Multi-scale approaches explained

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.^[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 :

g(x, t) = (1)/(\sqrt{ 2 \pi t) } \exp where t > 0 ,

• truncated exponential kernels (filters with one real pole in the s-plane):

h(x) = \exp({ - a x}) if x \geq 0 and 0 otherwise where a > 0 h(x) = \exp({b x}) 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) = p f _{in} (x) + q f _{in} (x -1 ) where p, q > 0 f _{out} (x) = p f _{in} (x) + q f _{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} (t ^{n})/(n!) for n \geq 0 where t\geq 0 p(n, t) = e ^{-t} (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 ^{[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]}

See also

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

References

1. http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.
2. http://www.dicklyon.com/tech/Scans/ICASSP87_ScaleSpace-Lyon.pdf Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987.
3. http://www.nada.kth.se/cvap/abstracts/cvap189.html Lindeberg, T. and Fagerstrom, F.: Scale-space with causal time direction, Proc. 4th European Conference on Computer Vision, Cambridge, England, April 1996. Springer-Verlag LNCS Vol 1064, pages 229--240.
4. http://citeseer.ist.psu.edu/young95recursive.html Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139-151.
5. http://citeseer.ist.psu.edu/deriche93recursively.html Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.