IRIS RECOGNITION BASED ON HILBERT–HUANG TRANSFORM 1 ...

Iris Recognition Based on Hilbert–Huang Transform 627
By the EMD algorithm, any signal x(t) can be decomposed into finite IMFs,
cj(t)(j =1, 2,...,n), and a residue r(t), where n is the number of IMFs, i.e.
x(t) =
n
cj(t)+r(t). (1)
j=1
Having obtained the IMFs by EMD, we can apply the Hilbert transform to each
IMF, cj(t), to produce its analytic signal zj(t) =cj(t) +iH[cj(t)] = aj(t)e iθj(t) .
Therefore, x(t) can also be expressed as
x(t) =Re
n
aj(t)e iθj (t) + r(t). (2)
j=1
Equation (2) enables us to represent the amplitude and the instantaneous frequency
as functions of time in a three-dimensional plot, in which the amplitude is contoured
on the time–frequency plane. The time–frequency distribution of amplitude
is designated as the Hilbert spectrum, denoted by H(f,t) whichgivesatime–
frequency–amplitude distribution of a signal x(t). HHT brings sharp localizations
both in frequency and time domains, so it is very effective for analyzing nonlinear
and nonstationary data.
With the Hilbert spectrum defined, the Hilbert marginal spectrum can be
defined as
h(f) =
T
0
H(f,t)dt. (3)
The Hilbert marginal spectrum offers a measure of total amplitude (or energy)
contribution from each frequency component.
3.2. Main frequency and main frequency center
It is found that the Hilbert marginal spectrum h(f) has some properties, which can
be used to extract features for iris recognition. Specifically, the main frequency center
of the Hilbert marginal spectrum can be served as a feature to identify different
irises. The “main frequency” and “main frequency center” concepts proposed by us
have been clear described and discussed in our previous works. 26,27 We have shown
that the main frequency center can characterize the approximate period very well.
Here, we only review the definitions of main frequency, main frequency center and
other related concepts as follows.
Definition 1 (Main frequency). Let x(t) be an arbitrary time series and h(f)
be its Hilbert marginal spectrum, then fm is called as the main frequency of x(t), if
h(fm) ≥ h(f), ∀f.
Definition 2 (Average Hilbert marginal spectrum of signal series). Let
X = {xj(t)|j =1, 2,...,N}, whereeachxj(t) isatimeseries,andhj(f) bethe