403 lines
13 KiB
TeX
403 lines
13 KiB
TeX
% bibtotoc[numbered] : Literaturv. wird in Inhaltsv. aufgenommen
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% abstracton : Abstract mit Ueberschrift
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]{scrbook}
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%%%%%%%%%%%%%%% Literaturverzeichnisstil %%%%%%%%%%%%%%%
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% \input{settings/hyphenation} %% Manchmal bricht latex nicht richtig um. hier trennregeln rein.
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% \includeonly{%
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% % files/0_titlepage.tex
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% % files/1_0_introduction,%
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% % files/2_0_knownDCJ,%
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% % files/3_0_DCJIndels,%
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% % files/4_0_DCJIndels_1comps,%
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% files/5_0_DCJIndels_2comps,%
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% % files/6_0_implementation,%
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% % files/7_0_evaluation%
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% % ,files/8_0_conclusion%
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% }
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%%%%%%%%%%%%%%% PANDOC-nedded defs %%%%%%%%%%
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% {}{}
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% \makeatother
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%%%%%%%%%%%%%%% Hauptdokument %%%%%%%%%%%%%%%
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\begin{document}
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% ###### Autoref definitions (hyperref package)#####
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\def\subtableautorefname{Table}
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\def\algorithmautorefname{Algorithm}
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\def\chapterautorefname{Chapter}
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\def\sectionautorefname{Section}
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\def\definitionautorefname{Definition}
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\def\exampleautorefname{Example}
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\def\observationautorefname{Observation}
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\def\propositionautorefname{Proposition}
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\def\lemmaautorefname{Lemma}
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% in diesem Dokument nicht verwendet:
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% \def\subsectionautorefname{Subsection}
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% \def\Subsubsectionautorefname{Subsubsection}
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%%%%%%%%%%%%%%% Deckblatt %%%%%%%%%%%%%%%
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% \input{files/0_deckblatt/title}
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\pagestyle{empty} % Rueckseite leer
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%
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%%%%%%%%%%%%%%% Verzeichnisse %%%%%%%%%%%%%%%
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\frontmatter % Abstrakte Gliederungsebene: Anfang des Buches
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\tableofcontents % Rueckseite leer
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%\lstlistoflistings % fuer listingsverzeichnis mit package listings
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%%%%%%%%%%%%%%% Hauptteil %%%%%%%%%%%%%%%
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% Insgesamt ca. 60-100 Seiten Davon mindesten 50% Eigene Arbeit
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\mainmatter %Abstrakte Gliederungsebene: Hauptteil des Buches
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\pagestyle{fancy}
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\pagenumbering{arabic}
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\chapter*{How to read this Thesis}
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As a guide through the nomenclature used in the formulas we prepend this
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chapter.
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Unless otherwise noted the following holds:
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\begin{itemize}
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\tightlist
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\item
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lowercase letters \(x,y,z\)\\
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refer to real variables and represent a point in 3D-Space.
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\item
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lowercase letters \(u,v,w\)\\
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refer to real variables between \(0\) and \(1\) used as coefficients
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in a 3D B-Spline grid.
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\item
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other lowercase letters\\
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refer to other scalar (real) variables.
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\item
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lowercase \textbf{bold} letters (e.g. \(\vec{x},\vec{y}\))\\
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refer to 3D coordinates
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\item
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uppercase \textbf{BOLD} letters (e.g. \(\vec{D}, \vec{M}\))\\
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refer to Matrices
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\end{itemize}
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\chapter{Introduction}\label{introduction}
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In this Master Thesis we try to extend a previously proposed concept of
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predicting the evolvability of \acf{FFD} given a
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Deformation-Matrix\cite{anrichterEvol}. In the original publication the
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author used random sampled points weighted with \acf{RBF} to deform the
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mesh and defined three different criteria that can be calculated prior
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to using an evolutional optimisation algorithm to asses the quality and
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potential of such optimisation.
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We will replicate the same setup on the same meshes but use \acf{FFD}
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instead of \acf{RBF} to create a deformation and evaluate if the
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evolution-criteria still work as a predictor given the different
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deformation scheme.
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\section{\texorpdfstring{What is \acf{FFD}?}{What is ?}}\label{what-is}
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First of all we have to establish how a \ac{FFD} works and why this is a
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good tool for deforming meshes in the first place. For simplicity we
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only summarize the 1D-case from \cite{spitzmuller1996bezier} here and go
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into the extension to the 3D case in chapter \ref{3dffd}.
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Given an arbitrary number of points \(p_i\) alongside a line, we map a
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scalar value \(\tau_i \in [0,1[\) to each point with
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\(\tau_i < \tau_{i+1} \forall i\). Given a degree of the target
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polynomial \(d\) we define the curve \(N_{i,d,\tau_i}(u)\) as follows:
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\begin{equation} \label{eqn:ffd1d1}
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N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
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\end{equation}
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and
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\begin{equation} \label{eqn:ffd1d2}
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N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
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\end{equation}
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If we now multiply every \(p_i\) with the corresponding
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\(N_{i,d,\tau_i}(u)\) we get the contribution of each point \(p_i\) to
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the final curve-point parameterized only by \(u \in [0,1[\). As can be
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seen from \eqref{eqn:ffd1d2} we only access points \([i..i+d]\) for any
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given \(i\)\footnote{one more for each recursive step.}, which gives us,
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in combination with choosing \(p_i\) and \(\tau_i\) in order, only a
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local interference of \(d+1\) points.
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We can even derive this equation straightforward for an arbitrary
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\(N\)\footnote{\emph{Warning:} in the case of \(d=1\) the
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recursion-formula yields a \(0\) denominator, but \(N\) is also \(0\).
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The right solution for this case is a derivative of \(0\)}:
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\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]
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For a B-Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
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derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).
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Another interesting property of these recursive polynomials is that they
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are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
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linearly between \(\tau_i\) and \(\tau_{i+d}\) and out linearly between
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\(\tau_{i+1}\) and \(\tau_{i+d+1}\) as can bee seen from the two
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coefficients in every step of the recursion.
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\subsection{\texorpdfstring{Why is \ac{FFD} a good deformation
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function?}{Why is a good deformation function?}}\label{why-is-a-good-deformation-function}
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The usage of \ac{FFD} as a tool for manipulating follows directly from
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the properties of the polynomials and the correspondence to the control
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points. Having only a few control points gives the user a nicer
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high-level-interface, as she only needs to move these points and the
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model follows in an intuitive manner. The deformation is smooth as the
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underlying polygon is smooth as well and affects as many vertices of the
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model as needed. Moreover the changes are always local so one risks not
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any change that a user cannot immediately see.
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But there are also disadvantages of this approach. The user loses the
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ability to directly influence vertices and even seemingly simple tasks
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as creating a plateau can be difficult to
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achieve\cite[chapter~3.2]{hsu1991dmffd}\todo{cite [24] aus \ref{anrichterEvol}}.
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This disadvantages led to the formulation of
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\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
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interacts with the surface-mesh. All interactions will be applied
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proportionally to the control-points that make up the parametrization of
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the interaction-point itself yielding a smooth deformation of the
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surface \emph{at} the surface without seemingly arbitrary scattered
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control-points. Moreover this increases the efficiency of an
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evolutionary optimization\todo{cite [25] aus \ref{anrichterEvol}}, which
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we will use later on.
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But this approach also has downsides as can be seen in
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\cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the
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tessellation of the invisible grid has a major impact on the deformation
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itself.
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All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a
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high-polygon mesh albeit the downsides.
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\section{What is evaluational
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optimization?}\label{what-is-evaluational-optimization}
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\section{Wieso ist evo-Opt so cool?}\label{wieso-ist-evo-opt-so-cool}
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\section{Evolvierbarkeitskriterien}\label{evolvierbarkeitskriterien}
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\begin{itemize}
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\tightlist
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\item
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Konditionszahl etc.
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\end{itemize}
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\chapter{Hauptteil}\label{hauptteil}
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\section{Was ist FFD?}\label{was-ist-ffd}
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\label{3dffd}
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\begin{itemize}
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\tightlist
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\item
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Definition
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\item
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Wieso Newton-Optimierung?
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\item
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Was folgt daraus?
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\end{itemize}
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\section{Szenarien vorstellen}\label{szenarien-vorstellen}
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\subsection{1D}\label{d}
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\subsubsection{Optimierungszenario}\label{optimierungszenario}
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\begin{itemize}
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\tightlist
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\item
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Ebene -\textgreater{} Template-Fit
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\end{itemize}
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\subsubsection{Matching in 1D}\label{matching-in-1d}
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\begin{itemize}
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\tightlist
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\item
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Trivial
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\end{itemize}
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\subsubsection{Besonderheiten der
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Auswertung}\label{besonderheiten-der-auswertung}
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\begin{itemize}
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\tightlist
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\item
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Analytische Lösung einzig beste
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\item
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Ergebnis auch bei Rauschen konstant?
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\item
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normierter 1-Vektor auf den Gradienten addieren
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\begin{itemize}
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\tightlist
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\item
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Kegel entsteht
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\end{itemize}
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\end{itemize}
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\subsection{3D}\label{d-1}
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\subsubsection{Optimierungsszenario}\label{optimierungsszenario}
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\begin{itemize}
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\tightlist
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\item
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Ball zu Mario
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\end{itemize}
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\subsubsection{Matching in 3D}\label{matching-in-3d}
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\begin{itemize}
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\tightlist
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\item
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alternierende Optimierung
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\end{itemize}
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\subsubsection{Besonderheiten der
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Optimierung}\label{besonderheiten-der-optimierung}
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\begin{itemize}
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\tightlist
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\item
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Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
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\item
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Kriterien trotzdem gut
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\end{itemize}
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\chapter{Evaluation}\label{evaluation}
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\section{Spearman/Pearson-Metriken}\label{spearmanpearson-metriken}
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\begin{itemize}
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\tightlist
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\item
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Was ist das?
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\item
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Wieso sollte uns das interessieren?
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\item
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Wieso reicht Monotonie?
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\item
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Haben wir das gezeigt?
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\item
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Stastik, Bilder, blah!
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\end{itemize}
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\chapter{Schluss}\label{schluss}
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HAHA .. als ob -.-
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\backmatter
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\cleardoublepage
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\renewcommand\thesection{\Roman{section}}
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\addtocontents{toc}{\protect\setcounter{tocdepth}{1}}
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\setcounter{section}{1} % reset section to 1 so its stars I, II, III,...
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\chapter*{Appendix}
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\addcontentsline{toc}{chapter}{\protect\numberline{}Appendix}
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\pagenumbering{roman}
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%%%%%%%%%%%%%%% Literaturverzeichnis %%%%%%%%%%%%%%%
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\bibliographystyle{natdin} % \bibliographystyle{natdin}
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\bibliography{bibma}
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\addcontentsline{toc}{section}{\protect\numberline{\thesection}Bibliography} % Literaturverzeichnis in das Inhaltsverzeichnis aufnehmen
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\addtocounter{section}{1}
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\newpage
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%%%%%%%%%%%%%%% Anhang %%%%%%%%%%%%%%%
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% \clearpage %spaeter alles wieder rein
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% % \input{files/appendix}
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\input{settings/abkuerzungen}
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\addcontentsline{toc}{section}{\protect\numberline{\thesection}Abbreviations}
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\addtocounter{section}{1}
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\newpage
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\listofalgorithms
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\addcontentsline{toc}{section}{\protect\numberline{\thesection}List of Algorithms}
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\addtocounter{section}{1}
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\newpage
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\listoffigures
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\listoftables
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\listoftodos
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\addcontentsline{toc}{section}{\protect\numberline{\thesection}TODOs}
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\addtocounter{section}{1}
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\newpage
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% \printindex
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%%%%%%%%%%%%%%% Erklaerung %%%%%%%%%%%%%%%
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% *\input{settings/declaration}
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\include{files/erklaerung}
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\end{document}
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