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| 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: | | The writer's name is Christy Brookins. As a woman what she truly likes is fashion and she's been doing it for fairly a whilst. Invoicing is what I do. Some time ago she chose to reside in Alaska and her parents live close by.<br><br>my blog [http://kjhkkb.net/xe/notice/374835 free tarot readings] |
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| ==Scale-space theory for one-dimensional signals==
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| 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.<ref>[http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.]</ref> For ''continuous signals'', it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels:
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| <ul>
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| <li>
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| the ''Gaussian kernel''
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| :<math>g(x, t) = \frac{1}{\sqrt{2 \pi t}} \exp({-x^2/2 t})</math> where <math>t > 0</math>,
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| <li>
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| ''truncated exponential'' kernels (filters with one real pole in the ''s''-plane):
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| :<math>h(x)= \exp({-a x})</math> if <math>x \geq 0</math> and 0 otherwise where <math>a > 0</math>
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| :<math>h(x)= \exp({b x})</math> if <math>x \leq 0</math> and 0 otherwise where <math>b > 0</math>,
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| <li>
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| translations,
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| <li>
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| rescalings.
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| </ul>
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| For ''discrete signals'', we can, up to trivial translations and rescalings, decompose any discrete scale-space kernel into the following primitive operations:
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| <li>
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| the ''discrete Gaussian kernel''
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| :<math>T(n, t) = I_n(\alpha t) </math> where <math>\alpha, t > 0</math> where <math>I_n</math> are the modified Bessel functions of integer order,
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| <li>
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| ''generalized binomial kernels'' corresponding to linear smoothing of the form
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| :<math>f_{out}(x) = p f_{in}(x) + q f_{in}(x-1)</math> where <math>p, q > 0</math>
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| :<math>f_{out}(x) = p f_{in}(x) + q f_{in}(x+1)</math> where <math>p, q > 0</math>,
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| <li>
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| ''first-order recursive filters'' corresponding to linear smoothing of the form
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| :<math>f_{out}(x) = f_{in}(x) + \alpha f_{out}(x-1)</math> where <math>\alpha > 0</math>
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| :<math>f_{out}(x) = f_{in}(x) + \beta f_{out}(x+1)</math> where <math>\beta > 0</math>,
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| <li>the one-sided ''Poisson kernel''
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| :<math>p(n, t) = e^{-t} \frac{t^n}{n!}</math> for <math>n \geq 0</math> where <math>t\geq0</math>
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| :<math>p(n, t) = e^{-t} \frac{t^{-n}}{(-n)!}</math> for <math>n \leq 0</math> where <math>t\geq0</math>.
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| </ul>
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| 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:
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| 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'' <ref>[http://www.dicklyon.com/tech/Scans/ICASSP87_ScaleSpace-Lyon.pdf Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987.]</ref><ref>[http://www.nada.kth.se/cvap/abstracts/cvap189.html 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.]</ref> 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.<ref>[http://citeseer.ist.psu.edu/young95recursive.html Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139-151.]</ref><ref>[http://citeseer.ist.psu.edu/deriche93recursively.html Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993.]</ref>
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| ==See also==
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| * [[Scale space]]
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| * [[Scale space implementation]]
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| * [[Scale-space segmentation]]
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| ==References==
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| <references/>
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| [[Category:Image processing]]
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| [[Category:Computer vision]]
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The writer's name is Christy Brookins. As a woman what she truly likes is fashion and she's been doing it for fairly a whilst. Invoicing is what I do. Some time ago she chose to reside in Alaska and her parents live close by.
my blog free tarot readings