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:

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.^[1] For continuous signals, it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels:

• the Gaussian kernel :${\displaystyle g(x,t)={\frac {1}{\sqrt {2\pi t}}}\exp({-x^{2}/2t})}$ where ${\displaystyle t>0}$,
• truncated exponential kernels (filters with one real pole in the s-plane):

${\displaystyle h(x)=\exp({-ax})}$ if ${\displaystyle x\geq 0}$ and 0 otherwise where ${\displaystyle a>0}$

${\displaystyle h(x)=\exp({bx})}$ if ${\displaystyle x\leq 0}$ and 0 otherwise where ${\displaystyle b>0}$,

• 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

${\displaystyle T(n,t)=I_{n}(\alpha t)}$ where ${\displaystyle \alpha ,t>0}$ where ${\displaystyle I_{n}}$ are the modified Bessel functions of integer order,

• generalized binomial kernels corresponding to linear smoothing of the form

${\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x-1)}$ where ${\displaystyle p,q>0}$

${\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x+1)}$ where ${\displaystyle p,q>0}$,

• first-order recursive filters corresponding to linear smoothing of the form

${\displaystyle f_{out}(x)=f_{in}(x)+\alpha f_{out}(x-1)}$ where ${\displaystyle \alpha >0}$

${\displaystyle f_{out}(x)=f_{in}(x)+\beta f_{out}(x+1)}$ where ${\displaystyle \beta >0}$,

• the one-sided Poisson kernel

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

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

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 ^[2][3] 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.^[4][5]

• Scale space
• Scale space implementation
• Scale-space segmentation