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".[1][2]

A Julia set, a fractal related to the Mandelbrot set

A fractal that models the surface of a mountain (animation)

Formal definition

Consider only nonempty, compact, and finite metric spaces. For a space ${\displaystyle X}$, let ${\displaystyle P(X)}$ denote the space of Borel probability measures on ${\displaystyle X}$, with

${\displaystyle \delta :X\rightarrow P(X)}$

the embedding associating to ${\displaystyle x\in X}$ the point measure ${\displaystyle \delta _{x}}$. The support ${\displaystyle |\mu |}$ of a measure in P(X) is the smallest closed subset of measure 1.

If

${\displaystyle f:X_{1}\rightarrow X_{2}}$

is Borel measurable then the induced map

${\displaystyle f_{*}:P(X_{1})\rightarrow P(X_{2})}$

associates to ${\displaystyle \mu }$ the measure ${\displaystyle f_{*}(\mu )}$ defined by

${\displaystyle f_{*}(\mu )(B)=\mu (f^{-1}(B))}$

for all ${\displaystyle B}$ Borel in ${\displaystyle X_{2}}$.

Then the Hutchinson metric is given by

${\displaystyle d(\mu _{1},\mu _{2})=\sup \left\lbrace \int u(x)\,\mu _{1}(dx)-\int u(x)\,\mu _{2}(dx)\right\rbrace }$

where the ${\displaystyle \sup }$ is taken over all real-valued functions u with Lipschitz constant ${\displaystyle \leq 1\,.}$

Then ${\displaystyle \delta }$ is an isometric embedding of ${\displaystyle X}$ into ${\displaystyle P(X)}$, and if

${\displaystyle f:X_{1}\rightarrow X_{2}}$

is Lipschitz then

${\displaystyle f_{*}:P(X_{1})\rightarrow P(X_{2})}$

is Lipschitz with the same Lipschitz constant.[3]

See also

• Acoustic metric
• Apophysis (software)
• Complete metric
• Fractal image compression
• Image differencing
• Metric tensor
• Multifractal system

Sources and notes

1. Drakopoulos, V.; Nikolaou, N. P. (December 2004). "Efficient computation of the Hutchinson metric between digitized images". IEEE Transactions on Image Processing. 13 (12): 1581–1588. doi:10.1109/tip.2004.837550. PMID 15575153.
2. Hutchinson Metric in Fractal DNA Analysis -- a Neural Network Approach Archived August 18, 2011, at the Wayback Machine
3. "Invariant Measures for Set-Valued Dynamical Systems" Walter Miller; Ethan Akin Transactions of the American Mathematical Society, Vol. 351, No. 3. (March 1999), pp. 1203–1225]