TY - INPR
A1 - Kreitmeier, Wolfgang
T1 - Asymptotic optimality of scalar Gersho quantizers
N2 - In his famous paper Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type - or Gersho quantizers for short - exist for non-atomic scalar distributions. As a main result we prove that Gersho quantizers are asymptotically optimal.
KW - Maßtheorie
KW - Informationstheorie
KW - Signaltheorie
KW - Approximation
KW - Kodierung
KW - Quantisierung
KW - Asymptotically optimal quantization
KW - Quantization error
KW - Scalar quantization
KW - Gersho quantizer
KW - High rate quantization
Y1 - 2012
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-27080
N1 - This is a preprint of an article accepted for publication in Constructive Approximation, ISSN: 0176-4276 (print version) ISSN: 1432-0940 (electronic version) Copyright (c) by Springer. The final publication is available at link.springer.com URL: http://dx.doi.org/10.1007/s00365-013-9214-2
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TY - INPR
A1 - Kreitmeier, Wolfgang
A1 - Linder, Tamas
T1 - Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint
N2 - Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$)mismatch results to general values of the entropy order parameter $\alpha$
KW - Maßtheorie
KW - Quantisierung
KW - Entropie
KW - Asymptotic quantization theory
KW - distortion density
KW - entropy density
KW - quantizer mismatch
KW - Rényi-entropy
Y1 - 2011
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-26132
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
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 vector quantization in terms of Wasserstein distance
N2 - The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the R\'enyi-$\alpha$-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained $(\alpha = 1)$ and memory-size constrained $(\alpha = 0)$ quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gy\"{o}rgy and Linder.
KW - Maßtheorie
KW - Transporttheorie
KW - Quantisierung
KW - Entropie
KW - Wasserstein distance
KW - optimal quantization error
KW - codecell convexity
KW - R\'enyi-$\alpha$-entropy
Y1 - 2011
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-22502
N1 - This is a preprint of an article accepted for publication in the Journal of Multivariate Analysis ISSN 0047-259X. The original publication is available at http://www.elsevier.com/. The digital object identifier (DOI) of the definitive article is 10.1016/j.jmva.2011.04.005.
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TY - INPR
A1 - Kreitmeier, Wolfgang
T1 - Hausdorff measure of uniform self-similar fractals
N2 - Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on Rd. Denote by D the Hausdorff dimension, by HD(E) the Hausdorff measure and by diam (E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and HD = (E) D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where ω depends on c and HD < diam(E)D, if c is small enough.
KW - Maßtheorie
KW - Iteriertes Funktionensystem
KW - Hausdorff-Dimension
KW - Hausdorff-Maß
KW - Self-similar set
KW - Hausdorff measure
Y1 - 2009
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-17948
N1 - This is a preprint of an article accepted for publication in Analysis in Theory and Applications ISSN: 1672-4070 (print version) ISSN: 1573-8175 (electronic version) Copyright (c) by Springer. The original publication is available at www.springerlink.com
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 - Error bounds for high-resolution quantization with Rényi - α - entropy constraints
N2 - We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on Rd under constrained Rényi-α-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d, ∞]. For α ∈ [0,1+r/d[ we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error decays exponentially fast with the entropy bound and the exact decay rate is determined for all α ∈ [0, ∞].
KW - Maßtheorie
KW - Quantisierung
KW - Vektorquantisierung
KW - Entropie
KW - Vector quantization
KW - high-resolution quantization
KW - Rényi-α-entropy
KW - approximation of probabilities
Y1 - 2009
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-16647
ER -
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
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 -
TY - INPR
A1 - Kreitmeier, Wolfgang
T1 - Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure
N2 - For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.
KW - Maßtheorie
KW - Fraktale Dimension
KW - Iteriertes Funktionensystem
KW - Cantor-Menge
KW - Hausdorff-Dimension
KW - Hausdorff-Maß
KW - Homogeneous Cantor set
KW - Euler characteristic
KW - Euler exponent
KW - quantization dimension
KW - quantization coefficient
KW - Hausdorff dimension
Y1 - 2007
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-7374
N1 - Die Endfassung des Artikels kann beim Verfasser angefordert werden. Kontaktinformation: opus@uni-passau.de
ER -