Biomedical Engineering Reference
In-Depth Information
then 2 j φ 2 j
2 j n
is the orthonormal basis in vector space V 2 j .The
vector space V 2 j is the set of all possible approximations of a signal at resolution 2 j .
The signal x
(
t
)
for n
,
j
Z
(
t
)
can be projected on the orthonormal basis:
φ 2 j u
2 j n
φ 2 j t
2 j n
2 j
(
)=
(
) ,
A 2 j
t
x
u
(2.99)
=
n
at resolution 2 j
The discrete approximation of signal x
(
t
)
is defined by a set of
inner products:
A 2 j x
2 j n
(
n
)=
x
(
u
) ,
φ 2 j
(
u
)
(2.100)
The above formula is equivalent to the convolution of signal x
(
t
)
with the dilated
which is a low-pass filter. The computation of A 2 j x for con-
scaling function φ 2 j
(
t
)
secutive resolutions 2 j
followed by
uniform sampling at rate 2 j . In practice, the computations of A 2 j x can be realized
iteratively by convolving A 2 j + 1 x with
can be interpreted as a low-pass filtering of x
(
t
)
h
h
(
n
)
, such that
(
n
)=
h
(
n
)
is a mirror filter
of:
h
(
n
)=
φ 2 1
(
u
) ,
φ
(
u
n
)
(2.101)
and downsampling the result by 2. This algorithm can be expressed by the formula:
A 2 0 =
x
(
n
)
D 2 k = h ( 2 n k ) A 2 j + 1 x ( n )
(2.102)
A 2 j x
(
n
)=
where D 2 is the operator of downsampling by factor 2.
The low-pass filtering used in the algorithm has the effect that after each iteration
the consecutive approximations contain less information. To describe the residual
information lost in one step and measure the irregularities of signal at resolution 2 j
a function Ψ 2 j is constructed, such that:
2 j Ψ 2 j t
Ψ 2 j
(
t
)=
(2.103)
and the functions Ψ 2 j t
2 j n are a basis in a vector space O 2 j . The vector space
O 2 j is orthonormal to V 2 j and together they span the vector space V 2 j + 1 :
O 2 j
V 2 j and O 2 j
V 2 j
=
V 2 j + 1
(2.104)
The functionΨ
is called a wavelet and corresponds to the mother wavelet function
in equation 2.92. The projection of a signal x
(
t
)
(
t
)
on the space O 2 j is called the discrete
detail signal and is given by the product:
Ψ 2 j u
2 j n
D 2 j x
(
n
)=
x
(
u
) ,
(2.105)
These inner products can be computed conveniently from higher resolution discrete
approximation by the convolution:
D 2 j x
A 2 j + 1
(
n
)=
g
(
2 n
k
)
(
k
)
(2.106)
k
=
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