Some recent developments in quantization of fractal measures

TL;DR

This paper reviews recent advances in the quantization of fractal measures, highlighting methods, progress, and open problems in the field, especially for self-affine, Markov-type, and multiscale Moran measures.

Contribution

It introduces a three-step estimation procedure for quantization errors and surveys recent developments applying this method to various fractal measures.

Findings

Effective estimation procedure for quantization errors.

Progress in quantization of self-affine and Markov-type measures.

Identification of open problems in fractal measure quantization.

Abstract

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.