Image Processing Reference
In-Depth Information
A weaker property than associativity is bisymmetry:
[0
,
1]
4
,m
m
(
x, y
)
,m
(
z, t
)
=
m
m
(
x, z
)
,m
(
y, t
)
.
∀
(
x, y, z, t
)
∈
[8.70]
The means that satisfy this property and are continuous and strictly increasing have
the following general form:
[0
,
1]
2
,m
(
x, y
)=
k
−
1
k
(
x
)+
k
(
y
)
2
,
∀
(
x, y
)
∈
[8.71]
where
k
is a continuous, strictly increasing function of [0
,
1] into [0
,
1]. The function
k
can be interpreted as a change of scale or dynamics of the values to combine. These
values, once they have been transformed by
k
, are then combined using a simple arith-
metic mean, and the result is then changed back to the initial scale.
The most common means are obtained from functions
k
of the type:
[0
,
1]
,k
(
x
)=
x
α
,
∀
x
∈
with
α
∈
R
. The arithmetic mean (
x
+
y
)
/
2 is obtained for
α
=1, the quadratic mean
(
x
2
+
y
2
)
/
2 for
α
=2, the harmonic mean 2
xy/
(
x
+
y
) for
α
=
−
1, the geometric
mean
√
xy
for
α
=0. At the limit when
α
tends to
−∞
∞
or +
,
m
tends to the min
or the max. Table 8.1 sums up these results.
α
m
(
x, y
)
comment
−∞
min(
x, y
)
limit value
2
xy
x
+
y
−
1
harmonic mean
(
xy
)
−
1
/
2
0
geometric mean
x
+
y
2
+1
arithmetic mean
+2
x
2
+
y
2
2
quadratic mean
+
∞
max(
x, y
)
limit value
Table 8.1.
Examples of continuous, strictly increasing and bisymmetric means.
For the harmonic mean, we adopt the convention that
m
(0
,
0) = 0
Figure 8.8 shows a few examples of means.
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