Multi-scale approaches
# 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 :$g(x,t)={\frac {1}{\sqrt {2\pi t}}}\exp({-x^{2}/2t})$ 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

       $T(n,t)=I_{n}(\alpha t)$ where $\alpha ,t>0$ where $I_{n}$ are the modified Bessel functions of integer order,
* generalized binomial kernels corresponding to linear smoothing of the form

    $f_{out}(x)=pf_{in}(x)+qf_{in}(x-1)$ where $p,q>0$
    $f_{out}(x)=pf_{in}(x)+qf_{in}(x+1)$ where $p,q>0$,
* first-order recursive filters corresponding to linear smoothing of the form

    $f_{out}(x)=f_{in}(x)+\alpha f_{out}(x-1)$ where $\alpha >0$
    $f_{out}(x)=f_{in}(x)+\beta f_{out}(x+1)$ where $\beta >0$,
* the one-sided Poisson kernel

    $p(n,t)=e^{-t}{\frac {t^{n}}{n!}}$ for $n\geq 0$ where $t\geq 0$
    $p(n,t)=e^{-t}{\frac {t^{-n}}{(-n)!}}$ for $n\leq 0$ where $t\geq 0$.

From this classification, it is apparent that 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.

