Hutchinson metric

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." [ [ Efficient computation of the Hutchinson metric between digitized images] abstract] [ [ 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.


: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 constantle 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. [ [ 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

* [ Efficient Computation of the Hutchinson Metric Between Digitized Images]

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