 

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 


Abstract
We consider convolving a Gaussian of a varying scale ε against
a Borel measure μ on Euclidean δdimensional space. The
L^{q} 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 ε_{n} of scales for the Gaussians
g_{ε}(x)
=ε^{δ}e^{(x/ε)2}.
Let ∥f∥_{q} denote the L^{q} norm for Lebesgue measure. The
differences
∥g_{εn+1}∗μ∥_{q}
 ∥g_{εn}∗μ∥_{q}
between the norms at adjacent scales ε_{n} and ε_{n1}
can be made to grow more slowly than any positive power of n by
setting the ε_{n} 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.


Acknowledgements
This work was supported in part by DARPA Contract N000140310900.


Author information
Current Address: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA.
http://www.math.unm.edu/~loring/

