Li S.Z., Jain A.K. (eds.) Encyclopedia of Biometrics

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encoding, decoding, and authentication (see Figure 2 ).

The ﬁrst operating module, the watermark encoder,

embeds the watermark into the host data. The water-

mark can range from a random binary bit sequence to

biometric eigen-face coefﬁcients utilized for face recog-

nition. Additionally, the encoding module may embed a

secret key that enables the system to determine the

embedding location of the watermark in the host data.

The second operating module, the watermark decoder,

takes the watermarked host data as input and processes

them to extract the watermark. If a secret embedding

key was used during the encoding module, then the

same key is required for the decoding process. Depend-

ing on the algorithm employed, the original host data

may also be used explicitly to extract the watermark.

The ﬁnal module, authentication, compares the recov-

ered watermark with the original watermark to estimate

the similarity between the two sets. If the watermark is

biometric in nature, then this biometric data can be

optionally used by the authentication system.

Watermark encoding and decoding techniques fall

into two categories: spatial and transform domain

Iris Digital Watermarking. Table 2 Characteristics of biometric watermarking systems

Characteristic Description

Imperceptibility The degree to which the host image is visibly altered or distorted due to the presence of the

watermark. Watermarked images that bear no visible difference from their original host image are

said to contain imperceptible watermarks. Rarely, this characteristic may be evaluated beyond the

scope of human perception. In these cases, a question is raised as to whether or not it is possible to

reveal that an image contains a watermark through the aid of a machine or program (without access

to the original host image)

Robustness The ability of the watermark to be detected and extracted after the watermarked image has been

subjected to any variety of transformations (i.e., compression, filters, affine transformations, etc.)

Fragility The ability to detect any file transformation by way of the watermark. The detection might result from

an inability to extract the watermark or from an extracted watermark that in not intact

Capacity The amount of information that can be embedded in the host data of a watermarking system. This is a

function of the type and size of the host data that are being watermarked and the robustness of the

watermarking system in terms of detectability and extractability

Performability The degree to which the watermark affects the performance of the biometric system(s) in question. At

a minimum, biometric watermarks should not have an adverse affect on the performance of the

biometric system(s) that they protect. Here, performance can entail matching error rates, image

quality, efficiency of computation time, etc. Some biometric watermarking schemes may positively

affect the performance of the biometric system

Iris Digital Watermarking. Figure 2 Generalized block diagram of an iris digital watermarking process. Shaded blocks

indicate the main watermarking modules, while dashed blocks/lines indicate optional areas of processing that are

algorithm specific.

780

Iris Digital Watermarking

techniques. Each category has speciﬁc advantages

and disadvantages, but in general, spatial domain tech-

niques have lower complexity and offer higher robust-

ness to biometric replacement attacks in which the

host biometric region is replaced by an imposter’s

biometric data. On the other hand, transform domain

techniques are of higher complexity, but are more

robust to geometrical attacks such as rotation, scaling,

and translation.

A number of studies have investigated different

approaches to biometric watermarking. Ratha et al.

[5] propose an algorithm for biometric watermarking

to counter replay attacks in on-line ﬁngerprint authen-

tication systems. The authors modify the least signiﬁ-

cant bit (LSB) of the indices obtained as a result of

applying wavelet scale quantization (WSQ) compres-

sion. The indices are chosen based on an embedding

key that is used as a seed for a random number gener-

ator. Similarly, Noore et al. [6], utilize the Discrete

Wavelet Transform (DWT) to watermark ﬁngerprint

images with face and demographic text data. In [7],

Low et al. watermark a nonbiometric host image with

off-line handwritten signature in the form of a discre-

tized bit string. They experiment w ith three water-

marking techniques: LSB, Code Division Multiple

Access (CDMA) spread spectrum in the spatial domain,

and CMDA spread spectrum in a transform domain

such as the DWT. Their experiments show that CMDA

in the wavelet domain provides the most robust results

with respect to jpeg compression and image quality. In

[8], a Quantization Index Modulation (QIM) water-

marking technique is utilized to encode dynamic and

static handwriting signature features into the host sig-

nature from which the features are extracted. Embed-

ding locations are chosen based on the analysis of the

signature in two transform domains: Ridglet and

Radon-DCT (Discrete Cosine Transform). The authors

conclude that although the static and dynamic fea-

tures by themselves provide modest levels of security,

fusion of both feature types improves security and

performance.

Watermark Encoding and Decoding

The encoding portion of an iris digital watermarking

scheme involves embedding the watermark in the orig-

inal host data. The decoding portion, on the other

hand,involves extracting the watermark from the host

image. Bartlow et al. [9] present a watermarking tech-

nique modiﬁed from [10], based on amplitude modu-

lation in the spatial domain to protect iris biometric

systems. In the work, iris images are watermar ked with

simulated voice feature vectors. Equations 1, 2,and3

correspond to the amplitude modulation scheme pre-

sented by Kutter in [10]. Equation 4 represents the

adaptation required to watermark iris biometric data

as presented in [9].



Encoding- Amplitude modulation encodes the bits

of the watermark bymodifying pixel intensities in

images. Pixel intensities are modiﬁed bychanging

values B

, in the blue channel of the RGB spectrum.

These modiﬁcation s occurmultiple times over the

extent of the image and are either additive orsub-

tractive, depending on the value of the bit, s,and its

proportionality to the luminance, L

, as seen in

Equation 1.

þð2s 1ÞL

2embedding

locations

s 2bit

2Luminance

q 2Encoding strength

ð1Þ



Decoding- The decoding process estimates the

pixel value in the ‘‘encoded’’image by considering

a linear combination of the pixels in a cross-shaped

neighborhood around the encoded bit as seen in

Equation 2.

k¼c

iþk;j

k¼c

i;jþk

 2B

Þ c 2 neighbor size

ð2Þ

After decoding and arriving at an estimated pixel B

value, the difference between the estimated and water-

marked pixelsis averaged over all embedding locations

for that bit. Finally, thesign of this value indi cates the

bit (if positive = 1, if negative = 0). However, to

attenuate robustness to compression, cropping, and

afﬁne transformations, an adaptive thresholding meth-

od is introduced: two bits, 0 and 1, are appended to

every bit stream as seen in Equation 3.

bit ¼ 1;d

þd

0;otherwise:



average diff of all 0 reference bits

average diff of all 1 reference bits

average diff of current bit

ð3Þ

Iris Digital Watermarking

781



Adaptation to Iris Biometric Data- Iris images for

use in biometric systems are usually captured in the

grayscale format. The encoding process has to be

modiﬁed to take this into consideration. For exam-

ple, [11] modiﬁes the encoding equation to take in

local image information such as gradient, P

, and

standard deviation, P

, of the cross-shaped neigh-

borhood to adjust the watermarking strength.

Parameters A and B aid in adjusting the strength

of the standard deviation and gradient while mod-

ulating the bits to be encoded (for all experiments

A ¼ 100, B ¼ 1,000). The following equation

represents this adaptation.

ði; jÞ¼Pði; jÞþð2s  1ÞP

ði; jÞq

1 þ

ði; jÞ



1 þ

ði; jÞ



ð4Þ

represents the average pixels in a 5  5 cross-

shaped neighborhood centered around i, j. Finally,

the reconstructed image can be calculated by replacing

the watermarked bit with the pixel value obtained from

Equation 2.

A crucial issue while encoding the watermark in the

host image relates to the degree to which it affects

the matching performance of the host biometric. The

algorithm in [9] further extends [11] by including a

parameter that indicate s the proportion of the water-

mark encoded in the iris region of the image. Figure 3

visualizes the variation of this parameter by examining

two extreme cases: the top row shows the most percep-

tible case and the bottom row shows the least percepti-

ble case tested in the work. This location-speciﬁc

encoding option may be beneﬁcial not only to iris

digital watermarking but also to biometric watermark-

ing in general.

Application Scenarios and Attacks

Depending on the intended use, iris digital water-

marking systems are vulnerable to a series of applica-

tion scenarios and attacks. Application scenarios can

be thought of as normal usage patterns that a water-

marking system should realistically be expected to

withstand without serious side effects on the perfor-

mance of any of the characteristics outlined in Table 2.

Examples of application scenarios can include, but

are not limited to database (re)compression, partial

progressive decoding, and noisy channel transmission.

Each of these scenarios can have an effect on one or

more characteristics of a biometric watermarking sys-

tem. For instance, a highly compressed watermarked

image may lead to difﬁculties in the watermark extrac-

tion process, as a compression algorithm often signiﬁ-

cantly alters an image, which in turn alters the

watermark itself. Occasionally, operat ional environ-

ments supporting slow data transmission speeds may

force a system to progressively decode portions of an

image as it becomes available. This type of application

scenario can have an effect on the robustness of the

extraction process, the performance of the biometric

system(s) in question, and potentially the impercept-

ibility of the watermark. In [9], the authors study the

robustness of the decoder after the application of these

three scenarios. Figure 4 shows the performance of the

decoder as measured by the mean percentage bit error

considering 27 different paramet er combinations of

the watermarking system (parameters are speciﬁcally

outlined in Table 3).

The graphs demonstrate the ability of the tech-

nique to tolerate the application scenarios, and in

many cases, with little or no effect on the extraction

process. Although this is just one example, many tech-

niques such as those described in [3] are capable of

tolerating scenarios similar to these. However, the

expected application environment should be consid-

ered while selecting a speciﬁc watermarking technique,

since a given technique may be well-suited for one

application scenario but ill-suited for another [3].

Perhaps the most notable difference between the

general ﬁeld of digital watermarking and biometric

watermarking is its relationship to the measure of per-

formability. For obvious reasons, a biometric water-

marking system must minimize the effect it has on the

biometric system it protects. Issues such as matching

performance, image quality, computational efﬁciency,

and even legal repercussions must not be ignored. A

biometric watermarking system should not negatively

impact on the main modules of the biometric system(s)

in question, viz., the feature extraction and matching

modules. It should be noted that this effect could po-

tentially propagate itself in two ways. Perhaps, the most

obvious effect is when the host data are used in a

biometric system; here, the presence of the watermark

may impede the feature extraction process by adding

noise to the image. Naturally this can lead to inaccura-

cies in the matching module. A less obvious effect is

782

Iris Digital Watermarking

Iris Digital Watermarking. Figure 3 Perceptability of Watermarked Images. Encodings refers to the number of times a watermark is embedded into the host image.

Iris Digital Watermarking

783

Iris Digital Watermarking. Figure 4 Robustness of the decoder in three application scenarios: (a) Varying levels of

J2K compression (measured in bits per pixel (bpp); (b) J2K compression + varying levels of zero mean white Gaussian

noise (variances shown in legend); and (c) J2K compression + varying levels of partial progressive decoding

(% decoded shown in legend).

784

Iris Digital Watermarking

when a biometric feature vector serves as a watermark

and is also used in the authentication stage. In this

scenario, accurate extraction of the watermark is of

utmost importance as small changes in the values of

feature vectors can lead to signiﬁcant changes in authen-

tication results. Although little or no work exists study-

ing the latter of the two effects, we once again refer

to [9] where the effect of an amplitude modulation

watermarking system on biometric image quality

and matching performance is studied. Figure 5 shows

the effect on image quality and matching performance

of the underlying iris biometric system. The graphs

compare the quality and matching results from the

original host images with the images resulting after

watermark extraction and image reconstruction, and

the tables provide averaged results across the entire

dataset.

Much like the three application scenarios men-

tioned earlier, these results provide one example of a

biometric watermarking system that does not produce

unwanted side effects. In particular, image quality is

seldom affected with any level of signiﬁcance and

matching rates also do not suffer from performance

degradation.

Application scenarios aside, biometric watermark-

ing systems must also deal with attacks or malicious

attempts to subvert a system. Such attacks may involve

removal, alteration, or replacement of the embedded

watermark found in an image. Although some of these

may fall in the application scenarios described earlier,

examples of attacks include rotation, scaling, transla-

tion, cropping, masking, and (re)watermarking. As in

the case of application scenarios, different biometric

watermarking techniques can handle different attacks

with varying degrees of success. Although not speciﬁc

to biometric watermarking, Zheng et al. provide an

excellent description of so-called RST (rotation, scal-

ing, translation) invariant watermarking algorithms

in [3]. Often the ability to handle a given attack lies

in the domain in which a biometric watermarking

technique operates. For instance, rotation attacks are

handled with greater ease by watermarking techniques

that operate in transform domains (i.e., Fourier, DCT,

wavelet, etc.). This t ype of attack is arguably more

difﬁcult to handle by techniques that operate in the

spatial domain. Conversely, techniques in the spatial

domain are more likely to handle biometric replace-

ment attacks (i.e., replacing the watermarked iris or

face region of an image), as the biometric ROI can be

easily located in the spatial domain, but more difﬁcult

to localize in a transform domain.

Patents, Tools, and Commercial

Products

Searching the United States Patent Ofﬁce (USPO) for

‘‘Biometric Watermarking’’ yields well over 100 entries

of varying relevance to the ﬁeld. The most notable is

titled ‘‘Biometric Watermarks’’ and was issued in 2001

to GTE Service Corporation [12]. This patent outlines

the general schematic for a biometric watermarking

system. A more recent patent issued to Canon by

the USPO relates speciﬁcally to iris digital watermark-

ing and includes an embedded watermarking system

in a camera with the intended purpose of associating

a photographers biometric information with images

taken by the camera [13]. There are a few freely

available tools related to digital watermarking and

biometric watermarking exist. For example, Stirmark

Iris Digital Watermarking. Table 3 Watermarking parameter combinations examined in [9]. The ﬁrst item of the 3-tuple

represents the degree of pixel modulation; second item represents the number of times the watermark is embedded in

the image; the third item represents the proportion of the watermark embedded in the iris portion of the host image.

Further explanations can be found in [9]

0.10-

60-

0.67 2

0.10-

60-

0.33 3

0.10-

60-

0.00 4

0.10-

40-

0.67 5

0.10-

40-

0.33 6

0.10-

40-

0.00 7

0.10-

20-

0.67 8

0.10-

20-

0.33 9

0.10-

20-

0.00

10 0.06-

60-0.67

11 0.06-

60-0.33

12 0.06-

60-0.00

13 0.06-

40-0.67

14 0.06-

40-0.33

15 0.06-

40-0.00

16 0.06-

20-0.67

17 0.06-

20-0.33

18 0.06-

20-0.00

19 0.04-

60-0.67

20 0.04-

60-0.33

21 0.04-

60-0.00

22 0.04-

40-0.67

23 0.04-

40-0.33

24 0.04-

40-0.00

25 0.04-

20-0.67

26 0.04-

20-0.33

27 0.04-

20-0.00

Iris Digital Watermarking

785

Benchmark 4.0 is a software tool designed to perform

robustness testing of image watermarking algorithms

[14]. Another tool, Checkmark, also provides a bed

of attacks to evaluate the robustness of a watermark-

ing system [15]. Many commercial entities offer a

broad range of digital watermarking solutions that

can potentially fall under the category of biometric

watermarking. Perhaps the most w idely known of

such companies is Di giMarc Corporatio n based in

Oregon, US.

Summary

Iris digital watermarking is a technique utilized in

tandem with cryptographic systems to protect iris bio-

metric images. The watermarking scheme can be used

as a mechanism for proving ﬁle authenticity, tracking

chain of custody and data reproduction, or affording

a multimodal biometric option, thereby offering an

additional layer of security after data decryption.

A w ide range of watermarking system s exist that

Iris Digital Watermarking. Figure 5 Effect of watermarking on biometric performability: (a) Compares the quality of 200

original host data images vs. the reconstructed images after watermark extraction; (b) Compares the EER across 100

users before watermarking and after image reconstruction; (c) Average image quality before watermarking and after

reconstruction; (d) Average EER (%) before watermarking and after reconstruction across 100 subjects.

786

Iris Digital Watermarking

operate in any one of several domains (spatial, Fourier,

DCT, wavelet, etc.). A system’s expected application

proﬁle and threat model will dictate the choice of

watermarking algorithm, the nature of the watermark

to utilize, and a viable set of algorithmic parameters.

Carefully making these dec isions will result in a formi-

dable layer of postdecryption protection without

compromising on the performance aspects of the un-

derlying biometric system(s).

Related Entries

▶ Binding of Biometric and User Data

▶ Biometric Encryption

▶ Biometric System Design

References

1. Tirkel, A.Z., Rankin, G.A., van Schyndel, R.M., Ho, W.J., Mee,

N.R.A., Osborne, C.F.: Electronic Watermark pp. 666–673

(1993)

2. Artz, D.: Digital steganography: hiding data within data. Internet

Computing, IEEE 5(3), 75–80 (May/Jun 2001). DOI 10.1109/

4236.935180

3. Zheng, D., Liu, Y., Zhao, J., El-Saddik, A.: A Survey of RST

Invariant Image Watermarking Algorithms. ACM Comput.

Surv. 39(2), (2007)

4. Ahmad, S., Lu, Z.M.: A joint biometrics and watermarking based

framework for ﬁngerprinting, copyright protection, proof of

ownership, and security applications. Computational Intelli-

gence and Security Workshops, 2007. CISW 2007. International

Conference on pp. 676–679 (15-19 Dec. 2007)

5. Ratha, N.K., Connell, J.H., Bolle, R.M.: Secure data hiding in

wavelet compressed ﬁngerprint images. In: MULTIMEDIA ’00:

Proceedings of the 2000 ACM workshops on Multimedia,

pp. 127–130. ACM, New York, NY, USA (2000). DOI http://

doi.acm.org/10.1145/357744.357902

6. Noore, A., Singh, R., Vatsa, M., Houck, M.M.: Enhancing secu-

rity of ﬁngerprints through contextual biometric watermarking.

Forensic Science International 169, 188–194

7. Low, C.Y., Teoh, A.B.J., Tee, C.: A preliminary study on biometric

watermarking for ofﬂine handwritten signature. Telecommuni-

cations and Malaysia International Conference on Communica-

tions, 2007. ICT-MICC 2007. IEEE International Conference on

pp. 691–696 (14-17 May 2007)

8. Maiorana, E., Campisi, P., Neri, A.: Biometric signature authen-

tication using radon transform-based watermarking techniques.

Biometrics Symposium, 2007 pp. 1–6 (11-13 Sept. 2007). DOI

10.1109/BCC.2007.4430543

9. Bartlow, N., Kalka, N., Cukic, B., Ross, A.: Protecting iris images

through asymmetric digital watermarking. Automatic Identiﬁ-

cation Advanced Technologies, 2007 IEEE Workshop on

pp. 192–197 (7-8 June 2007). DOI 10.1109/AUTOID.2007.380618

10. Kutter, M., Jordan, F., Bossen, F.: Digital signature of color

images using amplitude modulation. In: Proc. SPIE EI, San

Jose, CA, pp. 518–526 (1997)

11. Jain, A.K., Uludag, U., Hsu, R.L.: Hiding a face in a ﬁngerprint

image. In: ICPR ’02: Proceedings of the 16 th International

Conference on Pattern Recognition (ICPR’02) Volume 3,

p. 30756. IEEE Computer Society, Washington, DC, USA (2002)

12. Musgrave, C.: Biometric watermarks. Patent 6,208,746, U.S.

Patent and Trademark Ofﬁce, Washington, D.C. (2001)

13. Morikawa, G., Tokura, G.: Picture taking apparatus and method

of controlling same. Patent 7,305,089, U.S. Patent and Trade-

mark Ofﬁce, Washington, D.C. (2007)

14. Petitcolas, F.A., Anderson, R.J., Kuhn, M.G.: Attacks on copy-

right marking systems. In: Information Hiding, pp. 218–238

(1998). citeseer.ist.psu.edu/petitcolas98attacks.html

15. Pereira, S., Voloshynovskiy, S., Madueno, M., Marchand-Maillet,

S., Pun, T.: Second generation benchmarking and application

oriented evaluation In: IHW ’01: Proceedings of the 4th

International Workshop on Information Hiding, pp. 340–353

Springer, London, UK (2001)

Iris Encoding and Recognition using

Gabor Wavelets

JOHN DAUGMAN

Cambridge University, Cambridge, UK

Synonyms

Daugman algorithm; IrisCode; Iris2pi

Definition

The method of encoding iris patterns that is used in all

current public deployments of iris recognition technol-

ogy is based on a set of mathematical functions called

▶ Gabor wavelets that analyze and extract the unique

texture of an iris. They encode it in terms of its phase

structure at multiple scales of analysis. When this

phase information is coarsely quantized, it creates a

random bit stream that is sufﬁc iently stable for a given

eye, yet random and diverse for different eyes, that iris

patterns can be recognized very rapidly and reliably

over large databases by a simple test of statistical inde-

pendence. The success of this biometric algorithm may

be attributed in part to certain important properties of

the Gabor wavelets as encoders, and to the simplicity

Iris Encoding and Recognition using Gabor Wavelets

787

and efﬁciency of searches for matches when pattern

information is represented in terms of such phase bit

strings.

Introduction

Different biometric modalities use diverse methods for

encoding the features on which they depend. The over-

all goal in designing biome tric encoders is always the

same – maximizing between-class variation while

minimizing within-class variation – but very different

strategies have been developed for representing the

chosen features and their random variation. Even w ith-

in a single modality, such as ﬁngerprint recognition,

some methods compile lists of discrete minutiae coor-

dinates and angles, while other methods encode global

ridge-ﬂow descriptions. Facial representations may

be two-dimensional (‘‘appearance-based’’) or three-

dimensional ‘‘model-based’’); may try to achieve some

degree of pose-invariance, illumination-invariance,

or expression-invariance; and the scale of analysis may

be global (e.g., eigenface decompositions) or focused

on local, high-resolution, detail (e.g., skin texture anal-

ysis). In the case of iris patterns, one does not ﬁnd any

easily enumerated lists of distinct features like the ﬁn-

gerprint minutiae, nor indeed any sets of features that

even possess established names. Rather, one ﬁnds a

plethora of textures spanning many scales of analysis,

a wide spatial frequency range, and which might be

described using many candidate image statistics.

Whereas a natural feature to mark in a ﬁngerprint is

a ridge ending or a bifurcation, in the case of iris

patterns we ﬁnd random featu res with no simple geo-

metric or graph-like structure, that adhere to no taxon-

omy, and that are deﬁned across many different size

ranges. We need a language rich enough to capture

subtleties like ‘‘mottling’’ and ‘‘modulation,’’ yet simple

enough that all instances are commensurable, and pow-

erful enough to deliver extremely rapid and conﬁdent

recognition decisions. What might be such a language?

Most naturally occurring textures lend themselves

well to mathematical description in terms of both

spectral (Fourier-based) and spatially localized proper-

ties. The spectral language captu res predominant

undulations and quasi-coherences that are the essence

of texture, whose interwoven appearance reveals why

this word shares a root with ‘‘textile:’’ texere – ‘‘to

weave.’’ But at the same time, spatial variation in the

undulations destroys any simple coherence, and breaks

symmetries. To describe natural textures effectively

and efﬁciently, for examp le in order to build a biomet-

ric recognition system based on iris textures, we need

a language that is able to specify both the spatial and

the spectral proper ties well.

The upper panels of Fig. 1 illustrate two human iris

patterns, Caucasian on the left, oriental on the right.

They reveal rich and unique textural structure, and

also characteristic ethnic differences in their appear-

ance. For example, the Caucasian iris has ﬁner detail

and longer radial correlations, whereas the oriental

iris has somewhat coarser and more isotropic (less

elongated) features, which are also more concentrated

near the pupil. All of these general aspects of iris

patterns – their undulatory textures, their spatial vari-

ation, their statistical structure in correlation distances,

and, above all, their rich randomness, motivated the

author nearly two decades ago to develop a method for

encoding and recognizing iris patterns using a particu-

lar mathematical family of localized undulations called

▶ Gabor wavelets. Today, that algorithm [1, 2] remains

the metho d used in all public deployments of iris

recognition. This chapter reviews some of the essential

aspects of this approach to image analysis, encoding,

and pattern recognition.

Gabor Wavelets as a Complete

Image Basis

Classical signal processing divided broadly into analy-

sis performed in the signal domain (time in the case of

time-varying signals like sound waveforms, or space in

the case of images), versus analysis performed in the

Fourier domain in which the signal is represented as a

linear combination, or superposition, of global sinu-

soids. Although a bedrock of signal processing , Fourier

analysis is hampered by its excessively global perspec-

tive: in the Fourier domain, every coefﬁcient associated

with a particular frequency component summarizes

the presence of that frequency component over the

entire extent of the signal. Similarly, every local point

in the signal has an inﬂuence on every coefﬁcient in the

Fourier domain (i.e., on every frequency component).

This extreme, reciprocal, mapping ‘‘from local to global’’

and ‘‘from global to local’’ across the two domains

is reﬂected in the fact that their respective independent

variables are themselves reciprocals: time versus

788

Iris Encoding and Recognition using Gabor Wavelets

(temporal) frequency, as Hertz (1/s); and for images,

space versus spatial frequency (cycles per degree).

In the 1980s a number of mathematicians, most of

them Francophone, began to formulate a new synthesis

of these two domai ns based on a kind of compromise

between the global (spectral) and local (punctate) per-

spectives. It came to be called wavelet theory (translated

from a French neologism “ondelette’’ – a small wave).

The key insight of wavelet theorists like Yves Meyer [3]

and Ingrid Daubechies [4] was that it was possible to

construct complete representations of functions by

superposition of a set of universal elementary

functions all of which were dilates and translates of a

single shape on ﬁnite support, forming a dyadic set.

Typically the successive dilation (stretching) factors

were powers of 2, and the translation intervals scaled

reciprocally, so that self-similarity was preserved across

all scales. Originally there were ﬁve requirements for

such sets of basis functions to be deemed wavelets: they

must all be (1) dilates and (2) translates of each other;

they must have (3) strictly compact support; (4) all of

their derivatives must exist; and (5) they must be

mutually orthogonal (i.e., have zero inner products

with each other). These constraints and admissibility

Iris Encoding and Recognition using Gabor Wavelets. Figure 1 The upper two panels illustrate the rich textures

found in the iris, and also some typical ethnic differences: on the left is a Caucasian iris, and on the right an oriental iris.

Some significant differences include the longer radial correlation distances and finer detail in (especially blue-eyed)

Caucasians, and the tendency of oriental eyes to have more of their iris texture near the pupil. The lower two panels

show reconstructions of the upper two images using only a sparse discrete set of 2D Gabor wavelets, incorporating

just six spatial frequencies (one octave apart) and six orientations, as seen in Fig. 2.

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