One reason correlation and convolution are of interest is
that they can be used for pattern recognition and
generation.
For example, the correlation 
will have
peaks wherever the pattern 
occurs in field 
(or
vice versa); occurrences of patterns less similar to 
(in an inner-product sense) will cause lesser peaks.
Thus correlation 
returns an activity
pattern representing the spatial distribution in 
of
fields resembling 
.
This operation is approximately reversible. Suppose that
is a radial field, such as a Gaussian, with a single
narrow, sharp maximum.
Convolving 
with a pattern 
has the effect of
blurring 
by 
(i.e. smoothing 
by a
window of shape 
):
[ ()(s)
= _(s-u) (u) u . ]
Further, if 
is first displaced by r, then the
effect of the convolution is to blur 
and displace it
by r:
[ (T_r )
= T_r() . ]
[The 
operation translates (displaces) a field by
r:
.]
Finally, since convolution is bilinear, if 
is a field
containing a number of sharp peaks at various displacements
, then 
will produce a field
containing blurred copies of 
at corresponding
displacements:
[
= ( _k T_r_k )
= _k (T_r_k )
= _k T_r_k () .
]
(The convolution of a superposition is a superposition of the
convolutions.)
Such an operation could be used for constructing a
representation of the environment for motion planning.
For example, if 
is the shape of an obstacle retrieved
from memory, and 
is a map of the location of obstacles
of this kind in the environment, then 
represents the approximate boundaries of such obstacles in
the environment.