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{{Expert-subject|Mathematics|date=November 2008}} | |||
[[Image:Julia set (indigo).png|thumb|A [[Julia set]], a fractal related to the Mandelbrot set]] | |||
[[Image:Animated fractal mountain.gif|right|thumb|200px|A fractal that models the surface of a mountain (animation)]] | |||
In [[mathematics]], the '''Hutchinson metric''' is a function which [[metric (mathematics)|measures]] "the discrepancy between two [[image]]s for use in [[fractal]] [[image processing]]" and "can also be applied to describe the similarity between [[DNA]] sequences expressed as real or complex [[genomic]] signals."<ref> [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1355938 Efficient computation of the Hutchinson metric between digitized images] abstract</ref><ref> [http://isis.pub.ro/iafa2003/files/3-5.pdf HUTCHINSON METRIC IN FRACTAL DNA ANALYSIS -- A NEURAL NETWORK APPROACH]</ref> | |||
==Formal definition== | |||
Consider only nonempty, [[compact space|compact]], and finite [[metric space]]s. For a space <math>X \,</math>, let <math>P(X) \,</math> denote the space of Borel probability measures on <math>X \, </math>, with | |||
:<math>\delta : X \rightarrow P(X) \,</math> | |||
the embedding associating to <math>x \in X</math> the point measure <math>\delta_x \,</math>. The support <math>|\mu| \,</math> of a measure in P(X) is the smallest closed subset of measure 1. | |||
If | |||
:<math>f : X_1 \rightarrow X_2 \,</math> | |||
is [[Borel measure|Borel measurable]] then the induced map | |||
:<math>f_* : P(X_1) \rightarrow P(X_2) \,</math> | |||
associates to <math>\mu \,</math> the measure <math> f_*(\mu) \,</math> defined by | |||
:<math>f_*(\mu)(B)= \mu(f^{-1}(B)) \,</math> | |||
for all <math>B \, </math> Borel in <math>X_2 \, </math>. | |||
Then the '''Hutchinson metric''' is given by | |||
:<math>d(\mu_1,\mu_2)=\sup \left \lbrace \int u(x) \, \mu_1(dx) - \int u(x) \, \mu_2(dx) \right \rbrace</math> | |||
where the <math>\sup</math> is taken over all real-valued functions ''u'' with [[Lipschitz constant]] | |||
<math>\le 1 \,.</math> | |||
Then <math>\delta \, </math> is an [[isometric embedding]] of <math>X \, </math> into <math>P(X) \, </math>, and if | |||
:<math>f : X_1 \rightarrow X_2 \, </math> | |||
is Lipschitz then | |||
:<math>f_* : P(X_1) \rightarrow P(X_2) \, </math> | |||
is Lipschitz with the same Lipschitz constant.<ref> [http://links.jstor.org/sici?sici=0002-9947%28199903%29351%3A3%3C1203%3AIMFSDS%3E2.0.CO%3B2-L Invariant Measures for Set-Valued Dynamical Systems Walter Miller; Ethan Akin Transactions of the American Mathematical Society, Vol. 351, No. 3. (Mar., 1999), pp. 1203-1225] </ref> | |||
==See also== | |||
*[[Acoustic metric]] | |||
*[[Apophysis (software)]] | |||
*[[Complete metric]] | |||
*[[Fractal compression|Fractal image compression]] | |||
*[[Image differencing]] | |||
*[[Metric tensor]] | |||
*[[Multifractal system]] | |||
==Sources and notes== | |||
{{Reflist}} | |||
==Further reading== | |||
*[http://ieeexplore.ieee.org/iel5/83/29774/01355938.pdf Efficient Computation of the Hutchinson Metric Between Digitized Images] | |||
[[Category:Metric geometry]] | |||
[[Category:Topology]] |
Revision as of 01:54, 26 January 2014
In mathematics, the Hutchinson metric is a function which measures "the discrepancy between two images for use in fractal image processing" and "can also be applied to describe the similarity between DNA sequences expressed as real or complex genomic signals."[1][2]
Formal definition
Consider only nonempty, compact, and finite metric spaces. For a space , let denote the space of Borel probability measures on , with
the embedding associating to the point measure . The support of a measure in P(X) is the smallest closed subset of measure 1.
If
is Borel measurable then the induced map
associates to the measure defined by
Then the Hutchinson metric is given by
where the is taken over all real-valued functions u with Lipschitz constant
Then is an isometric embedding of into , and if
is Lipschitz then
is Lipschitz with the same Lipschitz constant.[3]
See also
- Acoustic metric
- Apophysis (software)
- Complete metric
- Fractal image compression
- Image differencing
- Metric tensor
- Multifractal system
Sources and notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Further reading
- ↑ Efficient computation of the Hutchinson metric between digitized images abstract
- ↑ HUTCHINSON METRIC IN FRACTAL DNA ANALYSIS -- A NEURAL NETWORK APPROACH
- ↑ Invariant Measures for Set-Valued Dynamical Systems Walter Miller; Ethan Akin Transactions of the American Mathematical Society, Vol. 351, No. 3. (Mar., 1999), pp. 1203-1225