Hutchinson metric explained

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]}

Formal definition

Consider only nonempty, compact, and finite metric spaces. For such a space

X , let P(X) denote the space of Borel probability measures on X , with\delta : X → 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} → X _{2} is Borel measurable then the induced mapf _{*} : P(X _{1)} → P(X _{2)}

associates to

\mu the measure f _{*(\mu)} defined byf _{*(\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} → X _{2} is Lipschitz then f _{*} : P(X _{1)} → 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

Notes and References

1. Drakopoulos . V. . Nikolaou . N. P. . Efficient computation of the Hutchinson metric between digitized images . IEEE Transactions on Image Processing . 13 . 12 . 1581–1588 . December 2004 . 10.1109/tip.2004.837550. 15575153 .
2. http://isis.pub.ro/iafa2003/files/3-5.pdf Hutchinson Metric in Fractal DNA Analysis -- a Neural Network Approach
3. https://www.jstor.org/stable/117923 "Invariant Measures for Set-Valued Dynamical Systems"