Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters.
satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation: In fact, it can be shown that the Gaussian kernel is a unique choice given several different combinations of subsets of these scale-space axioms:[1][2][3][4][5][6][7][8][9][10][11] most of the axioms (linearity, shift-invariance, semigroup) correspond to scaling being a semigroup of shift-invariant linear operator, which is satisfied by a number of families integral transforms, while "non-creation of local extrema"[4] for one-dimensional signals or "non-enhancement of local extrema"[4][7][10] for higher-dimensional signals are the crucial axioms which relate scale-spaces to smoothing (formally, parabolic partial differential equations), and hence select for the Gaussian.
Separability is, however, not counted as a scale-space axiom, since it is a coordinate dependent property related to issues of implementation.
In addition, the requirement of separability in combination with rotational symmetry per se fixates the smoothing kernel to be a Gaussian.
[10][11] In addition to variabilities over scale, which original scale-space theory was designed to handle, this generalized scale-space theory also comprises other types of variabilities, including image deformations caused by viewing variations, approximated by local affine transformations, and relative motions between objects in the world and the observer, approximated by local Galilean transformations.
In this theory, rotational symmetry is not imposed as a necessary scale-space axiom and is instead replaced by requirements of affine and/or Galilean covariance.