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 formula_1 of signal formula_2 obtained by smoothing with the Gaussian kernel formula_3 satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation:
formula_4
where formula_5 and formula_6 are signals while formula_7 and formula_8 are constants,
formula_9
where formula_10 denotes the shift (translation) operator formula_11
formula_12
with the associated "cascade smoothing property"
formula_13
formula_15
formula_16 at spatial maxima and formula_17 at spatial minima,
formula_18 for some function formula_6,
formula_20
for some functions formula_21 and formula_22 where formula_23 denotes the Fourier transform of formula_24,
formula_25,
formula_26.
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:
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. formula_27. 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.
There exists a generalization of the Gaussian scale-space theory to more general affine and spatio-temporal scale-spaces. 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. The generalized scale-space theory leads to predictions about receptive field profiles in good qualitative agreement with receptive field profiles measured by cell recordings in biological vision.
In the computer vision, image processing and signal processing literature there are many other multi-scale approaches, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; please see the article on related multi-scale approaches. There has also been work on discrete scale-space concepts that carry the scale-space properties over to the discrete domain; see the article on scale space implementation for examples and references.