Digital Signal Processing Reference
In-Depth Information
4.3.4 Undecimated Multiscale Morphological Transform
Mathematical morphology has, up to now, been considered another way to analyze
data, in competition with linear methods. But from a multiscale point of view (Starck
et al. 1998; Goutsias and Heijmans 2000; Heijmans and Goutsias 2000), mathemat-
ical morphology or linear methods are just filters allowing us to go from a given
resolution to a coarser one, and the multiscale coefficients are then analyzed in the
same way.
By choosing a set of structuring elements
B
j
having a size increasing with
j
,we
can define an undecimated morphological multiscale transform by
c
j
+
1
[
l
]
=M
j
(
c
j
)[
l
]
(4.15)
w
j
+
1
[
l
]
=
c
j
[
l
]
−
c
j
+
1
[
l
]
,
where
M
j
is a morphological filter (closing, opening, etc.) using the structuring el-
ement
B
B
j
is a box of size (2
j
+
×
(2
j
+
j
. An example of
1)
1). Because the detail
coefficients
w
j
+
1
are obtained by calculating a simple difference between the scaling
coefficients
c
j
and
c
j
+
1
, the reconstruction is straightforward and identical to that of
the starlet transform (see Section 3.5). An exact reconstruction of
c
0
is then given
by
j
=
1
w
j
[
l
]
J
c
0
[
l
]
=
c
J
[
l
]
+
,
(4.16)
where
J
is the number of scales used in the analysis step. Each scale has the same
number of samples as the original data. The redundancy factor is therefore
J
+
1,
as for the starlet transform.
4.4 MULTIRESOLUTION BASED ON THE MEDIAN TRANSFORM
4.4.1 Multiscale Median Transform
The search for new multiresolution tools has been motivated so far by problems
related to the wavelet transform. It would be more desirable for a point structure
(represented as an isolated or outlier sample in a signal or image) to be present only
at the first (finest) scale. It would also be desirable for a positive structure in the
signal or image not to create negative values in the coefficient domain. We will see
how such an algorithm can be arrived at using nonlinear filters such as the median
filter.
The median filter is nonlinear and offers advantages for robust smoothing (i.e.,
the effects of outlier sample values are mitigated). In two dimensions (2-D), denote
the median filtered version of a discrete image
f
, with a square
L
×
L
neighbor-
hood, as Med(
f
1. The iteration counter will be
denoted by
j
, and
J
is the user-specified number of resolution scales. The multi-
scale median transform (MMT) algorithm is (Starck et al. 1996)
,
L
). Let
L
=
2
s
+
1; initially,
s
=
