TY - INPR A1 - Kreitmeier, Wolfgang T1 - Optimal quantization for uniform distributions on Cantor-like sets N2 - In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar $N-$adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first $N$ quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed. KW - Maßtheorie KW - Quantisierung KW - Iteriertes Funktionensystem KW - Fraktale Dimension KW - optimal quantization KW - quantization dimension KW - quantization coefficient KW - self-similar probabilities Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-12449 ER - TY - INPR A1 - Kreitmeier, Wolfgang A1 - Linder, Tamas T1 - High-Resolution Scalar Quantization with Rényi Entropy Constraint N2 - We consider optimal scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha$. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for $\alpha=0$ (fixed-rate quantization) and $\alpha=1$ (entropy-constrained quantization). These results have recently been extended to quantization with R\'enyi entropy constraint of order $\alpha \ge r+1$. Here we consider the more challenging case $\alpha\in [-\infty,0)\cup (0,1)$ and for a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive. KW - Maßtheorie KW - Quantisierung KW - Entropie KW - Companding KW - high-resolution asymptotics KW - optimal quantization KW - Rényi entropy Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-23787 N1 - This is a preprint of an article accepted for publication in the IEEE Transactions on Information Theory Journal, ISSN: 0018-9448. The original publication is available at http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18 ER - TY - INPR A1 - Kreitmeier, Wolfgang T1 - Optimal quantization for the one-dimensional uniform distribution with Rényi -α-entropy constraints N2 - We establish the optimal quantization problem for probabilities under constrained Rényi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases α = 0 (restricted codebook size) and α = 1 (restricted Shannon entropy). KW - Maßtheorie KW - Quantisierung KW - Entropie KW - optimal quantization KW - uniform distribution KW - Rényi-α-entropy Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-16983 ER - TY - INPR A1 - Kreitmeier, Wolfgang T1 - Optimal quantization of probabilities concentrated on small balls N2 - We consider probability distributions which are uniformly distributed on a disjoint union of balls with equal radius. For small enough radius the optimal quantization error is calculated explicitly in terms of the ball centroids. We apply the results to special self-similar measures. KW - Maßtheorie KW - Quantisierung KW - Iteriertes Funktionensystem KW - Schwerpunkt KW - optimal quantization KW - centroid KW - self-similar probabilities Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-12010 ER -