21 research outputs found
Denoising and enhancement of digital images : variational methods, integrodifferential equations, and wavelets
The topics of this thesis are methods for denoising, enhancement, and simplification of digital image data. Special emphasis lies on the relations and structural similarities between several classes of methods which are motivated from different contexts. In particular, one can distinguish the methods treated in this thesis in three classes: For variational approaches and partial differential equations, the notion of the derivative is the tool of choice to model regularity of the data and the desired result. A general framework for such approaches is proposed that involve all partial derivatives of a prescribed order and experimentally are capable of leading to piecewise polynomial approximations of the given data. The second class of methods uses wavelets to represent the data which makes it possible to understand the filtering as very simple pointwise application of a nonlinear function. To view these wavelets as derivatives of smoothing kernels is the basis for relating these methods to integrodifferential equations which are investigated here. In the third case, values of the image in a neighbourhood are averaged where the weights of this averaging can be adapted respecting different criteria. By refinement of the pixel grid and transfer to scaling limits, connections to partial differential equations become visible here, too. They are described in the framework explained before. Numerical aspects of the simplification of images are presented with respect to the NDS energy function, a unifying approach that allows to model many of the aforementioned methods. The behaviour of the filtering methods is documented with numerical examples.Gegenstand der vorliegenden Arbeit sind Verfahren zum Entrauschen, qualitativen Verbessern und Vereinfachen digitaler Bilddaten. Besonderes Augenmerk liegt dabei auf den Beziehungen und der strukturellen Ähnlichkeit zwischen unterschiedlich motivierten Verfahrensklassen. Insbesondere lassen sich die hier behandelten Methoden in drei Klassen einordnen: Bei den Variationsansätzen und partiellen Differentialgleichungen steht der Begriff der Ableitung im Mittelpunkt, um Regularität der Daten und des gewünschten Resultats zu modellieren. Hier wird ein einheitlicher Rahmen für solche Ansätze angegeben, die alle partiellen Ableitungen einer vorgegebenen Ordnung involvieren und experimentell auf stückweise polynomielle Approximationen der gegebenen Daten führen können. Die zweite Klasse von Methoden nutzt Wavelets zur Repräsentation von Daten, mit deren Hilfe sich Filterung als sehr einfache punktweise Anwendung einer nichtlinearen Funktion verstehen lässt. Diese Wavelets als Ableitungen von Glättungskernen aufzufassen bildet die Grundlage für die hier untersuchte Verbindung dieser Verfahren zu Integrodifferentialgleichungen. Im dritten Fall werden Werte des Bildes in einer Nachbarschaft gemittelt, wobei die Gewichtung bei dieser Mittelung adaptiv nach verschiedenen Kriterien angepasst werden kann. Durch Verfeinern des Pixelgitters und Übergang zu Skalierungslimites werden auch hier Verbindungen zu partiellen Differentialgleichungen sichtbar, die in den vorher dargestellten Rahmen eingeordnet werden. Numerische Aspekte beim Vereinfachen von Bildern werden anhand der NDS-Energiefunktion dargestellt, eines einheitlichen Ansatzes, mit dessen Hilfe sich viele der vorgenannten Methoden realisieren lassen. Das Verhalten der einzelnen Filtermethoden wird dabei jeweils durch numerische Beispiele dokumentiert
Tableau Algorithms for Categorial Deduction and Parsing
Institute for Communicating and Collaborative SystemsIn this thesis we develop automated dedution mehanisms designed to keep complexity of
categorial parsing under control while preserving the levels of uniformity and
coverage one finds in labeled dedutive systems. First,we define the hierarhy of caluli whose
computational treatment is addresed in the thesis,review the main issues and linguistic motivations behind proof-theoretical features of each calculus and describe the correspondence between proofs and semantic interpretation with respect to lambda terms.
Next we introduce the rules and algorithms of a deductive system based on analytic tableaux which
covers the whole hierarchy of categorial calculi presented.Completenes and termination results are shown. We then impose syntactic constraints on the
calculi and elaborate label unification proceedures aimed at limiting the system's
complexity. Alternative proof-search strategies are discussed and a technique for recovering syntactic structure from tableau derivations is developed. In the last chapters we
compare our system with other methods used in
categorial deduction,discuss design issues,heuristics and extensions,and link
categorial deduction with theorem proving in recently developed logics of information
flow such as channel theory
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
Mathematical linguistics
but in fact this is still an early draft, version 0.56, August 1 2001. Please d