The
scale space representation of a signal obtained by
Gaussian smoothing
In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).
It is a widely used effect in graphics software, ...
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
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, in particular the notion of
wavelets
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
. 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
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
'' :
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 function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
s 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
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''ยท''y'', or simply ''xy'', ...
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'' 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.
Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.
/ref>
See also
* Scale space
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
* Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges o ...
* Scale-space segmentation
References
Image processing
Computer vision