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Non-enhancement of local extrema and infinitesimal generator.

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.