Digital Signal Processing Reference
In-Depth Information
columns
1
1
2
X
rows
d
(-)
21
(-)
1
h
X
Med[.]
1
2
X
1
1
2
X
v
(-)
21
Med[.]
1
Med[.]
1
2
X
a
: 1D Median filter applied on rows or columns
Med[.]
21 : Keep one column out of two
12
: Keep one row out of two
FIGURE 3.3
The nonlinear filter multiresolution structure used to generate the nonlinear decomposition.
(From M. Asghar and K.E. Barner,
IEEE Trans. Visualization Comput. Graphics
,7,76-93, Mar. 2001.
c
2001 IEEE. With permission.)
In these expressions
X
a
is the first-level approximation of
X
, and
X
h
,
X
1
, and
X
d
correspond to horizontal, vertical, and diagonal details, respectively. It is also
possible to perfectly reconstruct the original signal from the approximation
and detail coefficients.
The multiresolution structure can be relaxed to allow decompositions based
on nonlinear filters. As a simple modification, a 1D median filter can be used
in the multiresolution structure shown in Figure 3.2. In this case, the lowpass
filter
h
v
is replaced by the median filter. Since there is no highpass median
filter, some modification of the multiresolution structure is required. Instead
of highpass filtering the data, the detail signal can be obtained by subtracting
the median filtered data from the original data before decimation (Figure 3.3).
In this case, the level-one decomposed signal and detail signals are given by
(
n
)
=
(
↓
T
]
T
,
X
a
)
((
↓
)
)
2
MED[
2
MED[
X
]
(3.5)
v
=
(
↓
)
T
,
X
1
MED[
X
]
T
MED[
X
]
T
]
)((
↓
)
−
(
↓
)
2
2
MED[
2
(3.6)
X
h
=
(
↓
T
]
T
,
2
)
MED[
(
↓
2
)(
X
−
MED[
X
]
)
(3.7)
=
(
↓
)
T
X
d
T
T
]
2
)((
↓
2
)(
X
−
MED[
X
]
)
−
MED[
(
↓
2
)(
X
−
MED[
X
]
)
.
(3.8)
Search WWH ::
Custom Search