Hutchinson metric

In mathematics, the Hutchinson metric otherwise known as Kantorovich 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 such 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 ${\displaystyle P(X)}$ 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 ${\displaystyle u}$ 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

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