- Hutchinson metric
In

mathematics , the**Hutchinson metric**is a function which measures "the discrepancy between twoimage s for use infractal image processing " and "can also be applied to describe the similarity betweenDNA sequences expressed as real or complexgenomic signals." [*[*] [*http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1355938 Efficient computation of the Hutchinson metric between digitized images*] abstract*[*]*http://isis.pub.ro/iafa2003/files/3-5.pdf HUTCHINSON METRIC IN FRACTAL DNA ANALYSIS -- A NEURAL NETWORK APPROACH*]**Formal definition**Consider only nonempty, compact, and finite

metric space s. For a space $X\; ,$, let $P(X)\; ,$ denote the space of Borel probability measures on $X\; ,$, with:$delta\; :\; X\; ightarrow\; P(X)\; ,$

the embedding associating to $x\; in\; X$ the point measure $delta\_x\; ,$. The support $|mu|\; ,$ of a measure in P(X) is the smallest closed subset of measure 1.

If

:$f\; :\; X\_1\; ightarrow\; X\_2\; ,$

is Borel measurable then the induced map

:$f\_*\; :\; P(X\_1)\; ightarrow\; P(X\_2)\; ,$

associates to $mu\; ,$ the measure $f\_*(mu)\; ,$ defined by

:$f\_*(mu)(B)=\; mu(f^\{-1\}(B))\; ,$

for all $B\; ,$ Borel in $X\_2\; ,$.

Then the

**Hutchinson metric**is given by:$d(mu\_1,mu\_2)=sup\; left\; lbrace\; int\; u(x)\; ,\; mu\_1(dx)\; -\; int\; u(x)\; ,\; mu\_2(dx)\; ight\; brace$

where the $sup$ is taken over all real-valued functions "u" with

Lipschitz constant $le\; 1\; ,.$Then $delta\; ,$ is an

isometric embedding of $X\; ,$ into $P(X)\; ,$, and if:$f\; :\; X\_1\; ightarrow\; X\_2\; ,$

is Lipschitz then

:$f\_*\; :\; P(X\_1)\; ightarrow\; P(X\_2)\; ,$

is Lipschitz with the same Lipschitz constant. [

*[*]*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*]**ee also***

Acoustic metric

*Apophysis (software)

*Complete metric

*Fractal image compression

*Image differencing

*Metric tensor

*Multifractal system **ources and notes****Further reading*** [

*http://ieeexplore.ieee.org/iel5/83/29774/01355938.pdf Efficient Computation of the Hutchinson Metric Between Digitized Images*]

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