The process of convolution is symmetrical. It combines
two functions. We usually refer to one as the signal and the other as the
filter but it does not matter which is which. For filtered noise we usually
think of the function as a set of points and the filter as a circular,
spherical or hyperspherical footprint with higher values in the centre.
We get the same result if the filter is a single point with value 1.0 and
the function is a set of overlapping circular or spherical regions:
These regions could be made different sizes to achieve
the effect of better packing. We can visualize this as packed circles with
different diameters. But remember each circle is now smaller than the footprint
it represents. We can achieve better packing with non-uniform circles or
spheres but how should we place them to make filtered noise that is not
artificially shaped by our choice of non-uniform pattern? One answer is
to use a slice from a space of higher dimension.