Two closely-related bilinear operations that are especially
important for information processing are convolution
and correlation.
In the simplest case, correlation can be described as a
comparison of two fields at all possible relative positions.
More specifically, if 
is the correlation of two
one-dimensional fields 
and 
,
,
then 
reflects how well 
and 
match (in
an inner-product sense) when relatively displaced by
r.
Mathematically,
Higher dimensional correlations are the same, except that r is a relative displacement vector rather than a scalar.
Convolution,
,
is essentially the same as correlation, except that the field
is reflected before the comparison takes place:
Convolution is useful because: (1) its algebraic
properties are more like multiplication, and thus more
familiar, than correlation; and (2) many physical processes
(e.g. linear systems, such as dendritic nets) perform
convolutions.
