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

the Gaussian kernel : $g(x, t) = \frac{1}{\sqrt{2 \pi t}} \exp({-x^2/2 t})$ 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

: $p(n, t) = e^{-t} \frac{t^n}{n!}$ for $n \geq 0$ where $t\geq0$

: $p(n, t) = e^{-t} \frac{t^{-n}}{(-n)!}$ 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 [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.][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.] 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.[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.][http://citeseer.ist.psu.edu/deriche93recursively.html Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.]

References

Category:Image processing

Category:Computer vision

Multi-scale approaches

Scale-space theory for one-dimensional signals

See also

References