Scale-space axioms

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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 ^[1]^[2]^[3]^[4]^[5]^[6]^[7]^[8]^[9].

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.