Rényi dimension and Gaussian filtering. II.

zbMATH Open1153.28006arXivmath/0701795MaRDI QIDQ958355

Publication date: 3 December 2008

Published in: The New York Journal of Mathematics (Search for Journal in Brave)

Abstract: We consider convolving a Gaussian of a varying scale epsilon against a Borel measure mu on Euclidean delta-dimensional space. The Lq norm of the result is differentiable in epsilon. We calculate this derivative and show how the upper order of its growth relates to its lower Renyi dimension. We assume q is strictly between 1 and infty and that mu is finite with compact support. Consider choosing a sequence epsilon_n of scales for the Gaussians. The differences between the usual Lq norms at adjacent scales can be made to grow more slowly than any positive power of n by setting the epsilon_n by a power rule. The correct exponent in the power rule is determined by the lower Renyi dimension. We calculate and find bounds on the derivative of the Gaussian kernel versions of the correlation integral. We show that a Gaussian Kernel version of the Renyi entropy sum in continuous.

Full work available at URL: https://arxiv.org/abs/math/0701795

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zbMATH Keywords

regular variation Rényi dimension correlation dimension Gaussian kernel Laplacian pyramid asymptotic indices generalized fractal dimension

Mathematics Subject Classification ID

Fractals (28A80) Hausdorff and packing measures (28A78)