@unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {Entropy Density and Mismatch in High-Rate Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-26132}, year = {2011}, abstract = {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results to general values of the entropy order parameter \$\alpha\$}, subject = {Maßtheorie}, language = {de} } @unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {High-Resolution Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-23787}, year = {2011}, abstract = {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optimal quantizers via the companding approach, and is thus constructive.}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization for the one-dimensional uniform distribution with R{\´e}nyi -α-entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16983}, year = {2009}, abstract = {We establish the optimal quantization problem for probabilities under constrained R{\´e}nyi-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).}, subject = {Maßtheorie}, language = {de} } @phdthesis{Kreitmeier2006, author = {Kreitmeier, Wolfgang}, title = {Optimale Quantisierung verallgemeinerter Cantor-Verteilungen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-913}, school = {Universit{\"a}t Passau}, year = {2006}, abstract = {F{\"u}r verallgemeinerte Cantor-Verteilungen, die im Eindimensionalen mittels klassischer Wischkonstruktion bzw. in h{\"o}heren Dimensionen {\"u}ber iterierte Funktionensysteme definiert werden, wird das Problem der optimalen Quantisierung unter bestimmten Voraussetzungen vollst{\"a}ndig gel{\"o}st. Es werden die optimalen Codeb{\"u}cher bestimmt und Formeln f{\"u}r den optimalen Quantisierungsfehler bewiesen. Im eindimensionalen Fall wird eine Existenzcharakterisierung der Quantisierungsdimension gegeben und unter bestimmten Voraussetzungen die Nichtexistenz des Quantisierungskoeffizienten gezeigt. Auch in h{\"o}heren Dimensionen wird f{\"u}r die betrachteten Verteilungen bewiesen, dass der Quantisierungskoeffizient, bei existenter Quantisierungsdimension, nicht existiert. Die gewonnenen Resultate werden auf die Gleichverteilungen von modifizierten klassischen fraktalen Mengen, wie das Sierpinski-Dreieck, die Cantormenge und den Cantor-Staub angewandt.}, subject = {Maßtheorie}, language = {de} }