Biomedical Engineering Reference
In-Depth Information
wavelet
ψ
(
x
)
=
g
(
t
)
e
i
η
t
is then obtained with a Gaussian function
1
π
)
1
/
4
e
−
t
2
g
(
t
)
=
2
2
σ
(
σ
2
(see [14]).
Extension of Gabor wavelet to 2D is expressed as:
k
(
s
,
y
)
=
g
(
x
,
y
)
e
−
i
η
(
x
cos
α
k
)
+
y
sin
α
k
)
(6.24)
ψ
.
k
(
x
,
y
) constitute the wavelet
basis for expansion. An extra parameter
α
k
provides selectivity for the orien-
tation of the function. We observe here that the 2D Gabor wavelet has a non-
separable structure that provides more flexibility on orientation selection than
separable wavelet functions.
It is well known that optical sensitive cells in animal's visual cortex respond
selectively to stimuli with particular frequency and orientation [26]. Equation
(6.24) described a wavelet representation that naturally reflects this neurophysi-
ological phenomenon. Gabor expansion and Gabor wavelets have therefore been
widely used for visual discrimination tasks and especially texture recognition
[27, 28].
Different translation and scaling parameters of
ψ
6.2.3.2 Wavelet Packets
Unlike dyadic wavelet transform, wavelet packets decompose the low-frequency
component as well as the high-frequency component in every subbands [29].
Such adaptive expansion can be represented with binary trees where each sub-
band high- or low-frequency component is a node with two children correspond-
ing to the pair of high- and low-frequency expansion at the next scale. An admis-
sible tree for an adaptive expansion is therefore defined as a binary tree where
each node has either 0 or 2 children, as illustrated in Fig. 6.6(c). The number
of all different wavelet packet orthogonal basis (also called a wavelet packets
dictionary) is equal to the number of different admissible binary trees, which is
of the order of 2
2
J
, where
J
is the depth of decomposition [14].
Obviously, wavelet packets provide more flexibility on partitioning the
spatial-frequency domain, and therefore improve the separation of noise and
signal into different subbands in an approximated sense (this is referred to the
near-diagonalization of signal and noise). This property can greatly facilitate
