In
image processing and
computer vision, a
scale space
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
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 representation
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal th ...
s 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
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
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:
[Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.]
/ref>[Lindeberg, T.: On the axiomatic foundations of linear scale-space: Combining semi-group structure with causality vs. scale invariance. In: J. Sporring et al. (eds.) Gaussian Scale-Space Theory: Proc. PhD School on Scale-Space Theory, (Copenhagen, Denmark, May 1996), pages 75–98, Kluwer Academic Publishers, 1997.]
/ref>[Lindeberg, T. Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space, Journal of Mathematical Imaging and Vision, Volume 40, Number 1, 36-81, 2011.]
/ref>[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
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...
, 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 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.[Lindeberg, T. A computational theory of visual receptive fields, Biological Cybernetics, 107(6): 589-635, 2013.]
/ref>T. Lindeberg "Normative theory of visual receptive fields", Heliyon 7(1):e05897, 2021.
/ref>
In the computer vision, image processing and signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
literature there are many other multi-scale approaches, using wavelets
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
and a variety of other kernels, that do not exploit or require the same requirements as scale space
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
descriptions do; please see the article on related multi-scale approaches The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation. There are, however, also other types of "multi ...
. 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
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges o ...
for examples and references.
See also
* Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges o ...
References
Image processing
Computer vision