# Scale-space axioms

Scale-space axioms

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 has been described in the literature. 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 $L\left(x, y, t\right) = \left(T_t f\right)\left(x, y\right) = g\left(x, y, t\right)*f\left(x, y\right)$ of signal $f\left(x, y\right)$ obtained by smoothing with the Gaussian kernel $g\left(x, y, t\right)$ satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation:

*"linearity":$T_t\left(a f + b h\right) = a T_t f + b T_t h$where $f$ and $h$ are signals while $a$ and $b$ are constants,
*"shift invariance':$T_t S_\left\{\left(Delta x, Delta_y\right)\right\} f = S_\left\{\left(Delta x, Delta_y\right)\right\} T_t f$where $S_\left\{\left(Delta x, Delta_y\right)\right\}$ denotes the shift (translation) operator $\left(S_\left\{\left(Delta x, Delta_y\right)\right\} f\right)\left(x, y\right) = f\left(x-Delta x, y - Delta y\right)$
*the "semi-group structure:$g\left(x, y, t_1\right) * g\left(x, y, t_2\right) = g\left(x, y, t_1 + t_2\right)$with the associated "cascade smoothing property":$L\left(x, y, t_2\right) = g\left(x, y, t_2 - t_1\right) * L\left(x, y, t_1\right)$
*existence of an "infinitesimal generator" $A$:$partial_t L\left(x, y, t\right) = \left(A L\right)\left(x, y, t\right)$
*"non-creation of local extrema" (zero-crossings) in one dimension,
*"non-enhancement of local extrema" in any number of dimensions:$partial_t L\left(x, y, t\right) leq 0$ at spatial maxima and $partial_t L\left(x, y, t\right) geq 0$ at spatial minima,
*"rotational symmetry":$g\left(x, y, t\right) = h\left(x^2+y^2, t\right)$ for some function $h$,
*"scale invariance:$hat\left\{g\right\}\left(omega_x, omega_y, t\right) = hat\left\{h\right\}\left(frac\left\{omega_x\right\}\left\{varphi\left(t\right)\right\}, frac\left\{omega_x\right\}\left\{varphi\left(t\right)\right\}\right)$for some functions $varphi$ and $hat\left\{h\right\}$ where $hat\left\{g\right\}$ denotes the Fourier transform of $g$,
*"positivity"::$g\left(x, y, t\right) geq 0$,
*"normalization"::$int_\left\{x=-infty\right\}^\left\{infty\right\} int_\left\{y=-infty\right\}^\left\{infty\right\} g\left(x, y, t\right) , dx , dy = 1$.
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 [Koenderink, Jan "The structure of images", Biological Cybernetics, 50:363–370, 1984] [ [http://portal.acm.org/citation.cfm?id=11298&dl=GUIDE&coll=GUIDE J. Babaud, A. P. Witkin, M. Baudin, and R. O. Duda, Uniqueness of the Gaussian kernel for scale-space filtering. IEEE Trans. Pattern Anal. Machine Intell. 8(1), 26–33, 1986.] ] [ [http://portal.acm.org/citation.cfm?id=11297&dl=ACM&coll=ACM A. Yuille, T.A. Poggio: Scaling theorems for zero crossings. IEEE Trans. Pattern Analysis & Machine Intelligence, Vol. PAMI-8, no. 1, pp. 15–25, Jan. 1986.] ] [ [http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.] ] [ [http://www.nada.kth.se/~tony/book.html Lindeberg, Tony, Scale-Space Theory in Computer Vision, Kluwer, 1994] ,] [ [http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=628701 Pauwels, E., van Gool, L., Fiddelaers, P. and Moons, T.: An extended class of scale-invariant and recursive scale space filters, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 7, pp. 691–701, 1995.] ] [ [ftp://ftp.nada.kth.se/CVAP/reports/cvap159.pdf Lindeberg, T.: On the axiomatic foundations of linear scale-space: Combining semi-group structure with causailty 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.] ] [Florack, Luc, Image Structure, Kluwer Academic Publishers, 1997.] [ [http://portal.acm.org/citation.cfm?id=607668&dl=ACM&coll=ACM Weickert, J. Linear scale space has first been proposed in Japan. Journal of Mathematical Imaging and Vision, 10(3):237–252, 1999.] ] .

The Gaussian kernel is also separable in Cartesian coordinates, i.e. $g\left(x, y, t\right) = g\left(x, t\right) , g\left(y, t\right)$. 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.

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.

ee also

* scale space
* scale space implementation

References

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