New York Journal of Mathematics
Volume 14 (2008) 577-599


Terry A. Loring

Rényi dimension and Gaussian filtering. II

Published: October 23, 2008
Keywords: Asymptotic indices, Rényi dimension, generalized fractal dimension, regular variation, Laplacian pyramid, correlation dimension, gaussian kernel
Subject: 28A80, 28A78

We consider convolving a Gaussian of a varying scale ε against a Borel measure μ on Euclidean δ-dimensional space. The Lq norm of the result is differentiable in ε. We calculate this derivative and show how the upper order of its growth relates to the lower Rényi dimension of μ. We assume q is strictly between 1 and ∞ and that μ is finite with compact support.

Consider choosing a sequence εn of scales for the Gaussians

gε(x) =εe-(|x|/ε)2.
Let ∥f∥q denote the Lq norm for Lebesgue measure. The differences
|∥gεn+1∗μ∥q - ∥gεn∗μ∥q|
between the norms at adjacent scales εn and εn-1 can be made to grow more slowly than any positive power of n by setting the εn by a power rule. The correct exponent in the power rule is determined by the lower Rényi 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 Rényi entropy sum is continuous.


This work was supported in part by DARPA Contract N00014-03-1-0900.

Author information

Current Address: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA.