TY - THES A1 - Seidl, Andreas T1 - Cylindrical Decomposition Under Application-Oriented Paradigms T1 - Zylindrische Dekomposition unter anwendungsorientierten Paradigmen N2 - Quantifier elimination (QE) is a powerful tool for problem solving. Once a problem is expressed as a formula, such a method converts it to a simpler, quantifier-free equivalent, thus solving the problem. Particularly many problems live in the domain of real numbers, which makes real QE very interesting. Among the so far implemented methods, QE by cylindrical algebraic decomposition (CAD) is the most important complete method. The aim of this thesis is to develop CAD-based algorithms, which can solve more problems in practice and/or provide more interesting information as output. An algorithm that satisfies these standards would concentrate on generic cases and postpone special and degenerated ones to be treated separately or to be abandoned completely. It would give a solution, which is locally correct for a region the user is interested in. It would give answers, which can provide much valuable information in particular for decision problems. It would combine these methods with more specialized ones, for subcases that allow for. It would exploit degrees of freedom in the algorithms by deciding to proceed in a way that promises to be efficient. It is the focus of this dissertation to treat these challenges. Algorithms described here are implemented in the computer logic system REDLOG and ship with the computer algebra system REDUCE. KW - Quantorenelimination KW - cylindrical algebraic decomposition KW - CAD KW - cylindrical subdecomposition KW - SCAD KW - quantifier elimination KW - QE Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-816 ER - TY - THES A1 - Walsh, Florian T1 - Computing the Binomial Part of Polynomial Ideals N2 - Given an ideal in a polynomial ring over a field, we present a complete algorithm to compute its binomial part. Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-15096 ER - TY - THES A1 - Graf, Simone T1 - Kamerakalibrierung mit radialer Verzeichnung – die radiale essentielle Matrix T1 - Camera Calibration with Radial Distortion - the Radial Essential Matrix N2 - In der Bildverarbeitung wird die beobachtende Kamera meist als Lochkamera modelliert: ein Modell, das zahlreiche theoretische Vorteile bietet. So kann etwa das Abbildungsverhalten als projektive Abbildung aufgefasst werden. In einem Stereokamerasystem dieses Modells stehen korrespondierende Punkte – das sind Bildpunkte desselben 3D-Punktes – in einem linearen Zusammenhang, der auch ohne Kenntnis der Kameraparameter aus beobachteten Korrespondenzen geschätzt werden kann. Für die meisten Kameras, insbesondere für solche mit Weitwinkelobjektiven, ist die Modellannahme einer Lochkamera allerdings sichtbar unzureichend. Deshalb müssen zusätzlich zur Lochkamera noch Verzeichnungsabbildungen ins Modell integriert werden. In dieser Arbeit wird gezeigt, dass bei polynomialer radialer Verzeichnung die Parameter der Projektionsabbildung die Verzeichnungsparameter bestimmen. Dieses theoretische Ergebnis fließt in Algorithmen zur Kamerakalibrierung, d.h. zur Bestimmung der Parameter eines Kameramodells, ein. Diese wurden experimentell getestet und mit bestehenden Verfahren verglichen. Weiterhin wird die radiale essentielle Matrix eingeführt, die die Beziehung von korrespondierenden Punkten im Stereokamerafall bei radialer Verzeichnung beschreibt. Es werden vier Algorithmen vorgestellt, die diese theoretische Beziehung verwerten. Sie geben an, wie aus korrespondierenden Punkten die radiale essentielle Matrix geschätzt werden kann und welche Kameraparameter daraus gewonnen werden können. Damit ist beispielsweise eine Nachkalibrierung möglich. Auch diese Verfahren wurden implementiert und evaluiert. Umgekehrt ist bei bekannter radialer essentieller Matrix eine Einschränkung des Suchraums für korrespondierende Punkte möglich, die für die Rekonstruktion benötigt werden. KW - Optische Messtechnik KW - Kalibrieren KW - Stereokamera KW - Kamera KW - Korrespondenzproblem KW - Bildverarbeitung KW - Bildkorrelatio KW - Kamerakalibrierung KW - radiale Verzeichnung KW - epipolare Einschränkung KW - camera calibration KW - radial distortion KW - epipolar constraint Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-12711 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 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. ER - TY - INPR A1 - Kreitmeier, Wolfgang T1 - Optimal Quantization for Dyadic Homogeneous Cantor Distributions N2 - 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. KW - Maßtheorie KW - Fraktale Dimension KW - Iteriertes Funktionensystem KW - Cantor-Menge KW - Hausdorff-Dimension KW - Hausdorff-Maß KW - Quantization KW - homogeneous Cantor measures KW - Quantization dimension KW - Quantization coefficient Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-3845 N1 - Die Endfassung des Artikels kann beim Verfasser angefordert werden. Kontaktinformation: opus@uni-passau.de ER - TY - THES A1 - Ali, Rashid T1 - Weyl Gröbner Basis Cryptosystems N2 - In this thesis, we shall consider a certain class of algebraic cryptosystems called Gröbner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gröbner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks that do not require the computation of Gröbner basis for breaking the system, for example, the attacks based on linear algebra. To secure these (commutative) Gröbner basis cryptosystems against various attacks, among others, Ackermann and Kreuzer introduced a general class of Gröbner Basis Cryptosystems that are based on the difficulty of computing module Gröbner bases over general non-commutative rings. The objective of this research is to describe a special class of such cryptosystems by introducing the Weyl Gröbner Basis Cryptosystems. We divide this class of cryptosystems in two parts namely the (left) Weyl Gröbner Basis Cryptosystems and Two-Sided Weyl Gröbner Basis Cryptosystems. We suggest to use Gröbner bases for left and two-sided ideals in Weyl algebras to construct specific instances of such cryptosystems. We analyse the resistance of these cryptosystems to the standard attacks and provide computational evidence that secure Weyl Gröbner Basis Cryptosystems can be built using left (resp. two-sided) Gröbner bases in Weyl algebras. KW - Gröbner-Basis KW - Weyl-Algebra KW - Kryptologie KW - Public Key Cryptosystem KW - Non commutative Gröbner Basis Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-23195 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 - 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 - THES A1 - Kreitmeier, Wolfgang T1 - Optimale Quantisierung verallgemeinerter Cantor-Verteilungen N2 - Für verallgemeinerte Cantor-Verteilungen, die im Eindimensionalen mittels klassischer Wischkonstruktion bzw. in höheren Dimensionen über iterierte Funktionensysteme definiert werden, wird das Problem der optimalen Quantisierung unter bestimmten Voraussetzungen vollständig gelöst. Es werden die optimalen Codebücher bestimmt und Formeln für den optimalen Quantisierungsfehler bewiesen. Im eindimensionalen Fall wird eine Existenzcharakterisierung der Quantisierungsdimension gegeben und unter bestimmten Voraussetzungen die Nichtexistenz des Quantisierungskoeffizienten gezeigt. Auch in höheren Dimensionen wird für 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. KW - Maßtheorie KW - Fraktale Dimension KW - Iteriertes Funktionensystem KW - Sierpinski-Menge KW - Cantor-Menge KW - Hausdorff-Dimension KW - Hausdorff-Maß KW - Optimale Quantisierung KW - homogene Cantormaße KW - Quantisierungsdimension KW - Quantisierungskoeffizient KW - Sierpinski-Dreieck KW - Optimal Quantization KW - homogeneous Cantor measures KW - Quantization dimension KW - Quantization coefficient KW - Sierpinski Gasket Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-913 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 -