Biomedical Engineering Reference
In-Depth Information
(a)
(b)
(c)
(d)
Figure 6.1: Example of spatial-frequency tiling of various transformations.
x
-
axis: spatial resolution and
y
-axis: frequency resolution. (a) Discrete sampling
(no frequency localization), (b) Fourier transform (no temporal localization).
(c) windowed Fourier transform (constant Heisenberg boxes), and (d) wavelet
transform (variable Heisenberg boxes).
It is normalized
ψ
=
1, and centered in the neighborhood of
t
=
0. A fam-
ily of time-frequency atoms is obtained by scaling
ψ
by
s
and translating it
by
u
:
t
−
u
s
1
√
s
ψ
ψ
u
,
s
(
t
)
=
(6.6)
.
A continuous wavelet transform decomposes a signal over dilated and translated
wavelet functions. The wavelet transform of a signal
f
∈
L
2
(
R
) at time
u
and
scale
s
is performed as:
√
s
ψ
∗
t
−
u
dt
=
0
.
+∞
Wf
(
u
,
s
)
=
f
,ψ
u
,
s
=
1
f
(
t
)
(6.7)
s
−∞
Assuming that the energy of
ˆ
ψ
(
ω
) is concentrated in a positive frequency interval
centered at
η
, the time-frequency support of a wavelet atom
ψ
u
,
s
(
t
) is symboli-
cally represented by a Heisenberg rectangle centered at (
u
,η/
s
), with time and
frequency supports spread proportional to
s
and 1
/
s
respectively. When
s
varies,
the height and width of the rectangle change but its area remains constant, as
illustrated by Fig. 6.1 (d).
For the purpose of multiscale analysis, it is often convenient to introduce the
scaling function
φ
, which is an aggregation of wavelet functions at scales larger
than 1. The scaling function
φ
and the wavelet function
ψ
are related through
the following relations:
+∞
ˆ
φ
(
ω
)
ˆ
ψ
(
s
ω
)
2
ds
s
.
2
(6.8)
=
1
