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Abstract — This paper investigates the differences

between fractal encoding and other existing image

enhancement techniques, such as bicubic interpolation, for

iris image enhancement and their performance benefits

when matching iris images from the ICE database. The

algorithm used for iris recognition is the freely available

Libor Masek Matlab code. Commercially available

software was used for fractal encoding.

Index Terms—Fractal, Bicubic Interpolation, Libor Masek.

Iris Recognition, ICE.

I. INTRODUCTION

RIS recognition has proven to be a viable and highly

successful form of identification. The complex and

personally distinct composition of the iris allows for high

discrimination between individuals. The iris is protected from

the structurally damaging of the outside environment and the

unique iris pattern remains consistent over an individual‟s

lifetime [1], [2]. Acquisition of an iris image for recognition is

also less invasive, can be performed from greater distances and

can be more readily applied to less cooperative situations in

comparison to other biometric techniques. For all of these

reasons, irises represent ideal biometric discriminators.

Iris images can be collected at resolutions other than that of

the training data. This can be attributed to such variables as

the distance between the sensor and the iris and the dilation of

the iris during acquisition. This introduces the need to

perform extrapolation for those images that are of lower

resolution than the training images. Resizing techniques such

as bilinear interpolation and bicubic interpolation have been

used in the past to match acquired images with resolution. The

scale invariance of the fractal encoding algorithm allows the

image to be resized to any resolution [2]. This paper seeks to

also compare the performance of fractal extrapolation to

bilinear and bicubic interpolation in reconstructing an iris

image that has been reduced to half its original resolution. The

image scale reduction was performed with bicubic down

sampling.

II. THEORY

Fractals are in general complex multidimensional structures

which can be described by a set of recursive equations. These

structures exhibit self-similarity, the property in which the

whole structure is approximately similar in shape to its parts

however not scale. A simple example is the Koch curve, a

portion of which is shown in Figures 1-4 with increasing levels

of recursion. This example Koch curve is generated with the

Matlab Koch function [3].

Figure 1. One iteration of the Koch curve generator function shows one

large triangle.

Figure 2. Two iterations of the Koch curve generator function can be

viewed as the original large triangle connected by smaller triangles, some of

which are rotated.

Figure 3. Three iterations of the Koch curve generator function.

Figure 4. Four iterations of the Koch curve generator function.

The Koch curve figures demonstrate that a simple function can

be used recursively to create more detailed structures. While

patterns are readily identified visually in the Koch curve,

other, higher dimensional fractals are more appropriate for

modeling real world phenomena.

I

Fractal Encoding of Low Resolution Iris Imagery

for Improved Matching

Theodore Trebaol and Professor Marios Savvides

Carnegie Mellon University

5000 Forbes Avenue

Pittsburgh, PA 15213 USA

ttrebaol@andrew.cmu.edu, marios.savvides@ri.cmu.edu

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III. FRACTAL ENHANCEMENT

Fractal-like geometry appears in natural objects such as

clouds, coast lines and irises [4], [5]. Self-similarity in nature

has been displayed classically with plants and crustaceans as

shown in Figures 5 and 6 respectively however the self-

similarity of the iris has not yet been fully utilized for

biometric identification applications. Due to the self-similar

nature of fractals, any portion of a fractal may be magnified

resulting in a new pattern with equally detailed features as the

original fractal.

Figure 5. Each frond of the fern is similar to the next.

Figure 6. Each successive chamber of the Nautilus shell is a

logarithmically scaled and rotated version of the first.

Given an image that shows fractal-like geometry, an inverse

operation can be performed that is capable of efficiently

encoding the image by calculating its fractal coefficients [6].

The typical fractal equation is given by the affine

transformation:

i

i

i

i

ii

ii

i

g

f

e

z

y

x

s

dc

ba

z

y

x

W

00

0

0

(1)

Coefficients a, b, c, and d are the coordinate scale

parameters, s is the scale parameter, e and f are the location

luminance values and g is the luminance offset parameter, i is

the index of the image section being considered [7].

To calculate the fractal coefficients for an image, the image

is first segmented into large domain and smaller non-

overlapping region blocks. The region blocks are compared to

the domain block set to check for approximate similarity. If

approximate similarity exists, the affine transform coefficients

are then calculated for generating the region block from the

larger domain block. Once the entire image is analyzed with

this method, the resultant collection of coefficients represents

the encoded version of the original image. Since these

coefficients correspond to a recursive algorithm, a fractal

encoded image can be enlarged to any resolution [8].

IV. IRIS RECOGNITION ALGORITHM

The iris recognition algorithm used was the freely available

Libor Masek Matlab code. This algorithm performs an open

source version of the Daugman IrisCode algorithm and obtains

similar performance.

The Libor Masek algorithm starts by applying the circular

Hough transform to locate the iris and the linear Hough

transform to locate the eyelids. The iris region is polar

unwrapped and the eyelid region is used create a noise mask.

The segmented iris is then encoded using a 1-D log-Gabor

filter, creating a binary iris template. The Hamming distance

is calculated from the iris templates, using the noise mask to

exclude regions associated with the eyelids. Eight shifts are

performed in either direction and the minimum Hamming

distance is determined. The Hamming distance (HD) is given

by:

N

j

jj YXORXN

HD1

)(1

(2)

where N is in the number of pixels, Xj and Yj are the value of

pixel j from images X and Y in the database [1]. We calculate

the Hamming distance between each iris image to generate a

similarity matrix.

The HD varies from 0.0 to 1.0, where 0.0 is positively

correlated and 1.0 is negatively correlated. A HD of 0.5

corresponds to an uncorrelated match. Although a HD of 0.0

is expected for highly correlated images, an experimental

value of ~0.3 is common in previous works using both the

Daugman and Libor Masek algorithms [1], [9].

V. DATA

We use the ICE-Left iris database, 1528 of the total 2953

ICE images. ICE-Left has 120 classes with between 1 and 31

images per class. When polar unwrapped, the images are

360x61 pixels with 8 bits per pixel.

VI. RESULTS

A. Visual Comparison

With the goal of providing higher quality images from low

resolution imagery, it is worthwhile to perform a visual

verification as well as perform objective experiments. An

original, polar unwrapped image from the ICE-Left database is

shown in Figure 7. This image displays the finer stoma fibers

of the iris however when this image is enlarged with bicubic

interpolation, as shown in Figure 8, those fine details blur

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together and with the background. When fractal enhanced, the

patterns exhibited in the stoma can be used to preserve the

sharp edges as shown in Figure 9. Fractal enhancement also

better preserves the pupil-iris boundary in comparison to

bicubic interpolation as can be seen in the bottom right regions

in Figure 8 and Figure 9.

Figure 7. Sample image from ICE-Left.

Figure 8. Enlarged lower right region of sample image after bicubic

interpolation.

Figure 9. Enlarged lower right region of sample image after fractal

enhancement.

Figure 10. A second sample image from the ICE-Left database with an egg-

shaped feature adjacent to the midsection.

Figure 11. Enlarged region around the egg-shaped feature of the second

sample image after bicubic interpolation.

Figure 12. Enlarged region around the egg-shaped feature of the second

sample image after fractal enhancement.

Figure 10 shows another image from the ICE-Left database

with distinct features. The region around the egg-shaped

feature is enlarged with bicubic interpolation and fractal

enhancement in Figures 10 and 11 respectively. These images

clearly demonstrate the ability of fractal enhancement to

preserve the sharp boundaries between tissues in the iris. The

original image shows two dark dots on the inside left and right

edges if the egg-shaped feature. Bicubic interpolation blurred

these portions due to the bright edges surrounding them.

Fractal enhancement not only preserved these dark spots but

also all of the edges surrounding the egg-shaped feature.

B. Objective Comparison

We first generate a similarity matrix S0 that compares the

original size (360x61 pixels) ICE-Left dataset with itself.

Each value S0(i,j) corresponds to the HD between templates

generated by the Libor Masek algorithm from images i and j

from the original dataset. We then define our low-resolution

dataset by down-sampling the original images to

approximately half the original resolution (180x30 pixels)

using bicubic interpolation. We then create four new datasets

by applying bicubic interpolation, bilinear interpolation and

three versions of fractal enhancement based on different

assumptions to up-sample low-resolution dataset back to the

original size (360x61). We do not expect bilinear interpolation

to perform well, however we evaluate it in order to

demonstrate the significance between the performance bicubic

interpolation and fractal enhancement.

We then generate five additional similarity matrices S1 – S5

that compare the enhanced datasets to the original dataset.

Each value Sn(i,j) corresponds the HD between templates

generated by the Libor Masek algorithm from images image i

from the nth

dataset with image j from the original dataset.

Since we seek to resolve issues with comparing new low

resolution data against a higher resolution database, it is more

appropriate to compare each of the five new datasets to the

original dataset rather than compare them to themselves

individually. Receiver Operating Characteristic (ROC) curves

are plotted for all six databases in Figure 13 and summarized

with Equal Error Rates in Table 1.

Figure 13. Receiver Operator Characteristic curves for each dataset.

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Dataset EER

Original ICE-Left 0.013240

Bicubic Interpolation 0.013376

Bilinear Interpolation 0.013625

Fractal Scheme 1 0.013308

Fractal Scheme 2 0.014332

Fractal Scheme 3 0.014468

Table 1. Equal Error Rate for each dataset.

VII. CONCLUSION

As can be seen in Figure 13, not all fractal enhancement

schemes are equal. We tried our method with three different

sets of initial assumptions and obtained varying results.

Unexpectedly, two sets of assumptions that governed our

fractal enhancement scheme resulted in poor results in

comparison to bicubic and bilinear interpolation. One result,

labeled „Fractal 1‟ in Figure 13 outperformed bicubic

interpolation by a reasonable margin at lower False

Acceptance Rates (FAR) however all enhancement techniques

eventually converge after FARs of 10-2

. The margin between

bicubic interpolation and bilinear interpolation in the plotted

FAR region suggest that progress has been made towards

identifying alternate image enhancement techniques beyond

bicubic interpolation for the specific application of iris

recognition. The margin of performance between the Fractal 1

method and bicubic interpolation encourages further research

into this area.

REFERENCES

[1] L. Masek, Recognition of Human Iris Patterns for biometric

Identification, Honors Thesis, University of Western Australia, 2003.

[2] R. P. Wildes, “Iris Recognition: An Emerging Biometric Technology,”

Proceedings of the IEEE, Vol. 85, Sept. 1997, pp. 1348-1363.

[3] Salman Durrani, Koch, 2000, Available:

http://www.mathworks.com/matlabcentral/fileexchange/301

[4] P. S. Lee and H. T. Ewe, “Individual recognition based on human iris

using fractal dimension approach,” ICBA 2004, pp. 467-474.

[5] M. Frame, B. Mandelbrot, and N. Neger, Fractal Geometry, May 2007,

Available: http://classes.yale.edu/fractals/

[6] X. Wu and D. J. Jackson, “Novel fractal image-encoding algorithm

based on a full-binary-tree searchless iterated function system,” Optical

Engineering. vol 40, no. 10, Oct 2005, pp. 107002-1-13.

[7] S. Ongwattanakul, X. Wu, D. J. Jackson, “A new searchless fractal

image encoding method for a real-time image compression device,”

ISCAS 2004, pp. 957-960.

[8] American Computer Science Association Inc. “Fractals and Benoit

Mandelbrot,” 2005, Available: http://acsa2000.net/frac/

[9] J. Daugman, “How Iris Recognition Works,” ICIP 2002, pp. 33-36.