In the definitions of correlation and convolution,
Eqs. 1 and 2,
the expressions s-r and r-s show us that these operations
are sensitive to distance and direction in the domains of the
fields, that is, they depend on the coordinates over which
the fields are defined.
For example, if 
results from 
by a coordinate
transformation, 
, then the results
of convolving 
with a Gaussian 
will not be the
same as the results of convolving 
with 
. The
convolution 
averages over regions that
are circular in 
's domain, whereas
averages over circular regions in
's domain.
For example, because of the logmap transformation between the retina
and VI, a Gaussian convolution in VI will not have the effect
of a Gaussian convolution in retinal coordinates or vice
versa.
This sensitivity of convolutions and correlations to the
coordinate system can be a problem that needs to be solved or
a computational resource that can be exploited.
Suppose we have two domains 
and 
such that
fields over 
are transformations of fields over
; let 
be the coordinate
transformation (an isomorphism).
For example, 
and 
might be two brain regions
(such as the retina and VI), or one or the other might be an
external region (such as physical space around the body).
Let 
and 
be two fields over 
and suppose
we want to compute the convolution 
; for example we might want to do a
Gaussian convolution in retinal space. However, suppose that
the convolution is to be computed by means of fields defined
over the transformed domain 
.
We are given the transformed 
and want
to compute 
so that
.
We can get this by changing the integration variable of the
convolution
(assumed to be scalar to keep the example simple):
If we define the connectivity field
[ A_uv = [h^-1(u)-h^-1(v)]h'[h^-1(v)]
,]
then the convolution integral becomes
[ _u = _' A_uv _v v, ]
which is the integral operator,
.
This is a linear operator, but not a convolution, which means
that it is still implemented by a simple pattern of
connectivity, but that it is not a single pattern
duplicated throughout the region.
(If, as is often the case, the transformation h is a
homeomorphism, then it will preserve the topology of 
,
which means that a local convolution 
in 
will
translate into local connections A in 
.)
We remark without proof that if the domains are of more than
one dimension, then the connectivity kernel is defined
[ A_uv = [h^-1(u)-h^-1(v)]; J[h^-1(v)] , ]
where 
is the Jacobian of 
evaluated at
v.
Now, conversely, suppose we do a convolution 
in the transformed coordinates; what is
its effect in the original coordinates?
By a similar derivation we find that
where the kernel is defined
[ C_xy = [h(x)-h(y)];J[h(y)] . ]
In effect, the convolution kernel 
is projected
backward through the transformation h.
For example, if, like the logmap transformation, h expands
the space in the center of the visual field and compresses it
at the periphery, then the back-transformation of 
will result in a C that defines small receptive fields near
the center of the visual field, and large ones near its
periphery.