January 14, 2003 17:35 WSPC/181-IJWMIP 00002
International Journal of Wavelets, Multiresolution
and Information Processing
Vol. 1, No. 1 (2003) 1–17
c
World Scientific Publishing Company
DEMODULATION BY COMPLEX-VALUED WAVELETS FOR
STOCHASTIC PATTERN RECOGNITION
JOHN DAUGMAN
The Computer Laboratory, University of Cambridge, Cambridge, UK
john.daugman@cl.cam.ac.uk
Received 18 November 2002
Samples from stochastic signals having sufficient complexity need reveal only a little
unexpected shared structure, in order to reject the hypothesis that they are indepen-
dent. The mere failure of a test of statistical independence can thereby serve as a basis
for recognizing stochastic patterns, provided they possess enough degrees-of-freedom,
because all unrelated ones would pass such a test. This paper discusses exploitation of
this statistical principle, combined with wavelet image coding methods to extract phase
descriptions of incoherent patterns. Demodulation and coarse quantization of the phase
information creates decision environments characterized by well-separated clusters, and
this lends itself to rapid and reliable pattern recognition.
Keywords: Demodulation; phase; Gabor wavelets; iris; biometric; pattern recognition.
1. Introduction
The central issue in pattern recognition is the relationship between within-class
variability and between-class variability. These are determined by the dimensions
of variation (degrees-of-freedom) spanned by the pattern classes. Ideally the within-
class variability should be small and the between-class variability large, since this
creates optimal separation among the pattern classes. The reliability of pattern
recognition decisions depends upon the separation, or amount of overlap, among the
different pattern classes; we desire that the spacings between the clusters be larger
than the diameters of the clusters. This statistical separation in turn depends partly
upon the representation chosen for defining the classes. It is desirable to find image
representations which lend themselves optimally to these statistical requirements
of pattern recognition. This paper discusses the coupling of wavelet image coding
with a test of statistical independence on extracted phase information, in order
to obtain a demonstrably robust and reliable algorithm for recognizing stochastic
patterns of high dimensionality.
1
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2 J. Daugman
2. Complex-Valued 2D Wavelets for Image Analysis
The use of multi-resolution two-dimensional wavelets in image analysis and com-
puter vision has attracted much interest in recent years. The particular family of
2D wavelets that will be discussed here are closed under dilation, translation, and
rotation, and they also form a complete image basis, or frame. They are also self-
Fourier (equivalent in form to their own Fourier transforms), and they compose a
set that is also closed under convolution (the convolution of any two members of
thesetisalsoamemberoftheset).
If we take Ψ(x, y) to be any chosen generic 2D wavelet, which may be called a
mother wavelet, then we can generate from this function a complete self-similar
family of parametrized daughter wavelets Ψ
mpqθ
(x, y) through the generating
relation
Ψ
mpqθ
(x, y)=2
−2m
Ψ(x
0
,y
0
) , (2.1)
where the substituted variables (x
0
,y
0
) incorporate dilations of the wavelet in size
by 2
m
, translations in position (p, q), and rotations through angle θ:
x
0
=2
−m
[x cos(θ)+y sin(θ)] − p, (2.2)
y
0
=2
−m
[−x sin(θ)+y cos(θ)] − q. (2.3)
A particular choice for Ψ(x, y) which possesses several interesting and useful
properties is the complex-valued 2D “Gabor” wavelet, so named because it is a
generalization
2
of the 1D elementary functions originally discussed by Gabor.
8
The
2D wavelet form is defined as follows:
Ψ(x, y)=e
−π[(x−x
0
)
2
/α
2
+(y−y
0
)
2
/β
2
]
e
−2πi[u
0
(x−x
0
)+v
0
(y−y
0
)]
, (2.4)
where (x
0
,y
0
) specify wavelet position, (α, β) specify effective width and length,
and (u
0
,v
0
) specify a modulation wave-vector which can be interpreted in polar
coordinates as spatial frequency ω
0
=
p
u
2
0
+ v
2
0
and orientation (or direction)
θ
0
= arctan(v
0
/u
0
). Plots of the real and imaginary parts of such wavelets are
shown in Fig. 1.
The 2D Fourier transform F (u, v) of the 2D Gabor wavelet Ψ(x, y) has exactly
the same functional form (i.e. this family of wavelets is self-Fourier), with parame-
ters just interchanged or inverted:
F (u, v)=e
−π[(u−u
0
)
2
α
2
+(v−v
0
)
2
β
2
]
e
2πi[x
0
(u−u
0
)+y
0
(v−v
0
)]
. (2.5)
Thus the 2D Fourier power spectrum of the Ψ(x, y) wavelet, F (u, v)F
∗
(u, v), is a
bivariate Gaussian centered on (u
0
,v
0
). Its spectral energy peak is at orientation
θ
0
and spatial frequency ω
0
as defined above, and so it serves to extract image
structure in a particular band of the 2D Fourier spectrum. The effective support
of this 2D spectral band-area is an ellipse centered at (u
0
,v
0
) and whose principle
axes are (1/α, 1/β).
January 14, 2003 17:35 WSPC/181-IJWMIP 00002
Demodulation by Complex-valued Wavelets 3
[0, 0] [1, 0]
[1, 1][0, 1]
Re
Im
Fig. 1. Pattern encoding by phase demodulation. Image structure is extracted as phase sequences
by projection onto multi-resolution complex-valued 2D wavelets.
It is noteworthy that as consequences of the similarity, shift, and modulation
theorems of 2D Fourier analysis, together with the rotation isomorphism of the
2D Fourier transform, all the effects of the generating relation (2.1) applied to a
2D Gabor mother wavelet Ψ(x, y) to produce a new daughter wavelet Ψ
mpqθ
(x, y)
will have corresponding or reciprocal effects on its 2D Fourier transform F(u, v)
without any change in functional form.
3
This set of 2D wavelets and their 2D Fourier
transforms is closed under the transformations of dilation, translation, rotation,
and convolution with any member of the set. Two further interesting properties
of these wavelets are the fact that they achieve the lower bound of the Weyl–
Heisenburg “uncertainty relation” for conjoint resolution in the 2D space and 2D
Fourier domains,
3
and the fact that they form an excellent model for the receptive
field profiles of individual neurones (the so-called “simple cells”) in the visual cortex
of mammalian brains.
2,3
Any image can be represented completely in terms of
such wavelets used as expansion functions. A complication arises from the fact
that these wavelets are non-orthogonal (the inner product of any two of them is
in general nonzero), and so the expansion coefficients required do not correspond
simply to the inner product projections of the image onto the wavelets. A relaxation
network solution to this problem of obtaining the correct expansion coefficients was
presented in Ref. 4.
Because these wavelets are complex valued, it is possible to use the real and
imaginary parts of their convolution (∗) with an image I(x, y) to extract a de-
scription of image structure in terms of local modulus and phase. These corre-
spond, respectively, to an “amplitude modulation” function A(x, y) and a “phase
January 14, 2003 17:35 WSPC/181-IJWMIP 00002
4 J. Daugman
modulation” function φ(x, y) of the spatial image coordinates:
A(x, y)=
p
(Re{Ψ(x, y) ∗ I(x, y)})
2
+(Im{Ψ(x, y) ∗ I(x, y)})
2
, (2.6)
φ(x, y)=tan
−1
(Im{Ψ(x, y) ∗ I(x, y)})
(Re{Ψ(x, y) ∗ I(x, y)})
. (2.7)
These polar descriptors are shown in the phasor diagram of Fig. 1. The modulus
of the phasor in the complex plane represents a given patch of an image in terms
of its local contrast amplitude A(x, y), and the angle of this phasor represents
the patch in terms of its local phase φ(x, y). This phase angle can be quantized
very coarsely, as shown in Fig. 1 using only two bits, for building descriptions of
a pattern that lend themselves to efficient and reliable pattern recognition by the
simple failure of a test of statistical independence.
We turn now to illustrating this principle in a particular application in image
analysis and computer vision, namely the recognition of iris patterns in a person’s
eye as a reliable method of automatic personal identification.
3. Biometric Recognition of Persons by Iris Patterns
The highest density of biometric degrees-of-freedom (forms of variability among
individuals) which are both stable over time and easily measured, is to be found in
the complex texture of the iris pattern of the eye. This protected internal organ,
whose pattern can be encoded from distances of up to almost a meter, reveals about
250 independent degrees-of-freedom of textural variation across individuals. One
way to calibrate the “information density” of the iris is by its human-population
entropy per unit area. As we will see, this works out to about 3.2 bits per square
millimeter on the iris, based upon 9.1 million paired IrisCode comparisons that
have been performed using these algorithms.
Fig. 2. Example of a human iris pattern, imaged in near infrared light at a distance of 30 cm.
Such patterns have high statistical dimensionality and can serve as unique, reliable identifiers.
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Demodulation by Complex-valued Wavelets 5
3.1. Properties of the iris
The iris is composed of elastic connective tissue, the trabecular meshwork, whose
prenatal morphogenesis is completed during the 8th month of gestation. It consists
of pectinate ligaments adhering into a tangled mesh revealing striations, ciliary
processes, crypts, rings, furrows, a corona, sometimes freckles, vasculature, and
other features. During the first year of life a blanket of chromatophore cells often
changes the color of the iris, but the available clinical evidence indicates that the
trabecular pattern itself is stable throughout the lifespan. Because the iris is a
protected internal organ of the eye, behind the cornea and the aqueous humor, it
is immune to the environment except for its pupillary reflex to light. (The elastic
deformations that occur with pupillary dilation and constriction are readily reversed
mathematically by the algorithms for localizing the inner and outer boundaries of
the iris.) Pupillary motion, even in the absence of illumination changes (termed
hippus), and the associated elastic deformations in the iris texture, provide one test
against photographic or other simulacra of a living iris in high security applications.
There are few systematic variations in the amount of detectable iris detail as a
function of ethnic identity or eye color; even dark-eyed persons reveal plenty of iris
detail when imaged with infrared light. Further discussion of anatomy, physiology,
and clinical aspects of the iris may be found in Ref. 1.
3.2. Localizing irises and analyzing their patterns
The two-dimensional modulations which create iris patterns are extracted by
demodulation
7
with complex-valued 2D wavelets, as discussed above and illustrated
in Fig. 1, albeit in polar rather than cartesian coordinates.
First it is necessary to localize precisely the inner and outer boundaries of the
iris, and to detect and exclude eyelids if they intrude. These detection operations
are accomplished by integro-differential operators of the form
max
(r,x
0
,y
0
)
G
σ
(r) ∗
∂
∂r
I
r,x
0
,y
0
I(x, y)
2πr
ds
, (3.1)
where contour integration parametrized for size and location coordinates r, x
0
,y
0
at a scale of analysis σ set by some blurring function G
σ
(r) (e.g. a Gaussian of scale
σ) is performed over the image data array I(x, y). The result of this optimization
search is the determination of the circle parameters r, x
0
,y
0
which best fit the inner
and outer boundaries of the iris.
Then a doubly-dimensionless coordinate system is defined which maps the tis-
sue in a manner that is invariant to changes in pupillary constriction and overall
iris image size, and hence also invariant to camera optical zoom factor and distance
to the eye. This coordinate system is pseudo-polar, although it does not assume
concentricity of the inner and outer boundaries of the iris since the pupil is nor-
mally somewhat nasal, and inferior, in the iris. The coordinate system compensates
automatically for the stretching of the iris tissue as the pupil dilates. The inner
