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, and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches: 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 : where , truncated exponential kernels (filters with one real pole in the s-plane): if and 0 otherwise where if and 0 otherwise where , 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 where where are the modified Bessel functions of integer order, generalized binomial kernels corresponding to linear smoothing of the form where where , first-order recursive filters corresponding to linear smoothing of the form where where , the one-sided Poisson kernel for where for where . 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.