsearch methods indicating all 2D1D search methods are significantlybetter than the original algorithm. A magnified view centered at FAR= 0.001 is shown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.10 The ROC curves a) of the 20 3D3D search methods compared to theoriginal method. There is very little overlap between error bars of theROCs for the original algorithm and any of the perturbation searchmethods indicating all 3D3D search methods are significantly betterthan the original algorithm. A magnified view centered at FAR =0.001 is shown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.11 The ROC curves a) of the 90 2D2D search methods compared to theoriginal method. There is very little overlap between the error barsof the ROC for the original algorithm and any of the perturbationsearch methods indicating all 2D2D search methods are significantlybetter than the original algorithm. A magnified view centered at FAR= 0.001 is shown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.12 The ROC curves a) of the 6D search method compared to the orig-inal method. There is very little overlap between the error bars ofthe ROC for the original algorithm and the 6D perturbation searchmethod indicating the 6D search method is significantly better thanthe original algorithm. A magnified view centered at FAR = 0.001 isshown in b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.13 The ROC curves of the best performing search methods. The curvesare nearly identical indicating that no significant performance differ-ence exists between the best search methods. . . . . . . . . . . . . . . 55

4.14 The true match score distributions of the best performing searchmethods. Note that the methods have similar performance with re-gards to shifting of the original score distribution. . . . . . . . . . . . 56

v

TABLES

3.1 DECISION PROCESS FOR ROC ANALYSIS . . . . . . . . . . . . 11

4.1 COMPARISON OF ORIGINAL ALGORITHM TO MINIMUM SCOREMETHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 APPROXIMATION QUALITY FOR THE TEN BEST SEARCHMETHODS ACCORDING TO THE METRIC . . . . . . . . . . . . . 44

4.3 P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN2D1D METHODS AND THE MINIMUM SCORE METHOD . . . . 45

4.4 P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN3D3D METHODS AND THE MINIMUM SCORE METHOD . . . . 49

4.5 P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN2D2D METHODS AND THE MINIMUM SCORE METHOD ANDTHE 6D METHOD AND MINIMUM SCORE METHOD . . . . . . . 50

4.6 COMPARISON OF THE BEST PERFORMING SEARCH METH-ODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

vi

List of Algorithms

4.1 SearchSubspace : Prune and search method for searching a subspace . 344.2 Searching the Perturbation Space for the Best Perturbation . . . . . . 354.3 3D3D Search method . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 2D1D Search method . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 2D2D Search method . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.6 6D Search method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vii

CHAPTER 1

INTRODUCTION

The viability of using an image of the texture of the human iris to identify in-

dividuals was conceived by Leonard Flom and Aran Safir in 1987 [10]. Iris texture

is a measurable quantity of an individual, a biometric. Biometrics may come from

many sources, facial images, fingerprints, speech recordings and even videos of ca-

sual gait. To be useful, a biometric must contain information that leads to the

unique identification of the individual. Flom et al. initially proposed the presence

of identifying information in iris texture, and others later confirmed this hypothesis.

In [6], Daugman measures 249 degrees of freedom in phase information of a set of

4258 iris images, a huge amount of variability. Because of this, iris recognition has

proven to be an effective method of identifying individuals.

1.1 Description of Biometric Systems

An identification system can only identify individuals for whom it has biometric

information. This information is usually acquired in an enrollment stage in which a

subject presents a biometric to the system. The biometric is combined with identity

information and stored in a template. Later, when the system attempts to identify

the individual, new biometric information will be compared to the template in order

to determine identity.

1

Figure 1.1. The verification scenario for a biometric identification system

1.1.1 Verification vs. Recognition

Biometric identification systems operate in one of two scenarios: verification or

recognition. In the verification scenario, an individual presents the system with

a claimed identity and a biometric. For iris recognition, a picture is taken of the

subject’s eye. The system then retrieves previously enrolled information pertaining

to the claimed identity and compares it to the newly acquired sample in a one-to-one

match. If the quality of the match is high enough, the subject is confirmed to be

the claimed individual. Otherwise, the identity claim is rejected.

In a recognition scenario, the system attempts to identify individuals without the

aid of a claimed identity. For example, if all people must enter a building through

a checkpoint, each person will present a biometric to the system. The system must

then compare this measurement against all previously enrolled information from

all individuals, a one-to-many comparison. After comparisons, an identity decision

must be made. The system assigns the identity from the best match or unknown

2

Figure 1.2. The recognition system for a biometric identification system

identity to the individual. In what is referred to as a closed system, the individual is

assumed to be present in the system’s database and an identity is always assigned.

In an open system, the system may decide that the individual is not in the database

of previously enrolled people and not assign an identity.

All experiments presented hereafter are operating in the verification scenario.

1.2 Applications

Using the iris texture as a biometric feature for identification is beneficial even

without considering its intrinsic variability. The eye is an internal organ that is

externally visible; images of it are easily acquired.. By comparison, fingerprints are

located on an external organ, the skin. These are much more susceptible to damage

or modification. Also, altering the physical iris texture would be incredibly risky

and risks damaging the vision of the individual. Thus, as long as the subject is not

wearing contact lenses, the iris texture presented to the system can be considered

to be completely authentic. Combining the high risk of alteration and the body’s

inherent protection of the eye, the iris is found to change little over the course of

3

a person’s life indicating that less frequent re-enrollments would be needed in any

system.

Iris recognition is currently used for identification purposes in border crossings for

citizens of the United Kingdom with the UK Iris Recognition Immigration System

(UKIRIS) [3]. In this system, the user is able to claim identity and then verify it

with an iris image acquired at the crossing. Once identity is established, passport

information is retrieved and the user is allowed to enter the country.

India is currently building the largest biometric database on Earth. The Unique

Identification Authority of India (UIDAI) is issuing a unique identification number

to every citizen that requests one [2]. This number is linked to the citizen through the

use of both face and iris recognition. Not only will this build the largest database of

iris images to date, but it will also continue to generate data as verification of identity

through biometrics will be necessary to receive any benefits from the government.

1.2.1 Using Iris Texture for Identification

The majority of iris recognition systems are composed of three separate phases.

• In the first step, segmentation, the parts of the image containing the iris, theannular region between the pupil and the sclera, are found and demarcated.This entails finding the limbus and pupil boundary along with masking anyocclusion that may be present due to eyelids, eyelashes, or specular highlights.These parts of the eye are shown in Figure 1.3.

• Once the iris is found, it is usually normalized into a rectangular image as thisis a more convenient shape for application of different filters. After the imageis normalized, features are extracted and a template representing the image isgenerated. The transformation from the annulus to rectangle, along with thefeature extraction, is referred to as the normalization step.

• Finally, matching attempts to classify pairs of images into one of two classes,match and nonmatch.

As the normalization and matching phases of a recognition system rely upon

output from the segmentation step, a good segmentation algorithm is a critical

component of the overall system. If there is error or failure at this step, it is

4

Figure 1.3. A diagram of the parts of the human eye pertaining to iris recognitionalong with image artifacts

5

propagated throughout the rest of the system and performance degrades. If the

segmentation phase of a system is replaced with a better one, overall performance

of the recognition system would be expected to improve.

1.3 Hypothesis

Because the generated iris template is determined by the segmentation, different

(even minutely different) segmentations for a single image will result in the creation

of two different normalized images. While being very similar, the templates created

from these images will not be identical matches because each contains slightly differ-

ent information. As a result, it is possible to create multiple templates representing

different information from a single image. If each of the segmentations used to create

the templates appears to approximate the boundaries of the iris equally well, then

it is not obvious which of the templates best represents the image. Because multiple

templates generated from a single image may be considered accurate representations

of the iris, it is not clear which should be used for matching. This work proposes

that using multiple different segmentations to generate multiple probe templates

from a single image can benefit the matching algorithm. These probes are then

matched against a pre-enrolled gallery and the resulting match scores may be used

to improve verification performance.

1.4 Outline

Related work is presented in Chapter 2. Chapter 3 introduces receiver operating

characteristic (ROC) analysis and details statistical testing within that framework.

The experiments are explained and results are shown in Chapter 4. Finally, con-

cluding remarks are made in Chapter 5.

6

CHAPTER 2

RELATED WORK

This thesis proposes improvement of an iris recognition algorithm through the

perturbation of automated segmentations of probe images. Thus, an overview of

various segmentation methods would be beneficial to the understanding of this pro-

cess.

2.1 Segmentation Methods

For iris images acquired when the subject is level with the sensor and looking

directly at it, the boundaries of the limbus and pupil are well approximated by

circles. Daugman originally proposed an integro-differential operator (IDO) [7] to

determine the circular boundaries of the pupil and limbus. The IDO looks at all

possible circles and chooses the best boundary circle. The quality of a boundary

circle is determined by the integral of the gradient in the radial direction along

the path of the circle. The operator is run once to find the pupil boundary and a

second time to find the limbus boundary. Wildes et al. [23] proposed using a Hough

circle detector on an edge detected image to attempt to localize the iris. Masek [14]

provided a widely used implementation of this algorithm in Matlab. This work was

later extended in Liu et al. [13] in the IrisBEE recognition system. Since the Liu

system was first developed in 2005, it has been rewritten and modified for use in

these experiments.

7

If the iris images are obtained with the subject not gazing directly at the sensor,

circles fail to accurately approximate the boundaries of the pupil and limbus. In

these cases, ellipses are much better approximations of the actual boundaries. Zuo

et al. [26] determine the elliptical boundary of the pupil by first determining the

best pupil candidate boundary with a Hough circle detector and then refine the

circular boundary into an ellipse using contour fitting techniques. Smereka [20] also

presents a method of refining a circular pupil detection from a Hough transform into

an elliptical contour.

Realizing that neither circles nor ellipses perfectly represent the pupil and limbus

boundaries in any iris, researchers have developed a new method that attempts to fit

deformable contours, called active contours [4] [12], to the iris boundaries. Daugman

presents the benefits of such a segmentation in [8]. The active contour snakes model

[12], when working as intended, has the power to mark the boundaries better than

any other method. This increase in performance, however, comes at the cost of

much greater computational complexity.

2.2 Perturbation of Segmentation/Normalization Parameters

The word perturbation has two related, but slightly different, meanings in the

field of biometrics. Some work, views perturbations as a deviation from a perfect

model, in effect noise, that must be accounted for in order to improve matching

performance. In [25] Yoruk et al., the choice of distance metric used in a hand

recognition algorithm is directly influenced by the ability to deal with real world

pose perturbations.

Carrasco et al. also view perturbations as any noise that will occur later in

acquired data for face and voice recognition [5]. They attempt to artificially add

perturbations to the training data for the recognition algorithms. These perturba-

8

tions are added in the form of various noise and the recognition algorithms are run

in a standard way after this step.

The other view of perturbations is that of a tool to correct for imperfections

in a recognition algorithm. Because there are many points in a biometric system

where imperfections may be introduced, image acquisition, preprocessing, or feature

extraction for example, perturbations of probe data can be used to generate multiple

probes for analysis in a recognition system. This has the effect of moving the problem

of initially accounting for every kind of expected perturbation in the development

phase of a recognition system to problem of how to generate probe templates. This

has been shown to be successful in both the face and iris recognition fields.

2.2.1 Facial Domain

Work was initially done in the facial domain to attempt to determine impor-

tance of eye localization in face recognition algorithms. By perturbing correct eye

locations, Riopka et al. [17], noted that performance of all types of features were

negatively impacted if there were any errors in the localization of the eyes. This is

because the location of the eyes determines the angle of the head and has informa-

tion about the scale of the face. In most algorithms, the face image is normalized to

a predetermined orientation and scale using this information. Much like the local-

ization of the pupil and limbus boundaries, if the location of the eyes are incorrect,

the rest of the algorithm is likely to fail.

This work was continued in Min et al. [15] where eye locations returned by an

automatic localizer were perturbed to create multiple probe templates from a single

image. These probes were then matched to the gallery and score fusion was used to

improve performance with regard to the original localizations.

Riopka et al. also realized the performance enhancement potential in using

9

multiple probes generated by perturbing the original image. In [18], they analyze the

match scores returned by matching a probe against a gallery in order to determine if

it is likely there was a normalization error. If a normalization error likely occurred,

the probe image is perturbed, and the process is repeated.

A Perturbation Space Method is used by NEC’s face recognition software [11].

Imaoka et al. claim the method develops a shape model and various illumination

models given a single input image. These models are then used to perturb the

original image into different poses and illuminations. Features are extracted from

these probes and projected onto a feature space for matching.

2.2.2 Iris Domain

In what was claimed to be the first work dealing with perturbations of iris seg-

mentations to improve recognition performance, Thompson et al. [22] used pertur-

bations of the boundaries returned by an automatic segmentation program. Using

an optimal data set with no eyelid or eyelash occlusion, a large performance increase

was realized. This indicated that perturbing the iris segmentations was a possible

option for performance improvement even in non-ideal images. However, that work

was only a proof of concept and was infeasible in practice. Because every possi-

ble segmentation was considered for each match, the execution time was about 105

times greater than a single match and therefore cannot be used in any reasonable

verification scenario.

The work presented here is an extension of the earlier work in [22]. In these

experiments, a system is proposed that drastically reduces the running time to a

reasonable amount and attempts to show that this method also improves recognition

of non-ideal iris images.

10

CHAPTER 3

BACKGROUND

3.1 ROC Analysis

Receiver operating characteristic analysis provides a method for comparing the

performance of multiple systems as they perform similar tasks. Systems suitable for

ROC analysis are alarm type systems. That is, they classify continuous data vectors

into positive (alarm) and negative (non-alarm) classes. In attempting to do so, each

system will likely misclassify some instances of the negative class as the positive

class. This is referred to as false positive (FP), or type one, error. Misclassifying

a positive instance as a negative instance is a type two error referred to as a false

negative (FN) error. In addition to the two types of error, correct alarms are referred

to as true positives (TP) and correct non-alarms are referred to as true negatives

(TN). These decisions are outlined in Table 3.1.

TABLE 3.1

DECISION PROCESS FOR ROC ANALYSIS

Classification+ -

Truth+ TP FN- FP TN

11

After a system processes and classifies input data, it will have correctly classified

some percentage of all the positive class instances presented to it. This is the true

positive rate (TPR) of the system. The percentage of negative instances classified

as positive is the false positive rate of the system (FPR). The relationship between

these two rates represents the overall performance of the system. The key to ROC

analysis is noting that the TPR of a system can be increased by making the positive

classification requirements more lenient at the cost of also increasing the FPR. The

receiver operating characteristic is the plot of the TPR against the FPR as a single

parameter that manages the tradeoff is varied. The optimal point on the curve

representing the ROC is (TPR = 1.0, FPR = 0.0) and all systems can eventually

reach the point (TPR = 1.0, FPR = 1.0) by simply classifying every encountered

instance as a positive. Similarly, the point, (TPR = 0.0, FPR = 0.0) can also be

reached by classifying every instance as a negative. The better performing a system

is, the closer its ROC will approach the (1.0, 0.0) point.

A sample ROC is shown along with the underlying distribution in Figure 3.1. In

this example, the input data vectors are iris template match comparison scores. The

comparison scores are bitwise Hamming distances between two templates where a

score of zero indicates all bits in the templates are the same. A score of one will

indicate that all bits are different. Comparing images of different eyes should result

in a score near 0.5 and comparing images of the same eye is expected to yield a

much lower score. The classifier uses a decision threshold to classify scores less than

the threshold as positive and those greater than it as negative.

Spackman [21] is credited with first using ROC curves to analyze machine learn-

ing type systems. When comparing the ROCs of two or more systems, it is simple to

determine the best performer if one curve completely dominates the others. Curve

A is said to dominate curve B if for every FPR, the TPR of A is greater than the

12

(a) Score Histogram

(b) ROC Curve

Figure 3.1. A sample score histogram a) detailing the generation of a single pointon the ROC curve and the associated ROC curve b) with important points marked

13

TPR of B. Or in other words, every point on curve A is closer than corresponding

point on curve B to the point (1.0,0.0). In most practical cases, however, the per-

formances of the systems may be so similar that the ROC curves cross at points. If

this occurs, the above method of ordering by performance cannot be used and other

metrics are needed.

Three such metrics are presented here. The area under the ROC (AUROC)

is a global measurement of how closely a given ROC approximates the theoretical

optimal ROC. As the optimum has all of its points lie on the line segment connecting

(0.0, 0.0) and (0.0, 1.0) or the line segment connecting (0.0, 1.0) and (1.0, 1.0), the

area under the curve will be 1.0. If AUROC is used as the performance metric, the

best performing ROC will have the AUROC closest to 1.0.

The equal error rate (EER) of a ROC is the point on the curve where the type

one error (FPR) is equal to the type two error (FNR). Better performing systems

will have lower equal error rates.

Practical applications of the systems being tested may require the FPR to be

fixed. In such circumstances the challenge is to maximize the TPR. For applications

such as this, it is beneficial to examine the TPR at a specified FPR (T@F) for each

of the ROC curves. The best performer will be the one with the greatest T@F.

Given a set of input data, each system being compared can generate a ROC and

the metric of choice can be calculated for each curve and compared to determine

the best performer. While, this method will correctly establish the best performer

on this input data, it does not necessarily provide any information on which system

would be expected to be the best performer in an operational context. In other

words, the performance results may just be an anomaly based on the provided

data. The problem then becomes how to approximate performance in an operational

context given a limited set of input data.

14

3.1.1 Bootstrapping

In order to estimate the true performance of a system given limited input data,

bootstrapping should be utilized [9]. Bootstrapping is the sampling of the provided

data in order to artificially create a new data set. The sampling is done with replace-

ment and with regard to the original distribution of the data. For example, if 10% of

the input data consists of positive instances, then 10% of each bootstrapped sample

will be positive examples. Once the sampling is completed, each bootstrapped sam-

ple is input to the system and a ROC is generated. If B bootstraps are performed,

B ROC curves will be generated.

The bootstrapping method requires B times more execution time than a single

sample to determine a distribution of the sought performance metric. How many

bootstraps should be performed? More bootstraps imply more accurate estimates.

However, execution time constraints limit the number that can be performed. Wu

et al. [24] have noted that little gain is made with regard to the quality of the

metric estimate after the number of bootstraps reaches a certain limit. This limit

is determined by examining the coefficient of variance (CV) across L samples of

the standard deviation of the distribution of the sought performance metric across

B bootstraps. A target CV of 0.02 is proposed in [24]. In order to calculate the

CV, a distribution of the standard deviations must be created requiring L inde-

pendent executions of the B bootstraps. It is important to note that the number

of bootstraps is dependent upon both the input data and the system being tested.

More accurate systems will require fewer bootstraps than a less accurate system.

Performing too many bootstraps is not a concern because additional bootstraps will

increase the quality of the estimate of the performance metric. With a sufficient

number of bootstraps completed, the performance of the system is well estimated

by the average performance of all of the bootstraps.

15

3.1.2 Significance Testing

As the estimated performance of the system is determined by taking the average

of a distribution of performance measurements provided by the bootstraps, each of

these estimates also has an uncertainty associated with it. This uncertainty can be

visualized a number of ways and is depicted on ROC plots through the use of error

bars.

If no assumption is made about the metric’s distribution, confidence intervals

can be derived from the histogram of the data [9]. In practice, 95% confidence

intervals are often used because they allow for simple determination of statistically

significant differences between two estimates. If the 95% confidence intervals of two

estimates do not overlap, the two estimates can safely be said to be significantly

different for essentially every significance level.

Another method of viewing uncertainty arises when the metric’s distribution

is known to be normal or assumed to be so by a test of normality such as the

Shapiro-Wilk test [19]. If this is the case, the mean and standard deviation of the

distribution provides all of the information needed for confidence interval estima-

tion. For example, the 95% confidence interval on an estimate θ̂ whose underlying

distribution is normal is simply [θ̂−2σθ, θ̂+2σθ], where σθ is the standard deviation

of distribution.

As stated above, statistical significance of the difference between two estimates

is easily determined if the the 95% confidence intervals do not overlap. But what if

they do overlap? At this point, a statistical test is needed to determine significance.

Further, although, a binary decision on significance can be determined if the con-

fidence intervals do not overlap, no information on the strength of the decision is

known. Using a statistical test allows for the calculation of a p-value related to the

certainty of the decision. The knowledge of normality of the estimate greatly eases

16

the complexity of this test.

The following scenario is presented: the performances of two systems are being

compared. Each of the estimated metrics has an underlying normal distribution

whose means, θ1 and θ2, and standard deviations, σ1 and σ2 are known. The null

hypothesis proposes that two estimates are equal, and the alternative is that they

are not equal:

H0 : θ1 = θ2

H1 : θ1 6= θ2

(3.1)

The above is two-tailed test of the means of two normal distributions with known

standard deviations. This is easily accomplished through the use of a Z-test. The

Z-test computes a statistic with an expected standard normal distribution and com-

pares the value with the percentiles of the standard normal distribution in order to

determine significance. For uncorrelated estimates, the Z-score is computed as fol-

lows:

Z =θ̂1 − θ̂2√σ̂21 + σ̂2

2

(3.2)

However, in practice, because both systems are attempting to accurately classify

the same input data, a positive correlation most likely exists between the perfor-

mance metrics. The calculation of the Z-score taking the positive correlation into

account is [24]:

Z =θ̂1 − θ̂2√

σ̂21 + σ̂2

2 − 2r̂θ1θ2σ̂1σ̂2(3.3)

where rθ1θ2 is the sample correlation of the sample means.

If the correlation is not taken into account, the resulting Z-score will be lower

than it should be. This, in turn, will result in acceptance of the null hypothesis

in cases where it should actually be rejected (type one error). The correlation

coefficient itself is non-trivial to compute due to bootstrapping.

17

3.1.3 Calculating the Correlation Coefficient of Two Estimates

Because bootstrapping is used to calculate the distribution of the performance

metric for a system, a slightly different distribution would be found if the test was

run again. Extending this concept to testing two systems, the performance of system

A on one bootstrap of the original data cannot be expected to be correlated with the

performance of system B on a different bootstrap of the data because the input to

the systems has changed. To solve this problem, both systems must be provided the

same input data. This is accomplished through synchronized bootstrapping [24].

Synchronized bootstrapping is a process of selecting the same input data for each

system being tested. The same number of bootstraps are carried out resulting in

distributions of the performance metric for each system that contain the correlated

performance if any exists. After synchronized bootstrapping to determine the esti-

mates of the performance metric, θ̂1 and θ̂2, with standard deviations, σ1 and σ2,

the correlation coefficient can be determined as follows:

rθ1θ2 =

∑ni=1(θ1i − θ̂1)(θ2i − θ̂2)

(n− 1)σ1σ2(3.4)

3.1.4 ROC Use in Experiments

In the context of the experiments presented later, the systems being tested are

iris recognition algorithms. For a given comparison between iris images A and B, a

similarity (match) score is produced. The classification into positive (called accept

or match in this context) and negative (reject or nonmatch in this context) classes

is the result of a simple threshold about a set score. If the provided match score

is less than the threshold, the pair of images are considered a match. If the score

is greater than or equal to the threshold, the pair is a nonmatch. To generate the

points on the ROC curve, the threshold is varied uniformly across a range of values.

Performance is measured using the true accept rate at a false accept rate of 0.001.

18

For a given ROC, 2000 bootstraps performed. Through the previously discussed

procedures above, it was determined that 2000 bootstraps were enough to lower the

CV of the standard error of the statistic of interest to about 1.5% for the best and

worst algorithms.

19

CHAPTER 4

PERTURBATION EXPERIMENTS

4.1 Overview of the IrisBEE Recognition System

The following experiments utilize a Daugman-like [7] recognition system that is

a reimplementation of a system originally built by Masek [14]. The system, IrisBEE

[13], was originally developed by Xiaomei Liu in 2005 and has been rewritten and

extensively modified.

The system first segments the provided iris image by finding the pupil boundary,

assumed to be a circle, in an edge map using a Hough circle transform. After finding

the pupil, another Hough transform is used to locate the circle representing the

limbic boundary. For each candidate circle in both steps, contrast tests are used to

determine which are the best approximations of the actual boundaries. A two-piece

linear eyelid detector then attempts to mask out eyelids. It does this by fitting

two lines to each of the upper and lower eyelids. If these lines intersect the limbus

boundary, everything above the upper eyelid and below the lower eyelid is masked.

Finally, specular highlights are masked.

With the segmentation determined, the image is normalized by transforming

the (x, y) coordinates into the polar-like doubly dimensionless coordinate system

originally proposed by Daugman [7]. This creates a rectangular image containing

only the iris data that is well suited to filtering and is referred to as the normalized

image. A segmented image along with the associated normalized image can be

20

seen in Figure 4.3a). Each row of the normalized image is convolved with a one-

dimensional log Gabor filter. The complex output of this filtering step is then binned

into one of the four bins based upon the phase information. This has the effect of

encoding the normalized image into a binary image with two bits per pixel.

The matcher in this system calculates a bitwise Hamming distance between two

encoded images. The Hamming distance is a similarity score in the range of [0, 1]

with lower scores indicating higher similarity.

4.2 Experiment Context

A probe to gallery matching experiment is conducted. The gallery is built by

using the oldest image available for each of a subject’s left and right eyes. This is

done to simulate an enrollment with subsequent images taken as probes at some later

time. The provided probes are compared only to gallery images on the same side

of the face. This not only saves computation time, but also represents a screening

step where metadata about the image can be used to ignore obvious nonmatches.

A verification scenario is used as the context for the matching experiment. As such,

performance will be analyzed through ROC analysis rather than cumulative match

curve analysis.

4.3 Data Set

The ICE2005 [16] data set is used in these experiments with certain images

removed if segmentation errors were present and subjects removed if too few images

were present after the segmentation errors were removed. The images were all

captured on a LG 2200 iris camera. The camera uses near infrared illumination to

capture a 640x480 image of the iris. The image is contrast stretched to the range

[0, 255]. All of the images are of frontally presented irises. 2678 images of 175 unique

21

irises from 98 unique subjects are present in the data set.

4.4 Baseline Results

With the probe sets and gallery sets defined, a fully automated baseline experi-

ment is performed. The images are taken as provided and are first segmented using

the previously discussed algorithm. Each probe is then compared to each gallery

image. The resulting distributions of the match and nonmatch scores are used to

compute the ROC curves for analysis in Figure 4.1.

4.5 Introduction to Perturbations

4.5.1 Definition

In the initial step of the matching algorithm, the software first segments the

iris region of the eye image. It does this by fitting a circle to the pupil boundary

and another circle to the boundary of the limbus. Each of these circles is defined

by three parameters : two values to define the pixel location of the center, (x, y),

and a third value, r, to define the radius. Thus a segmentation of an iris is defined

by six parameters, (xp, yp, rp) for the pupil boundary and (xl, yl, rl) for the limbus

boundary as shown in Figure 4.2. A perturbation of this segmentation is an alter-

ation of one of these values by some amount, δ. By altering any combination of

these values, a different normalized image can be obtained to use in the next step

of the matching algorithm. For all parameters, only perturbations of up to three

pixels are used. With this constraint, given an initial segmentation, each of the six

parameters may be perturbed by seven different values, {−3,−2,−1, 0, 1, 2, 3}, for a

total of 76 or 117649 possible perturbations. These 117649 perturbations create the

perturbation space for a given segmentation. The coordinates of the space are the

values of each of the perturbation parameters (δxp , δyp , δrp , δxl , δyl , δrl). The value,

P (dxp , dyp , drp , dxl , dyl , drl) where each d ∈ {−3,−2,−1, 0, 1, 2, 3}, of each location

22

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

0.5

Original ScoreDistribution

Match Score Bins

%inbin Match

Nonmatch

(a) Score Histogram.

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

Baseline ROC

False Accept Rate

True

Acc

ept R

ate

Original

(b) ROC Curve.

Figure 4.1. The original algorithm’s match and nonmatch histograms a) and theassociated ROC curve b). Error bars represent a 2σ (approximately 95%) confidenceinterval

23

Figure 4.2. A diagram showing the segmentation parameters that determine theboundary circles of the pupil and limbus

in the space is the true match score for the template created by that perturbation.

4.5.2 Motivation

To begin, an assumption must be made that the software provides an adequate

segmentation. That is, the circular boundaries provided by the segmentation algo-

rithm are good approximations to the actual pupil and limbus boundaries. Small

perturbations of the provided segmentation can then be used to attempt to find a

”better” segmentation. If the segmentation provided by the software is poor, then

any small alteration made to this segmentation will also be a poor segmentation.

24

What does it mean to find a ”better” segmentation? When the word better

is used in this context, it is not referring to the quality of the fit of the circular

boundaries to the actual boundaries of the iris. Instead, a better segmentation is

defined as a segmentation that creates a template that better matches the associated

gallery image. The gallery image was pre-enrolled with the default segmentation

provided by the software.

For example, consider the desired comparison between probe image A and true

match gallery image B. Even if the actual pupil and limbus boundaries in image A

were perfect circles and the segmentation algorithm found these exact circles, this

may not be the case in image B. In the gallery image B, the pupil boundary may

be slightly elliptical. Because of this, fitting any circle to this boundary may cut

through the pupil on one side and include a small amount of the iris texture on

the other side. Because the gallery image was enrolled with the software provided

segmentation, this segmentation is set and cannot be adjusted. Thus, even though

the segmentation for image A is perfect, it may be beneficial to consider a segmen-

tation where the pupil boundary intersects the pupil on one side and contains some

of the iris texture on the other side. If this segmentation has a lower match score

to the true gallery image, this segmentation is considered better than the software

provided segmentation. An example of such a comparison is shown in Figure 4.3.

In essence, perturbations can allow a provided probe image to better match the

corresponding gallery image even if the gallery image was enrolled with a slightly

flawed segmentation.

4.6 Revised Experiment Flow

The original experiment is expanded to use segmentation perturbations. Given

the defined gallery set and a probe image, the probe image is perturbed multiple

25

(a) Original segmentation with the normalized image.

(b) Optimum segmentation with the normalized image.

Figure 4.3. An example perturbed segmentation. The original segmentation withassociated normalized image is shown to compare to the optimum segmentation

26

times to yield new segmentations. Each of these generated segmentations is then

normalized and a template is generated. These templates are then treated as indi-

vidual probes and independently matched against every gallery image. This system

is illustrated in Figure 4.4.

4.7 Determining the Best Perturbation

4.7.1 Minimum Score Method

When presented with a probe image, every possible perturbation within the

already defined space is generated. Each of perturbed segmentations is then nor-

malized and filtered resulting in a total of 117649 perturbed probe templates for each

original template. Each probe template is then matched to the each of the gallery

images. Because ROC analysis is used here, the true match scores are extracted

and the perturbation with the lowest true match score is used as the representative

perturbation for score distribution analysis. Because all possible perturbations are

used and the minimum true match score from the set is used, this method represents

the optimum match score distribution which the perturbation method can generate.

4.7.2 Results

The minimum score method outperforms the original recognition algorithm. The

results are presented in Table 4.1 and Figure 4.5. With a p-value of 0, the minimum

score method is statistically significantly better than the original algorithm at both

the α = 0.05 and α = 0.01 levels with a strong confidence.

4.7.3 Runtime

As previously shown, the minimum score perturbation method greatly improves

the overall matching performance. However, this improvement comes at a tremen-

dous computational cost. After the initial segmentation step, every other step must

27

Figure 4.4. A flowchart of the minimum score method. All possible perturbationswithin the defined space are created and matched against the gallery. Then theminimum score is selected as the true match score.

28

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.1

0.2

0.3

0.4

0.5

Minimum Score Method Score Distribution

Match Score Bins

%in

bin Minimum Score Method Match

Minimum Score Method MatchOriginal MatchOriginal Nonmatch

(a) Score Histogram.

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

MinimumandOriginal ROCs

False Accept Rate

True

Acc

eptR

ate

Minimum Score MethodOriginal

(b) ROC Curve.

Figure 4.5. The score distributions a) and resulting ROC b) for the minimum scoremethod. The distributions and ROC for the original algorithm. Error bars representa 2σ (approximately 95%) confidence interval

29

TABLE 4.1

COMPARISON OF ORIGINAL ALGORITHM TO MINIMUM SCORE

METHOD

Algorithm T@F p-valueOriginal 0.983493

0.0Minimum 0.994079

be repeated 117649 times due to the independent matching of each perturbation.

Although the generation of a template given a segmentation is much faster than the

segmentation step, the combined time needed for all 117649 perturbations domi-

nates the total execution time and renders this method infeasible when operated in

a single threaded execution environment. Thus, in order to allow for perturbations

to be effectively used for matching, either the entire process must be massively par-

allelized or approximations of the minimum score method must be used that utilize

fewer perturbations as independent probes. In this thesis, the parallelization of the

minimum score perturbation algorithm is not addressed and various approximations

are explored.

4.8 Segmentation Perturbation Space Search Methods

When a software provided probe segmentation and gallery segmentation are

examined, we expect that perturbing the probe segmentation in some ways will

improve the score and in other ways will degrade the score. Further, if a good probe

segmentation is found, it can be expected that the similar perturbed segmentations

will also perform well because the generated templates should not be significantly

different. This intuition yields a view of the perturbation space that contains a

number of local minima and a sought single global minimum perturbation. With

30

the space organized in this way, an intelligent search of the space should quickly

find a perturbed segmentation that performs better than the one initially provided

by the software.

4.8.1 Description of Search Method

The intelligent search method proposed is a prune and search algorithm. A set

of parameters,

∆ = {δ1, ..., δm} m ∈ {1, 2, 3, 6}

δi ∈ {δxp , δyp , δrp , δxl , δyl , δrl}

is chosen to be searched. The chosen parameters are allowed to vary while fixing

the others to constant values,

C = {c1, ..., cn} n = 6−m

ci ∈ {cxp , cyp , crp , cxl , cyl , crl}

initially all ci = 0

This creates a subspace of the original space that is then searched. For example, if

three parameters, δxp , δxl , and δrl , are allowed to vary, a three dimensional subspace

of the perturbation space is created, P (∆, C) = P (δxp , cyp , crp , δxl , cyl , δrl). The

method divides this subspace and chooses a single representative perturbation from

each region. These are then used to generate templates which are matched against

the gallery. The perturbation with the lowest match score is chosen as the best and

the search recurses on that region. This continues until the regions can no longer be

divided. Figure 4.6 illustrates this process in a flowchart and pseudocode is provided

in Procedures 4.1 and 4.2.

Certain parameters are fixed while others vary to create the subspaces. The

fixed parameters initially take on the default value of zero. However, once they are

31

Figure 4.6. A flowchart of the subspace search methods. The search for the bestperturbation is restricted to a prune and search of a subspace of the original per-turbation space. Results from this search are used to alter refine the subspace as allparameters are searched.

32

determined, the found values are used instead of the default in later searches.

4.8.2 Challenges

Because local minima may exist within the perturbation space, the order in

which the parameters are searched may lead different searches to arrive at different

perturbations. Also, parameters are allowed to be searched one at a time, two at a

time, three at a time, or six at a time. Even searching three parameters at the same

time may not yield the same perturbation as searching along two of them and then

the remaining one immediately after.

Finding a local minimum is just one of the challenges present in this search

method. Even if the search method finds a local minimum and ignores the global

minimum, a perturbation that is better than the original segmentation is still found,

but may not be the best possible perturbation. The other major issue that arises

with this method is the approach to performing the search. Is it advantageous to

perform a one-dimensional search on each parameter fixing them in a certain order,

or is it better to attempt to fix all six parameters at once by performing a six-

dimensional search? Because it is difficult to reason which search methods should

find the best performing perturbations, many different methods are tested here to

empirically determine the best one.

4.8.3 Defining Search Methods

A number of search methods are defined based on how each chooses parameters

to build and search the subspaces. Pseudocode for each method is presented.

4.8.3.1 3D3D Method

The 3D3D method uses three of the parameters to create the subspace in the

first step of the search. The result of this subspace search is used to build the

33

Procedure 4.1 SearchSubspace : Prune and search method for searching a subspace

Given: P (∆, C)1: if |∆| = 1 then2: if |P (∆, C)| ≥ 3 then3: Choose representative from each half of subspace pL, pR {Figure 4.7a}4: if pL < pR then5: Prune P (∆, C)← left half of space6: RECURSE7: else8: Prune P (∆, C)← right half of space9: RECURSE

10: end if11: else12: return minimum of two remaining perturbations13: end if14: else if |∆| = 2 then15: if |P (∆, C)| ≥ 5 then16: Choose representative from region of subspace p1, ..., p4 {Figure 4.7b}17: Select minRep← min(p1, ..., p4)18: Prune P (∆, C)← region containing minRep19: RECURSE20: else21: return minimum of four remaining perturbations22: end if23: else if |∆| = 3 then24: if |P (∆, C)| ≥ 9 then25: Choose representative from region of subspace p1, ..., p8 {Figure 4.7c}26: Select minRep← min(p1, ..., p8)27: Prune P (∆, C)← region containing minRep28: RECURSE29: else30: return minimum of eight remaining perturbations31: end if32: else if |∆| = 6 then33: if |P (∆, C)| ≥ 65 then34: Choose representative from region of subspace p1, ..., p64 {No Figure}35: Select minRep← min(p1, ..., p64)36: Prune P (∆, C)← region containing minRep37: RECURSE38: else39: return minimum of sixty-four remaining perturbations40: end if41: end if

34

Procedure 4.2 Searching the Perturbation Space for the Best Perturbation

1: {cxp , cyp , crp , cxl , cyl , crl} ← {0, 0, 0, 0, 0, 0}2: repeat3: Choose ∆ and C4: Create P (∆, C)5: foundPerturbation← SearchSubspace(P (∆, C))6: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation {set the perturbation param-

eters to found values}7: until all parameters are searched8: return {cxp , cyp , crp , cxl , cyl , crl}

(a) 1D (b) 2D (c) 3D

Figure 4.7. Prune and search methods on the subspaces in one a), two b), and threec) dimension

35

subspace for the the second three-dimensional search. The approximate optimum

perturbation is returned from the second subspace search. Pseudocode is shown in

Procedure 4.3.

Procedure 4.3 3D3D Search method1: {cxp , cyp , crp , cxl , cyl , crl} ← {0, 0, 0, 0, 0, 0}2: Choose ∆ of size 3 and C3: Create P (∆, C)4: foundPerturbation← SearchSubspace(P (∆, C))5: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation {set the perturbation parame-

ters to found values}6:

7: Choose ∆ with three remaining parameters and C8: Create P (∆, C)9: foundPerturbation← SearchSubspace(P (∆, C))

10: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation11: return {cxp , cyp , crp , cxl , cyl , crl}

The decision to break up the process to search three parameters and then three

parameters is not a random decision. It is motivated by the desire to be able to

search the space of all perturbations one boundary at a time. Using the 3D3D

search method, the subspace of pupil perturbations, P (δxp , δyp , δrp , cxl , cyl , crl), may

be searched and then the subspace of limbus perturbations, P (cxp , cyp , crp , δxl , δyl , δrl)

may be searched with the pupil parameters fixed. This is, however, just one of the

many orders of searching the parameters that may be used in this method. For

completeness, every one of the(63

)(33

)= 20 possible orderings is tested.

4.8.3.2 2D1D Method

As the 3D3D method arises from the intuition that it would be beneficial to

first determine the pupil circle parameters and then determine the limbus circle pa-

rameters, we could arrive at other methods in a similar manner. By still desiring

to find the pupil circle parameters and then those for the limbus circle boundary,

the search could first determine the optimum center of the pupil circle and then

36

determine its radius in a second step. This would entail a two-dimensional search

for the center, using ∆ = {δxp , δyp}, followed by a one-dimensional search for the

radius, ∆ = {δxr}. The same process is then repeated for the limbus circle. Pro-

cedure 4.4 provides pseudocode for this process. Because this method utilizes a

two-dimensional search followed by a one-dimensional search, it is referred to as the

2D1D method. There are(62

)(41

)(32

)(11

)= 180 possible orderings of the parameters

in this method.

Procedure 4.4 2D1D Search method1: {cxp , cyp , crp , cxl , cyl , crl} ← {0, 0, 0, 0, 0, 0}2: Choose ∆ of size 2 and C3: Create P (∆, C)4: foundPerturbation← SearchSubspace(P (∆, C))5: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation {set the perturbation parame-

ters to found values}6:

7: Choose ∆ of size 1 and C8: Create P (∆, C)9: foundPerturbation← SearchSubspace(P (∆, C))

10: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation11:

12: Choose ∆ of size 2 and C13: Create P (∆, C)14: foundPerturbation← SearchSubspace(P (∆, C))15: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation16:

17: Choose ∆ with the remaining parameter and C18: Create P (∆, C)19: foundPerturbation← SearchSubspace(P (∆, C))20: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation21: return {cxp , cyp , crp , cxl , cyl , crl}

4.8.3.3 2D2D Method

The 2D2D method first chooses two parameters to create a two-dimensional

subspace. The approximate optimum values for these parameters are then used to

build another two-dimensional subspace to be searched. The resulting parameter

37

values are then used in a final two-dimensional subspace search for the remaining

two parameters. Pseudocode is provided in Procedure 4.5.

Intuitively, this method could be used to find the optimum pupil circle center by

using ∆ = {δxp , δyp} in the first search followed by a search for the best limbus circle

center with ∆ = {δxl , δyl}. Finally the subspace containing the radii perturbations

may be jointly searched with ∆ = {δrp , δrl}. Using this method there are(62

)(42

)(22

)=

90 possible orderings.

Procedure 4.5 2D2D Search method1: {cxp , cyp , crp , cxl , cyl , crl} ← {0, 0, 0, 0, 0, 0}2: Choose ∆ of size 2 and C3: Create P (∆, C)4: foundPerturbation← SearchSubspace(P (∆, C))5: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation {set the perturbation parame-

ters to found values}6:

7: Choose ∆ of size 2 and C8: Create P (∆, C)9: foundPerturbation← SearchSubspace(P (∆, C))

10: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation11:

12: Choose ∆ with the two remaining parameters and C13: Create P (∆, C)14: foundPerturbation← SearchSubspace(P (∆, C))15: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation16: return {cxp , cyp , crp , cxl , cyl , crl}

4.8.3.4 6D Method

Finally, the six-dimensional search method, 6D, may be used to attempt to

jointly determine the best parameters in one search. In this method, ∆ =

{δxp , δyp , δrp , δxl , δyl , δrl} and C = ∅, allowing all parameters to vary. Pseudocode

is presented in Procedure 4.6. There is only one order in which to perform the

six-dimensional search.

38

Procedure 4.6 6D Search method1: {cxp , cyp , crp , cxl , cyl , crl} ← {0, 0, 0, 0, 0, 0}2: Choose ∆ of size 6 and C = ∅3: Create P (∆, C)4: foundPerturbation← SearchSubspace(P (∆, C))5: {cxp , cyp , crp , cxl , cyl , crl} ← foundPerturbation {set the perturbation parame-

ters to found values}6: return {cxp , cyp , crp , cxl , cyl , crl}

4.8.4 Naming

The search methods are named according to the order in which they search the

parameter space. Parameters that are jointly searched will be included together in

a set of parentheses. For example, the name 2D1D(yp, yl)(xp)(rp, rl)(xl) names the

2D1D search method that first does a two-dimensional search of yp and yl, followed

by a one-dimensional search of xp, then a two-dimensional search of rp and rl, and

finally a one-dimensional search of xl. The 6D method does not contain any of this

information after it as there is only one possible way to perform the search.

4.8.5 Runtimes

The runtimes for the different search methods are analyzed by counting the num-

ber of perturbations that must be considered to find the the approximate optimum

perturbation. Because the same amount of work must be done for each perturbation

used regardless of search method, this is a useful measure for comparison against

the minimum score method.

In a one-dimensional subspace search, two of the seven perturbations in this

space are compared and the better one is chosen. The size of the subspace is halved

and the procedure recurses. Two of the remaining four are compared and the corre-

sponding subregion is chosen to recurse (Figure 4.7a). This last subspace will have

two elements in it, the representative from the last step and a remaining unchecked

39

perturbation. This match score is calculated for this perturbation and compared

against the previous representative. The minimum is chosen as the approximate

optimum perturbation. A total of five perturbations were used to arrive at this

result.

A two-dimensional subspace search attempts to find the best perturbation out

of forty-nine in the subspace. The subspace is divided into four regions by splitting

each of the dimensions in half. One representative from each of these regions is used

in the first comparison. The best perturbation is chosen and the corresponding

region is chosen to recurse (Figure 4.7b). This reduces the subspace to roughly

14

of the original size of sixteen. Four representative perturbations are once again

chosen and compared. The best subregion is determined and the procedure recurses.

At this point, the size of the subspace is reduced to four. These perturbations are

compared and the minimum is chosen as the best perturbation. Because one of these

was used as a representative in the previous step, only three new perturbations must

be calculated. In total, eleven perturbations are used to arrive at the approximate

optimal perturbation.

The three-dimensional subspace contains 73 = 343 perturbations which are di-

vided into eight regions (each parameter is split in half) as illustrated by Figure

4.7c. Thus eight representative elements are needed and the best region is chosen.

This reduces the size of the subspace by approximately 18

to 64 elements. Eight

representatives are again chosen and recursion occurs on the best subregion. This

leaves eight remaining perturbations and the minimum is chosen as the approxi-

mate optimal perturbation. This process required eight perturbations at each of the

three levels of recursion. One of the perturbations in the final recursive step is a

representative used in the previous level and would not need to be recalculated in

this step. Thus, a total of twenty-three perturbations are needed.

40

The six-dimensional method searches the entire space containing 76 = 117649

perturbations at one time. Dividing the space in half along each of the six parameters

yields sixty-four regions. Choosing the best region at this level reduces the space to

4096 perturbations. Another sixty-four representative elements are chosen and the

space is reduced to 64 elements. The minimum is chosen as the best perturbation.

This requires 64+64+63 = 191 perturbations (one of the perturbations in the final

level was calculated as a previous representative).

With this information the number of perturbations for each method can be

calculated. The 3D3D method consists of two three-dimensional searches for a total

of forty-six perturbations. The 2D1D method requires thirty-six perturbations (two

two-dimensional searches and two one-dimensional searches). The 2D2D requires

thirty-three by running three two-dimensional searches. Finally, the 6D method

requires 191 perturbations. All of these methods are significantly faster than the

minimum score method.

4.8.6 Determining the Best Method

With the search approximation methods defined, every possible ordering of the

segmentation parameters for each of the methods is tested. Given a probe image,

every one of the search methods is applied yielding 20 + 180 + 90 + 1 = 291 approx-

imations of the optimum perturbation for that image in addition to the optimum

perturbation found through the minimum score method. For each of these pertur-

bations, the true match score and nonmatch score distribution are stored. This is

done for every probe image in order to generate the match and nonmatch score

distributions for an approximation. Finally, ROC analysis is used to measure the

performance of the various approximation methods.

41

4.9 Performance Measurement

4.9.1 Quality of Approximation Metrics

Each of the approximation methods yields a true match score that can be com-

pared to the optimal score provided by the minimum score method. If the match

score from the original image is used as a baseline, it is impossible for the approxima-

tion methods to make the match score distribution any worse as the perturbation

will not be used if its performance is worse than the original image. Thus, the

various methods can only improve the distribution. As a result, one of the most

important metrics for grading an approximation is how close it comes to replicating

the minimum score method.

Let the true match score returned by the minimum score method, MinScore, be

the best possible performance that can be achieved by any approximation and let the

unperturbed true match score, OriginalScore, be the worst possible performance

that can be achieved. Then, for a given probe image, the quality of an approximation

method, ApproxQuality, may be determined by

ApproxQuality = 1− ApproxScore−MinScore

OriginalScore−MinScore(4.1)

where ApproxScore is the match score of the perturbation found by the search

method. The best possible score is 1.0 and represents finding the best possible

perturbation with the search method. When computed for every probe image, the

distributions of this performance metric can give information regarding the relative

performance of the different approximations.

Some sample distributions of ApproxQuality are shown in Figure 4.8. As these

distributions all appear to come from the same family of distributions, the mean is

used to compare them where better distributions will have means closer to 1.0. The

best performers arising from this comparison are shown in Table 4.2.

42

(a)

(b)

Figure 4.8. The approximation quality distributions for the 6D search method a)and the 2D2D(xp, xl)(yp)(yl, rl)(rp) method b). Note that the distribution of thebetter performer, 6D, has a mean closer to 1.0

.

43

TABLE 4.2

APPROXIMATION QUALITY FOR THE TEN BEST SEARCH METHODS

ACCORDING TO THE METRIC

Algorithm Approximation Quality6D 0.8715584

3D3D(xp, yp, rp)(xl, yl, rl) 0.83735943D3D(xp, yp, rl)(rp, xl, yl) 0.82280883D3D(xp, yp, xl)(rp, yl, rl) 0.81556773D3D(xp, yp, yl)(rp, xl, rl) 0.81182402D2D(xp, yp)(rp, rl)(xl, yl) 0.81031972D1D(xp, yp)(rl)(rp, xl)(yl) 0.79907772D1D(xp, yp)(rp)(yl, rl)(xl) 0.79833752D1D(xp, yp)(xl)(rp, rl)(yl) 0.79827292D2D(xp, yp)(rp, yl)(xl, rl) 0.7975337

4.9.2 ROC Analysis

A comparison of the various search methods is also performed using ROC analy-

sis. ROC curves were generated for all 291 search methods and confidence intervals

of two standard deviations were calculated using 2000 bootstraps of the score dis-

tributions. The ROC curves for each method are grouped according to the search

type. Every curve in the group is plotted against the original ROC curve in order to

demonstrate the statistically significant performance improvement over the original

recognition algorithm.

The ROC curves for the 2D1D search methods are presented in Figure 4.9.

Though some overlap of the error bars is present, there is still a statistically signif-

icant improvement present with every possible search order in this group. This is

verified by closer examination of the TAR at a FAR of 0.001 (T@F) for each of the

curves. With the null hypothesis that the T@Fs are equal and the alternative being

44

TABLE 4.3

P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN 2D1D

METHODS AND THE MINIMUM SCORE METHOD

Algorithm T@F T@F Difference p-value2D1D(xp, xl)(yp)(yl, rl)(rp) 0.992841 0.001780 0.0316362D1D(xp, xl)(yp)(rp, rl)(yl) 0.992718 0.001813 0.030362D1D(xp, xl)(yl)(yp, rp)(rl) 0.992625 0.001884 0.0293662D1D(xp, xl)(yl)(yp, rl)(rp) 0.992747 0.001884 0.02892D1D(yp, xl)(xp)(rp, rl)(yl) 0.992632 0.001954 0.0283842D1D(xp, xl)(rp)(yp, rl)(yl) 0.992703 0.001835 0.0283142D1D(yp, xl)(rp)(xp, rl)(yl) 0.992706 0.001872 0.0274822D1D(xp, xl)(yp)(xp, yl)(rl) 0.992755 0.001830 0.027422D1D(xp, rp)(yp)(xl, rl)(yl) 0.992641 0.001911 0.0263582D1D(yp, yl)(xp)(rp, rl)(xl) 0.992426 0.002179 0.020376

that they are not equal, the test statistic is calculated as described in Chapter 3. A

large value of the test statistic corresponds to a small p-value which provides strong

evidence that the null hypothesis should be rejected. For every ROC curve in Figure

4.9, the calculated p-value is 0. Thus, very strong evidence exists to reject the null

hypothesis and support the statistical significance of the performance difference.

With the statistical significance of the T@F difference established, the next im-

portant question is whether or not a statistically significant difference exists between

the minimum score method and the search methods. The T@F performance differ-

ences between the minimum score method and the 2D1D search methods are much

less than between the original algorithm and the search methods. These differences,

along with the associated p-values, are shown in Table 4.3. In this case the, the p-

value is good indicator of the performance of the search method as a higher p-value

indicates that less of a difference between the algorithms exists.

These results indicate that the the null hypothesis should be rejected at a level

45

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

2D1D ROCs

False Accept Rate

True

Acc

ept R

ate

2D1DOriginal

(a) ROC Curves

0.0008 0.0009 0.0010 0.0011 0.0012

0.95

0.96

0.97

0.98

0.99

1.00

2D1D ROCs

False Accept Rate

True

Acc

ept R

ate

2D1DOriginal

(b) Magnified View

Figure 4.9. The ROC curves a) of the 180 2D1D search methods compared to theoriginal method. There is very little overlap between the error bars of the ROCfor the original algorithm and any of the perturbation search methods indicatingall 2D1D search methods are significantly better than the original algorithm. Amagnified view centered at FAR = 0.001 is shown in b).

46

α = 0.05 level of significance, but cannot be rejected at the more restrictive signifi-

cance level of α = 0.01. Looking at the p-values alone would seem to indicate that

the minimum score method performs much better than all of the 2D1D search meth-

ods. However, examining the T@F difference shows that the performance difference

is actually quite small. This is counterintuitive as the small performance difference

would imply a large p-value, not a small one. The p-values, instead, are so small due

to the correlation between the different algorithms. Because all of the algorithms

being compared are very accurate, there is a very high correlation between their

results. Thus, even a tiny performance difference will greatly decrease the p-value.

Similar results were also found with the 3D3D methods, the 2D2D methods, and

the 6D method. The ROC curves for the 3D3D methods compared to the original

algorithm are presented in Figure 4.10. As was the case with the 2D1D search meth-

ods, when compared against the original algorithm, all p-values were 0 indicating

a statistically significant difference for every ordering. When compared with the

minimum score method, low p-values are again observed. The best performing or-

derings for the 3D3D methods when compared against the minimum score method

are shown in Table 4.4.

The p-values provide enough evidence to reject the null hypothesis at a α =

0.05 level of significance. Some of the methods, however, cannot be considered

statistically significantly different at a 0.01 level. From Table 4.4 though, it can be

seen that the 3D3D search methods are slightly poorer performers than the 2D1D

methods.

Finally, ROC curves are presented for the 2D2D methods and the 6D method.

The performance of the 2D2D methods are presented in Figure 4.11 and the 6D

method is presented in Figure 4.12. Like all of the methods presented previously,

the 2D2D methods and 6D method are statistically significantly better than the

47

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

3D3D ROCs

False Accept Rate

True

Acc

ept R

ate

3DOriginal

(a) ROC Curves

0.0008 0.0009 0.0010 0.0011 0.0012

0.95

0.96

0.97

0.98

0.99

1.00

3D3D ROCs

False Accept Rate

True

Acc

ept R

ate

3D3DOriginal

(b) Magnified View

Figure 4.10. The ROC curves a) of the 20 3D3D search methods compared to theoriginal method. There is very little overlap between error bars of the ROCs for theoriginal algorithm and any of the perturbation search methods indicating all 3D3Dsearch methods are significantly better than the original algorithm. A magnifiedview centered at FAR = 0.001 is shown in b).

48

TABLE 4.4

P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN 3D3D

METHODS AND THE MINIMUM SCORE METHOD

Algorithm T@F T@F Difference p-value3D3D(yp, rp, yl)(xp, xl, rl) 0.992524 0.002005 0.024973D3D(yp, xl, rl)(xp, rl, yl) 0.992449 0.002156 0.0161963D3D(xp, yp, yl)(rp, xl, rl) 0.992370 0.002188 0.0160483D3D(rp, xl, yl)(xp, yp, yl) 0.992094 0.002477 0.0134243D3D(xp, yl, rl)(yp, rp, xl) 0.992117 0.002443 0.0130583D3D(yp, rp, rl)(xp, xl, yl) 0.992010 0.002547 0.012263D3D(xp, rp, xl)(yp, yl, rl) 0.991987 0.002577 0.011383D3D(yp, xl, yl)(xp, rp, rl) 0.991936 0.002656 0.010923D3D(xp, yp, rl)(rp, xl, yl) 0.992043 0.002559 0.009753D3D(xp, rp, yl)(yp, xl, rl) 0.991967 0.002570 0.009608

original algorithm with p-values of 0. The results of testing the methods against

the minimum score method are shown in Table 4.5.

Once again, similar results are observed where the null hypothesis can be rejected

at a 0.05 significance level but not at the 0.01 level. If the p-value is used as a relative

performance measure, then the 6D method is the top performer.

4.10 Analysis of Results

The best performer according to ROC analysis from each of the search method

groups was chosen to determine which search method is best suited for approximat-

ing the minimum score method. From the previously discussed results, it is clear

that all 4 methods presented here do a good (if not statistically significant) job of

approximating the performance of the minimum score method.

The ROC curves of each of these are shown together in Figure 4.13. The un-

derlying true match score distributions are shown in Figure 4.14 plotted against the

49

TABLE 4.5

P-VALUES CALCULATED FROM T@F DIFFERENCES BETWEEN 2D2D

METHODS AND THE MINIMUM SCORE METHOD AND THE 6D METHOD

AND MINIMUM SCORE METHOD

Algorithm T@F T@F Difference p-value6D 0.992838 0.001782 0.03185

2D2D(xp, xl)(yp, rl)(rp, yl) 0.992750 0.001856 0.0271082D2D(xp, xl)(yp, rp)(yl, rl) 0.992638 0.002002 0.0215582D2D(yp, xl)(rp, yl)(xp, yl) 0.992359 0.002172 0.0195222D2D(yp, rp)(xp, rl)(xl, yl) 0.992343 0.002181 0.0186682D2D(rp, xl)(xp, yl)(yp, rl) 0.992226 0.002348 0.0173962D2D(yp, xl)(xp, yl)(rp, rl) 0.992372 0.002196 0.0167522D2D(xp, xl)(yl, rl)(yp, rp) 0.992432 0.002141 0.0166162D2D(yp, yl)(xp, rp)(xl, rl) 0.992334 0.002190 0.0166022D2D(xp, yp)(yl, rl)(xp, xl) 0.992400 0.002201 0.0165862D2D(yp, xl)(yl, rl)(xp, rp) 0.992345 0.002241 0.015714

50

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

2D2D ROCs

False Accept Rate

True

Acc

ept R

ate

2D2DOriginal

(a) ROC Curves

0.0008 0.0009 0.0010 0.0011 0.0012

0.95

0.96

0.97

0.98

0.99

1.00

2D2D ROCs

False Accept Rate

True

Acc

ept R

ate

2D2DOriginal

(b) Magnified View

Figure 4.11. The ROC curves a) of the 90 2D2D search methods compared to theoriginal method. There is very little overlap between the error bars of the ROCfor the original algorithm and any of the perturbation search methods indicatingall 2D2D search methods are significantly better than the original algorithm. Amagnified view centered at FAR = 0.001 is shown in b).

51

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

6D ROC

False Accept Rate

True

Acc

ept R

ate

6DOriginal

(a) ROC Curves

0.0008 0.0009 0.0010 0.0011 0.0012

0.95

0.96

0.97

0.98

0.99

1.00

6D ROC

False Accept Rate

True

Acc

ept R

ate

6DOriginal

(b) Magnified View

Figure 4.12. The ROC curves a) of the 6D search method compared to the originalmethod. There is very little overlap between the error bars of the ROC for theoriginal algorithm and the 6D perturbation search method indicating the 6D searchmethod is significantly better than the original algorithm. A magnified view centeredat FAR = 0.001 is shown in b).

52

TABLE 4.6

COMPARISON OF THE BEST PERFORMING SEARCH METHODS

Method 1 Method 2 T@F 1 T@F 2 p-value

2D1D2D1D 2D2D2D0.992841 0.992750 0.909928

(xp, xl)(yp)(yl, rl)(rp) (xp, xl)(yp, rl)(rp, yl)

2D1D2D1D 3D3D0.992841 0.992524 0.765092

(xp, xl)(yp)(yl, rl)(rp) (yp, rp, yl)(xp, xl, rl)

2D1D2D1D6D 0.992841 0.992838 0.974116

(xp, xl)(yp)(yl, rl)(rp)

2D2D2D 3D3D0.992750 0.992524 0.852752

(xp, xl)(yp, rl)(rp, yl) (yp, rp, yl)(xp, xl, rl)

2D2D2D6D 0.992750 0.992838 0.90184

(xp, xl)(yp, rl)(rp, yl)

3D3D6D 0.992524 0.992838 0.654802

(yp, rp, yl)(xp, xl, rl)

match scores from the original algorithm. The ROC curves and score distributions

overlap so much, that it appears difficult to choose a best method simply through

a visual inspection.

To further analyze any differences that may be present in these methods, an all

versus all difference of TAR at a FAR of 0.001 test was performed. The results are

shown in Table 4.6.

From these tests, there were no statistically significant differences found between

the performances of these search methods. In fact, the p-values of the tests very

strongly suggest that the performances are all about the same. However, evidence

suggests that 6D is likely the best. Referring back to the approximation quality

53

metric presented in Table 4.2, this search method outperformed the others in finding

perturbations that performed nearly as well as the minimum score method.

54

0.0000 0.0005 0.0010 0.0015 0.0020

0.92

0.94

0.96

0.98

1.00

Best Search Methods

False Accept Rate

True

Acc

ept R

ate

2D2D3D2D1D6D

Figure 4.13. The ROC curves of the best performing search methods. The curves arenearly identical indicating that no significant performance difference exists betweenthe best search methods.

55

0.0 0.1 0.2 0.3 0.4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Best Search Method Distributions

Match Score Bins

% in

bin

2D2D3D2D1D6DOriginal

Figure 4.14. The true match score distributions of the best performing search meth-ods. Note that the methods have similar performance with regards to shifting ofthe original score distribution.

56

CHAPTER 5

CONCLUDING REMARKS

This thesis proposed to improve the performance of an iris recognition algorithm

in a verification scenario by perturbing the segmentation information provided by the

software and creating multiple probe templates to match against gallery templates.

Initially, a huge number, 117649, of perturbed templates were created from a single

input image. These templates were then compared against the associated gallery

template and the minimum match score of all of the generated templates was chosen

as the match score for the probe image. This resulted in a statistically significant

improvement over the original algorithm.

Statistical significance was established through testing of the true accept rate

(TAR) at a false accept rate (FAR) of 0.001 in the ROC analysis of the different

algorithms. The background for this testing was provided in Chapter 3.

This improvement, however, came at a huge computational cost, because each

perturbation results in a template that must be independently matched against the

gallery. This results in a computational increase that is linear with regard to the

number of perturbations used. As a result, the minimum score method, the method

of creating all perturbed templates and then using the minimum match score, is

currently infeasible for use in most contexts.

Alternatives to creating every perturbed template were then explored. Envision-

ing the set of all perturbations as a space to be searched led to the idea that prune

57

and search methods may be able to arrive at best performing perturbations without

needing to create all 117649 of them for each probe image. With the segmenta-

tion defined by six parameters, various search methods were proposed that explored

subspaces of different dimensions spanned by different sets of parameters.

All search methods tried found perturbations that performed statistically signif-

icantly better than the original algorithm. This was also achieved with anywhere

from about 500 to 1000 times less computational cost than the minimum score

method. Having established that these methods were indeed better than the origi-

nal method, the next question was how well they approximated the minimum score

method.

The same statistical test was used to compare the best search methods with

the minimum score method. And while a significant difference was found at level

α = 0.05, the actual performance difference between the search methods and the

minimum score method was around 0.1%, about ten times less than the difference

between the search methods and the original algorithm. Thus, it was reasoned that

the statistically significant difference between the search methods and the minimum

score method arose mostly from the algorithms being incredibly highly correlated.

That is, the algorithms agreed so much that even a minute difference could be

considered statistically significant.

One of the search methods was chosen as the best performer. The six-dimensional

(6D) search method was found to perform the best in both the ROC analysis and

in the approximation quality metric that measured how much the raw match scores

approximated those of the minimum score method. Because this method performs

much better than the original algorithm and very nearly as well as the minimum

score method while needing to create only 191 perturbations, the 6D method is con-

sidered proof that segmentation perturbations may realistically be used to improve

58

the performance of an iris recognition algorithm.

5.1 Future Work

The perturbation method for recognition improvement can be applied to any of

a number of recognition systems. The only requirement is that the segmentation

algorithm must have easily perturbed parameters. In theory, this would include

advanced contour based segmentations [8] where each point on the contour could be

perturbed, but more practically is limited to shape based segmentation methods.

Along these lines, the obvious next step is to apply this method to ellipse based

segmentations. The process would be quite similar to perturbing circle based seg-

mentations. However, the parameter space for ellipses has more dimensions thus

requiring more time to perform each match.

With the increase in time required to find the best perturbation and match, a

parallel implementation of this process becomes increasingly beneficial. To address

this, an implementation of this concept has begun using the CUDA [1] library which

allows easily programmed general purpose computing on a GPU. Once completed,

this would allow for perturbation matching to occur in nearly as little time as single

template matching.

59

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