Digital Signal Processing Reference
In-Depth Information
Perfect reconstruction is possible from the approximation and detail compo-
nents only if the data is not decimated after the filtering operation. Decima-
tion, however, is advantageous in many applications as it reduces the size of
the data set.
Linear multiresolution techniques tend to smooth the edges in signals while
nonlinear techniques, such as median filters, are known to preserve edges. A
median filter, however, completely removes details with spatial span smaller
than half of the window size. Conversely, linear filters do not take into account
the spread of samples, and thus spread the effect of a pulse to its neighbor-
ing samples. The median affine filter provides a bridge between these two
extremes and provides additional flexibility to the user in that a single pa-
rameter is required to specify the degree of filter nonlinearity. This flexibility
allows the filter to behave as a linear filter, as a median filter, or as a hybrid
filter combining the properties of the linear and nonlinear filters. Employing
the median affine filter in the same structure yields
T
]
T
,
X
a
=
(
↓
2
)
MAFF[
((
↓
2
)
MAFF[
X
]
)
(3.9)
v
=
(
↓
)
T
,
X
1
MAFF[
X
]
T
MAFF[
X
]
T
]
)((
↓
)
−
(
↓
)
2
2
MAFF[
2
(3.10)
X
h
=
(
↓
T
]
T
,
)
(
↓
)(
−
)
2
MAFF[
2
X
MAFF[
X
]
(3.11)
=
(
↓
)
T
X
d
T
T
]
2
)((
↓
2
)(
X
−
MAFF[
X
]
)
−
MAFF[
(
↓
2
)(
X
−
MAFF[
X
]
)
(3.12)
The digital elevation model (DEM) data used to illustrate the performance
of the multiresolution techniques are from the Lake Charles data set. The
actual size of this data set is 1200
1200. However, for illustration purposes
only certain sections of the data set are rendered at one time. A size 150
×
150
portion of the data set is shown in Figure 3.4. The approximations of the Lake
Charles data set produced by the level-one biorthogonal wavelet, median
filter, median affine filter, and scale invariant median affine filter
9
decompo-
sitions are shown in Figure 3.5. As the figure shows, the original data set has a
number of sharp edges that are smeared in the wavelet decomposed surface.
In comparison, the median-based decomposition contains edges that preserve
sharp transitions. The hybrid behavior of the median affine filter produces a
decomposition that smoothes fine details while preserving edge integrity.
The differences in the behaviors of the multiresolution techniques are more
marked in the level-two decomposed approximations, which are shown in
Figure 3.6. The figures show that the median-based multiresolution decompo-
sition perfectly preserves large-scale edge features while completely removing
small details. At the other extreme, the wavelet decomposition significantly
smoothes fine details as well as large-scale edge features. The median affine
decomposition can be varied between these limiting cases through control
over the spread parameter
×
γ
. This hybrid response allows edges to be well
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