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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]]