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  <doc>
    <id>95</id>
    <completedYear>2008</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2008-07-14</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal quantization for uniform distributions on Cantor-like sets</title>
    <abstract language="deu">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-12449</identifier>
    <identifier type="opus3-id">1244</identifier>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Acta Applicandae Mathematicae, ISSN (Print) 0167-8019 ISSN (Online) 1572-9036 Copyright (c) by Springer. The original publication is available at www.springerlink.com. digital object identifier DOI: 10.1007/s10440-008-9278-3</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Iteriertes Funktionensystem</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Fraktale Dimension</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantization dimension</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantization coefficient</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>self-similar probabilities</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28A80">Fractals [See also 37Fxx]</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/95/quant_cant_02.pdf</file>
  </doc>
  <doc>
    <id>140</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2011-04-18</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal vector quantization in terms of Wasserstein distance</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-22502</identifier>
    <identifier type="opus3-id">2250</identifier>
    <note>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.</note>
    <enrichment key="SourceTitle">Journal of Multivariate Analysis. ISSN 0047-259X</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Transporttheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Wasserstein distance</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization error</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>codecell convexity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>R\'enyi-$\alpha$-entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="60B10">Convergence of probability measures</collection>
    <collection role="msc" number="60E05">Distributions: general theory</collection>
    <collection role="msc" number="62E17">Approximations to distributions (nonasymptotic)</collection>
    <collection role="msc" number="68P30">Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/140/quant_wass_rev1.pdf</file>
  </doc>
  <doc>
    <id>83</id>
    <completedYear>2005</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2007-10-31</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal Quantization for Dyadic Homogeneous Cantor Distributions</title>
    <abstract language="eng">For a large class of dyadic homogeneous Cantor distributions in \mathbb{R}, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to \frac{1}{3}. For these distributions we calculate the quantization errors explicitly.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-3845</identifier>
    <identifier type="opus3-id">384</identifier>
    <note>Die Endfassung des Artikels kann beim Verfasser angefordert werden. Kontaktinformation: opus@uni-passau.de</note>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Mathematische Nachrichten, Print ISSN:0025-584X, Online ISSN:1522-2616, Copyright © by Wiley http://www3.interscience.wiley.com/journal/60500208/home. The digital object identifier (DOI) of the definitive article is 10.1002/mana.200510680.</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Fraktale Dimension</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Iteriertes Funktionensystem</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Cantor-Menge</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Dimension</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Maß</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>homogeneous Cantor measures</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Quantization dimension</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Quantization coefficient</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28A80">Fractals [See also 37Fxx]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/83/dyadic_cantor.pdf</file>
  </doc>
  <doc>
    <id>145</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2011-07-08</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">High-Resolution Scalar Quantization with Rényi Entropy Constraint</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-23787</identifier>
    <identifier type="opus3-id">2378</identifier>
    <note>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</note>
    <enrichment key="SourceTitle">IEEE Transactions on Information Theory Journal, ISSN: 0018-9448</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <author>Tamas Linder</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Companding</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>high-resolution asymptotics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28D20">Entropy and other invariants</collection>
    <collection role="msc" number="41A46">Approximation by arbitrary nonlinear expressions; widths and entropy</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/145/renyi_scalar_final_doublecolumn.pdf</file>
  </doc>
  <doc>
    <id>119</id>
    <completedYear>2009</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2010-02-10</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Hausdorff measure of uniform self-similar fractals</title>
    <abstract language="eng">Let &lt;i&gt;d&lt;/i&gt; &amp;#8805; 1 be an integer and &lt;i&gt;E&lt;/i&gt; a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R&lt;sup&gt;d&lt;/sup&gt;. Denote by &lt;i&gt;D&lt;/i&gt; the Hausdorff dimension, by &lt;i&gt;H&lt;/i&gt;&lt;sup&gt;D&lt;/sup&gt;&lt;i&gt;(E)&lt;/i&gt; the Hausdorff measure and by diam &lt;i&gt;(E)&lt;/i&gt; the diameter of &lt;i&gt;E&lt;/i&gt;. If the UIFS is parametrised by its contracting factor &lt;i&gt;c&lt;/i&gt;, while the set &amp;omega; of fixed points of the UIFS does not depend on &lt;i&gt;c&lt;/i&gt;, we will show the existence of a positive constant depending only on &amp;omega;, such that the Hausdorff dimension is smaller than one and &lt;i&gt;H&lt;/i&gt;&lt;sup&gt;D&lt;/sup&gt; = &lt;i&gt;(E)&lt;/i&gt; &lt;sup&gt;D&lt;/sup&gt; if &lt;i&gt;c&lt;/i&gt; is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where &amp;omega; depends on &lt;i&gt;c&lt;/i&gt; and &lt;i&gt;H&lt;/i&gt;&lt;sup&gt;D&lt;/sup&gt; &lt; diam&lt;i&gt;(E)&lt;/i&gt;&lt;sup&gt;D&lt;/sup&gt;, if &lt;i&gt;c&lt;/i&gt; is small enough.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-17948</identifier>
    <identifier type="opus3-id">1794</identifier>
    <note>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</note>
    <enrichment key="SourceTitle">Analysis in Theory and Applications ISSN: 1672-4070 (print version) ISSN: 1573-8175 (electronic version)</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Iteriertes Funktionensystem</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Dimension</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Maß</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Self-similar set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hausdorff measure</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28A78">Hausdorff and packing measures</collection>
    <collection role="msc" number="28A80">Fractals [See also 37Fxx]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/119/haus_meas_v3.pdf</file>
  </doc>
  <doc>
    <id>114</id>
    <completedYear>2009</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2009-10-19</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">Optimal quantization for the one-dimensional uniform distribution with Rényi -α-entropy constraints</title>
    <abstract language="eng">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).</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-16983</identifier>
    <identifier type="opus3-id">1698</identifier>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Kybernetika ISSN 0023-5954 The original publication is available at http://www.kybernetika.cz/content.html</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung &lt;Nachrichtentechnik&gt;</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie &lt;Informationstheorie&gt;</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>uniform distribution</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi-&amp;#945;-entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="60Exx">Distribution theory [See also 62Exx, 62Hxx]</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/114/kreitmeier_optimal_quantization.pdf</file>
  </doc>
  <doc>
    <id>112</id>
    <completedYear>2009</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2009-10-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Error bounds for high-resolution quantization with Rényi - &amp;#945; - entropy constraints</title>
    <abstract language="eng">We consider the problem of optimal quantization with norm exponent r &gt; 0 for Borel probabilities on R&lt;sup&gt;d&lt;/sup&gt; under constrained Rényi-&amp;#945;-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 &amp;#945; = 0 (memory-constrained quantization) and &amp;#945; = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter &amp;#945; &amp;#8712; [1+r/d, &amp;#8734;]. For &amp;#945; &amp;#8712; [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 &amp;#945; &amp;#8712; [0, &amp;#8734;].</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-16647</identifier>
    <identifier type="opus3-id">1664</identifier>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Acta Mathematica Hungarica ISSN: 0236-5294 (print version) ISSN: 1588-2632 (electronic version) Copyright (c) by Springer. The original publication is available at www.springerlink.com, see also http://dx.doi.org/10.1007/s10474-010-9079-9</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung &lt;Nachrichtentechnik&gt;</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Vektorquantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie &lt;Informationstheorie&gt;</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Vector quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>high-resolution quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi-&amp;#945;-entropy</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>approximation of probabilities</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="60Exx">Distribution theory [See also 62Exx, 62Hxx]</collection>
    <collection role="msc" number="62E17">Approximations to distributions (nonasymptotic)</collection>
    <collection role="msc" number="68P30">Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/112/quant_renyi_v2.pdf</file>
  </doc>
  <doc>
    <id>93</id>
    <completedYear>2007</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2008-06-18</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal quantization of probabilities concentrated on small balls</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-12010</identifier>
    <identifier type="opus3-id">1201</identifier>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Forum Mathematicum, ISSN (Print) 0933-7741, ISSN (Online) 1435-5337 Copyright (c) by de Gruyter. http://www.degruyter.de/journals/forum/detailEn.cfm. The digital object identifier (DOI) of the definitive article is 10.1515/FORUM.2010.017</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Iteriertes Funktionensystem</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Schwerpunkt</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>centroid</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>self-similar probabilities</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28A80">Fractals [See also 37Fxx]</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/93/quant_ball_03.pdf</file>
  </doc>
  <doc>
    <id>86</id>
    <completedYear>2007</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2008-01-02</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-7374</identifier>
    <identifier type="opus3-id">737</identifier>
    <note>Die Endfassung des Artikels kann beim Verfasser angefordert werden. Kontaktinformation: opus@uni-passau.de</note>
    <enrichment key="SourceTitle">This is a preprint of an article accepted for publication in Journal of Mathematical Analysis and Applications, ISSN: 0022-247X. Copyright (c) by Elsevier. URL: http://www.elsevier.com/. The digital object identifier (DOI) of the definitive article is 10.1016/j.jmaa.2007.12.052</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Fraktale Dimension</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Iteriertes Funktionensystem</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Cantor-Menge</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Dimension</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Hausdorff-Maß</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Homogeneous Cantor set</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Euler characteristic</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Euler exponent</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantization dimension</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantization coefficient</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hausdorff dimension</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28A78">Hausdorff and packing measures</collection>
    <collection role="msc" number="28A80">Fractals [See also 37Fxx]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/86/kreitmeier_eul_quant.pdf</file>
  </doc>
  <doc>
    <id>160</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation>Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada</contributingCorporation>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2012-03-27</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint</title>
    <abstract language="eng">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 $\alpha$</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-26132</identifier>
    <identifier type="opus3-id">2613</identifier>
    <note>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</note>
    <enrichment key="SourceTitle">IEEE Transactions on Information Theory Journal, ISSN: 0018-9448</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <author>Tamas Linder</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Asymptotic quantization theory</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>distortion density</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>entropy density</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantizer mismatch</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi-entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28D20">Entropy and other invariants</collection>
    <collection role="msc" number="41A46">Approximation by arbitrary nonlinear expressions; widths and entropy</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/160/renyi_mismatch_final_doublecolumn.pdf</file>
  </doc>
  <doc>
    <id>193</id>
    <completedYear>2012</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2013-10-09</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Asymptotic optimality of scalar Gersho quantizers</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-27080</identifier>
    <identifier type="opus3-id">2708</identifier>
    <note>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</note>
    <enrichment key="SourceTitle">Constructive Approximation ISSN: 0176-4276 (print version) ISSN: 1432-0940 (electronic version)</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Informationstheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Signaltheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Approximation</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Kodierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>uncontrolled</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Asymptotically optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Quantization error</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Scalar quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Gersho quantizer</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>High rate quantization</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="41A29">Approximation with constraints</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/193/asym_gersho_quant_v3.pdf</file>
  </doc>
</export-example>
