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 $L(x,\; y,\; t)\; =\; (T\_t\; f)(x,\; y)\; =\; g(x,\; y,\; t)*f(x,\; y)$ of signal $f(x,\; y)$ obtained by smoothing with the Gaussian kernel $g(x,\; y,\; t)$ satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation:

*"linearity":$T\_t(a\; f\; +\; b\; h)\; =\; a\; T\_t\; f\; +\; b\; T\_t\; h$where $f$ and $h$ are signals while $a$ and $b$ are constants,
*"shift invariance':$T\_t\; S\_\{(Delta\; x,\; Delta\_y)\}\; f\; =\; S\_\{(Delta\; x,\; Delta\_y)\}\; T\_t\; f$where $S\_\{(Delta\; x,\; Delta\_y)\}$ denotes the shift (translation) operator $(S\_\{(Delta\; x,\; Delta\_y)\}\; f)(x,\; y)\; =\; f(x-Delta\; x,\; y\; -\; Delta\; y)$
*the "semi-group structure:$g(x,\; y,\; t\_1)\; *\; g(x,\; y,\; t\_2)\; =\; g(x,\; y,\; t\_1\; +\; t\_2)$with the associated "cascade smoothing property":$L(x,\; y,\; t\_2)\; =\; g(x,\; y,\; t\_2\; -\; t\_1)\; *\; L(x,\; y,\; t\_1)$
*existence of an "infinitesimal generator" $A$:$partial\_t\; L(x,\; y,\; t)\; =\; (A\; L)(x,\; y,\; t)$
*"non-creation of local extrema" (zero-crossings) in one dimension,
*"non-enhancement of local extrema" in any number of dimensions:$partial\_t\; L(x,\; y,\; t)\; leq\; 0$ at spatial maxima and $partial\_t\; L(x,\; y,\; t)\; geq\; 0$ at spatial minima,
*"rotational symmetry":$g(x,\; y,\; t)\; =\; h(x^2+y^2,\; t)$ for some function $h$,
*"scale invariance:$hat\{g\}(omega\_x,\; omega\_y,\; t)\; =\; hat\{h\}(frac\{omega\_x\}\{varphi(t)\},\; frac\{omega\_x\}\{varphi(t)\})$for some functions $varphi$ and $hat\{h\}$ where $hat\{g\}$ denotes the Fourier transform of $g$,
*"positivity"::$g(x,\; y,\; t)\; geq\; 0$,
*"normalization"::$int\_\{x=-infty\}^\{infty\}\; int\_\{y=-infty\}^\{infty\}\; g(x,\; y,\; t)\; ,\; dx\; ,\; dy\; =\; 1$.

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. $g(x,\; y,\; t)\; =\; g(x,\; t)\; ,\; g(y,\; t)$. 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