Abstract
We discuss two frequently calculated fractal dimensions, the capacity and information dimension and present efficient methods for their computation for sets embedded in ℝn. In particular we show how Monte Carlo calculation of areas and volumes can be used to compute the capacity using fewer box counting operations than straightforward box counting, and we discuss an efficient implementation of the method using a very fast one-dimensional sorting algorithm. Sets embedded in ℝ2 and ℝ3 are mapped to [0,1] using a folding map φ with the property that n·dim(φ(X)) = dim(X) where X c ℝn and dim is either the Hausdorff or the capacity dimension. Thus the problem of calculating the capacity dimension or the information dimension (in the case it coincides with the Hausdorff dimension) is reduced to computing these quantities for φ(X) in [0,1].
Partially supported by NSF grant DMS-8603703.
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Hunt, F., Sullivan, F. (1989). Methods of computing fractal dimensions. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086754
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DOI: https://doi.org/10.1007/BFb0086754
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