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

[math]\displaystyle{ h(x)= \exp({-a x}) }[/math] if [math]\displaystyle{ x \geq 0 }[/math] and 0 otherwise where [math]\displaystyle{ a \gt 0 }[/math]

[math]\displaystyle{ h(x)= \exp({b x}) }[/math] if [math]\displaystyle{ x \leq 0 }[/math] and 0 otherwise where [math]\displaystyle{ b \gt 0 }[/math],

• 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

[math]\displaystyle{ T(n, t) = I_n(\alpha t) }[/math] where [math]\displaystyle{ \alpha, t \gt 0 }[/math] where [math]\displaystyle{ I_n }[/math] are the modified Bessel functions of integer order,

• generalized binomial kernels corresponding to linear smoothing of the form

[math]\displaystyle{ f_{out}(x) = p f_{in}(x) + q f_{in}(x-1) }[/math] where [math]\displaystyle{ p, q \gt 0 }[/math]

[math]\displaystyle{ f_{out}(x) = p f_{in}(x) + q f_{in}(x+1) }[/math] where [math]\displaystyle{ p, q \gt 0 }[/math],

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

[math]\displaystyle{ f_{out}(x) = f_{in}(x) + \alpha f_{out}(x-1) }[/math] where [math]\displaystyle{ \alpha \gt 0 }[/math]

[math]\displaystyle{ f_{out}(x) = f_{in}(x) + \beta f_{out}(x+1) }[/math] where [math]\displaystyle{ \beta \gt 0 }[/math],

• the one-sided Poisson kernel

[math]\displaystyle{ p(n, t) = e^{-t} \frac{t^n}{n!} }[/math] for [math]\displaystyle{ n \geq 0 }[/math] where [math]\displaystyle{ t\geq0 }[/math]

[math]\displaystyle{ p(n, t) = e^{-t} \frac{t^{-n}}{(-n)!} }[/math] for [math]\displaystyle{ n \leq 0 }[/math] where [math]\displaystyle{ t\geq0 }[/math].

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.

See also

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