 Multiscale approaches

The scalespace representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scalespace axioms, which make it into a special form of multiscale representation. There are, however, also other types of 'multiscale 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:
Scalespace theory for onedimensional signals
For onedimensional signals, there exists quite a welldeveloped theory for continuous and discrete kernels that guarantee that new local extrema or zerocrossings cannot be created by a convolution operation ^{[1]}. For continuous signals, it holds that all scalespace kernels can be decomposed into the following sets of primitive smoothing kernels:
 the Gaussian kernel : where t > 0,
 truncated exponential kernels (filters with one real pole in the splane):
 h(x) = exp( − ax) if and 0 otherwise where a > 0
 h(x) = exp(bx) if and 0 otherwise where b > 0,
 translations,
 rescalings.
For discrete signals, we can, up to trivial translations and rescalings, decompose any discrete scalespace 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,
 firstorder 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 onesided Poisson kernel
 for where
 for where .
From this classification, it is apparent that it we require a continuous semigroup structure, there are only three classes of scalespace kernels with a continuous scale parameter; the Gaussian kernel which forms the scalespace of continuous signals, the discrete Gaussian kernel which forms the scalespace of discrete signals and the timecausal Poisson kernel that forms a temporal scalespace over discrete time. If we on the other hand sacrifice the continuous semigroup 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 onesided truncated exponential kernels and the firstorder recursive filters provide a way to define timecausal scalespaces ^{[2]}^{[3]} that allow for efficient numerical implementation and respect causality over time without access to the future. The firstorder recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scalespace properties ^{[4]}^{[5]}.
See also
References
 ^ Lindeberg, T., "Scalespace for discrete signals," PAMI(12), No. 3, March 1990, pp. 234254.
 ^ Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987.
 ^ Lindeberg, T. and Fagerstrom, F.: Scalespace with causal time direction, Proc. 4th European Conference on Computer Vision, Cambridge, England, april 1996. SpringerVerlag LNCS Vol 1064, pages 229240.
 ^ Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139151.
 ^ Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.
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