Image Processing Reference
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 )
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 )
( x, y )
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 ,
. 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
, m tends to the min
or the max. Table 8.1 sums up these results.
m ( x, y )
min( x, y )
x + y
( xy ) − 1 / 2
x + y
+2 x 2 + y 2
max( x, y )
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.