Hutchinson metric

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." [ [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 spaces. 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]