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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