1. Scale-space theory for one-dimensional signals

2. See also

3. References

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 : 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. 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

• Ma Wan Tung Wan Beach
• Mawarannahr
• Mawashi
• Mawddach
• Mawdryn Undead
• Mawdudi
• Mawdudi, Abu'l-A'la
• Mawenzi
• Mawile
• Mawla
• Maw lam
• Mawlamyine
• Mawlawiyah
• Mawlawiyyah
• Mawlid
• Maw lum
• Mawlānā
• Mawon
• Maw Pete
• Mawr
• Mawsil
• Mawson, Australian Capital Territory
• Mawson (disambiguation)
• Mawson Lakes, South Australia
• Mawson Peak