If the semi-group structure per se is combined with a strong continuity
requirement with respect to the scale parameter,
then it follows from well-known results in functional analysis
[] that
the scale-space family must have an infinitesimal generator.
In other words, if a transformation operator
from the input signal
to the scale-space representation at any scale t is defined by
,
then under reasonable regularity requirements
there exists a limit case of this operator
(the infinitesimal generator)
and the scale-space family satisfies the
differential equation

Lindeberg [, ] showed
that this structure implies that the scale-space family
must satisfy the diffusion equation if
combined with a slightly modified formulation of Koenderinks
causality requirement expressed as
non-enhancement of local extrema:

Non-enhancement of local extrema:
If for some scale level a point is
a non-degenerate local maximum for the scale-space representation
at that level
(regarded as a function of the space coordinates only) then its value
must not increase when the scale parameter increases. Analogously, if a
point is a non-degenerate local minimum then its value must not
decrease when the scale parameter increases.

Moreover, he showed that this scale-space formulation extends to
discrete data as well as to non-symmetric temporal and spatio-temporal
image domains.