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(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 hwhere 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 fwhere 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] [ [ 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.] ] [ [ 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.] ] [ [ Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234–254.] ] [ [ Lindeberg, Tony, Scale-Space Theory in Computer Vision, Kluwer, 1994] ,] [ [ 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.] ] [ [ 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.] [ [ 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


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