Hutchinson metric

From Wikipedia, the free encyclopedia

 This article or section is in need of attention from an expert on the subject.

WikiProject Mathematics or the Mathematics Portal may be able to help recruit one.

If a more appropriate WikiProject or portal exists, please adjust this template accordingly.
A Julia set, a fractal related to the Mandelbrot set
A Julia set, a fractal related to the Mandelbrot set
A fractal that models the surface of a mountain (animation)
A fractal that models the surface of a mountain (animation)

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]

Contents

[edit] 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 \rightarrow 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 \rightarrow X_2 \,

is Borel measurable then the induced map

f_* : P(X_1) \rightarrow 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) \right \rbrace

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 \rightarrow X_2 \,

is Lipschitz then

f_* : P(X_1) \rightarrow P(X_2) \,

is Lipschitz with the same Lipschitz constant.[3]

[edit] See also

[edit] Sources and notes

  1. ^ Efficient computation of the Hutchinson metric between digitized images abstract
  2. ^ HUTCHINSON METRIC IN FRACTAL DNA ANALYSIS -- A NEURAL NETWORK APPROACH
  3. ^ 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

[edit] Further reading