Image Processing Reference
In-Depth Information
L 0
L 1
L 2
L 3
Fig. 9.10. The
concentric annuli
illustrate the effective frequency bands captured by different
levels of a Laplacian pyramid
can be represented by
f
(
t
). There are symbols for each tone, and the time durations
of the tones are part of the symbols. Hence a given tone is played for a certain dura-
tion, followed by another tone with its duration, and so on. This way of bringing to
life a 1D function
f
is radically different than telling how much air pressure should
be produced at a given time, i.e., a straightforward time sampling of
f
(
t
). The sam-
pled joint time-frequency representations of functions have relatively recently been
given a formal mathematical frame [10,53,77,203,214]. This is remarkable because,
for several centuries humans have been synthesizing and analyzing certain music sig-
nals by using sequences of tones chosen from a limited set, differing from each other
either in their (basic) frequencies or durations. Below, we discuss time-frequency
sampling concept in further detail. We subsequently extend these results to 2D and
higher dimensional images.
Let
f
(
t
) be a 1D function and
w
(
t
) be a window function with limited support,
where the coordinate
t
varies continuously in ]
−∞
,
∞
[ .
is
f
(
t
0
,t
)=
f
(
t
Definition 9.2.
The local function around
t
0
t
0
)
w
(
t
)
, and the
local spectrum
is defined as the Fourier transform of the local function
f
:
F
(
t
0
,ω
0
)=
−
f
(
t
−
t
0
)
w
(
t
) exp(
−
iω
0
t
)
dt
=
w
(
t
) exp(
iω
0
t
)
,f
(
t
−
t
0
)
(9.32)
