In mathematics, the Hutchinson metric, otherwise known as the Kantorovich metric, is a function which measures "the discrepancy between two s 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".Hutchinson Metric in Fractal DNA Analysis -- a Neural Network Approach
Formal definition 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."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]
See also Wasserstein metric Acoustic metric Apophysis (software) Complete metric Fractal image compression Image differencing Metric tensor Multifractal system
Sources and notes