A first approach to granular metrics is to use the range space, Y, and
assume that it has an underlying metric. For 
, let
provides a modification of g so that
combines values of g(y'), 
,
in some appropriate way. The set 
is a sphere obtained by
the underlying metric.
A particurlar choice for 
might be
the mean value of the neighbourhood. This is equivalent of low-pass filtering g, which means that high frequencies will be supressed in the metric. This in turn means that we obtain a metric that is relatively robust to noise.
An other choice of is
Taking the
supremum of the neighbourhood might in some applications be
motivated. This is mainly true if our 
has a very good locality,
that is, it concentrates 
to some points in Y. In these
cases it is not desirable to smear these points by using a mean
value.
We can now define a metric on F(X,L) through
We will use 
to mean 
.
A normalisation of c can be given as
Constructions in [3] correspond to the use of
,
where we use
.
For natural reasons, 
is limited downwards by 0 and
since 
is the same as the supremum
[4]. Note that this is true regardless of the
used.
Similarly, is limited upwards by 
and
Since 
, 
will be constant over
the whole Y.
Suppose we have two images, 
and 
, that are transformed to
and 
, 
see Image . The mean value- is used in
this example.