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Hutchinson metric

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A Julia set, a fractal related to the Mandelbrot set
A fractal that models the surface of a mountain (animation)

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

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Consider only nonempty, compact, and finite metric spaces. For such a space , let denote the space of Borel probability measures on , with

the embedding associating to the point measure . The support of a measure in is the smallest closed subset of measure 1.

If is Borel measurable then the induced map

associates to the measure defined by

for all Borel in .

Then the Hutchinson metric is given by

where the is taken over all real-valued functions with Lipschitz constant

Then is an isometric embedding of into , and if is Lipschitz then is Lipschitz with the same Lipschitz constant.[3]

See also

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Sources and notes

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  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]