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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 [].

Tony Lindeberg
Tue Jul 1 14:57:47 MET DST 1997