# Multi-scale approaches

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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-scale approaches' in the areas of computer vision, image processing and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches:

## Scale-space theory for one-dimensional signals

For one-dimensional signals, there exists quite a well-developed theory for continuous and discrete kernels that guarantee that new local extrema or zero-crossings cannot be created by a convolution operation . For continuous signals, it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels:

• the Gaussian kernel  : $g(x, t) = \frac{1}{\sqrt{2 \pi t}} \exp({-x^2/2 t})$ where t > 0,
• truncated exponential kernels (filters with one real pole in the s-plane):
h(x) = exp( − ax) if $x \geq 0$ and 0 otherwise where a > 0
h(x) = exp(bx) if $x \leq 0$ and 0 otherwise where b > 0,
• translations,
• rescalings.

For discrete signals, we can, up to trivial translations and rescalings, decompose any discrete scale-space kernel into the following primitive operations:

• the discrete Gaussian kernel
T(n,t) = Int) where α,t > 0 where In are the modified Bessel functions of integer order,
• generalized binomial kernels corresponding to linear smoothing of the form
fout(x) = pfin(x) + qfin(x − 1) where p,q > 0
fout(x) = pfin(x) + qfin(x + 1) where p,q > 0,
• first-order recursive filters corresponding to linear smoothing of the form
fout(x) = fin(x) + αfout(x − 1) where α > 0
fout(x) = fin(x) + βfout(x + 1) where β > 0,
• the one-sided Poisson kernel $p(n, t) = e^{-t} \frac{t^n}{n!}$ for $n \geq 0$ where $t\geq0$ $p(n, t) = e^{-t} \frac{t^{-n}}{(-n)!}$ for $n \leq 0$ where $t\geq0$.

From this classification, it is apparent that it we require a continuous semi-group structure, there are only three classes of scale-space kernels with a continuous scale parameter; the Gaussian kernel which forms the scale-space of continuous signals, the discrete Gaussian kernel which forms the scale-space of discrete signals and the time-causal Poisson kernel that forms a temporal scale-space over discrete time. If we on the other hand sacrifice the continuous semi-group structure, there are more options:

For discrete signals, the use of generalized binomial kernels provides a formal basis for defining the smoothing operation in a pyramid. For temporal data, the one-sided truncated exponential kernels and the first-order recursive filters provide a way to define time-causal scale-spaces  that allow for efficient numerical implementation and respect causality over time without access to the future. The first-order recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scale-space properties .

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