In image processing and computer vision, a scale space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale space representations exist. A typical approach for choosing a particular type of scale space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters. A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision. Scale space axioms for the linear scale-space representation The linear scale space representation of signal obtained by smoothing with the Gaussian kernel satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation: linearity where and are signals while and are constants, shift invariance where denotes the shift (translation) operator semi-group structure with the associated cascade smoothing property existence of an infinitesimal generator non-creation of local extrema (zero-crossings) in one dimension, non-enhancement of local extrema in any number of dimensions at spatial maxima and at spatial minima, rotational symmetry for some function , scale invariance for some functions and where denotes the Fourier transform of , positivity , normalization . 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:<ref name=Lin13>[https://dx.doi.org/10.1016/B978-0-12-407701-0.00001-7 Lindeberg, T. Generalized axiomatic scale-space theory, Advances in Imaging and Electron Physics, Elsevier, volume 178, pages 1-96, 2013.]</ref> 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" for one-dimensional signals or "non-enhancement of local extrema" 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. The Gaussian kernel is also separable in Cartesian coordinates, i.e. . 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