Witkins observation shows that Gaussian convolution
satisfies certain sufficiency requirements for being a smoothing
operation. The first proof of the necessity of Gaussian
smoothing for generating a scale-space representation was given by
Koenderink [], who also gave a formal extension of the scale-space
theory to higher dimensions.
He introduced the concept of causality, which means that new
level surfaces

must not be created in the scale-space representation when the
scale parameter is increased.
By combining causality with the
notions of isotropy and homogeneity,
which essentially
mean that all spatial positions and all scale levels must be treated in
a similar manner, he showed that the scale-space representation must
satisfy the diffusion equation.
Related formulations have been expressed by
Yuille and Poggio [] and by Hummel [].