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

  h(x) = exp( − ax) if $x \geq 0$ and 0 otherwise where a > 0

  h(x) = exp(bx) 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(αt) where α,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) = pf_in(x) + qf_in(x − 1) where p,q > 0

f_out(x) = pf_in(x) + qf_in(x + 1) where p,q > 0,

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

f_out(x) = f_in(x) + αf_out(x − 1) where α > 0

f_out(x) = f_in(x) + βf_out(x + 1) where β > 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 it 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. ^ Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.
2. ^ Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987.
3. ^ 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. ^ Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139-151.
5. ^ Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.