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'' :$g(x, t) = \frac \exp()$ where $t > 0$,
* ''truncated exponential'' kernels (filters with one real pole in the ''s''-plane):
::$h(x)= \exp()$ if $x \geq 0$ and 0 otherwise where $a > 0$
::$h(x)= \exp()$ 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
:$f_(x) = p f_(x) + q f_(x-1)$ where $p, q > 0$
:$f_(x) = p f_(x) + q f_(x+1)$ where $p, q > 0$,
* ''first-order recursive filters'' corresponding to linear smoothing of the form
:$f_(x) = f_(x) + \alpha f_(x-1)$ where $\alpha > 0$
:$f_(x) = f_(x) + \beta f_(x+1)$ where $\beta > 0$,
* the one-sided ''Poisson kernel''
:$p(n, t) = e^ \frac$ for $n \geq 0$ where $t\geq0$
:$p(n, t) = e^ \frac{(-n)!}$ for $n \leq 0$ where $t\geq0$.

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.Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.
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See also

* Scale space
* Scale space implementation
* Scale-space segmentation

References

Image processing
Computer vision