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%%%%%%%%%%%%%%% Hauptteil %%%%%%%%%%%%%%%
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\chapter*{How to read this Thesis}

As a guide through the nomenclature used in the formulas we prepend this
chapter.

Unless otherwise noted the following holds:

\begin{itemize}
\tightlist
\item
  lowercase letters \(x,y,z\)\\
   refer to real variables and represent a point in 3D-Space.
\item
  lowercase letters \(u,v,w\)\\
   refer to real variables between \(0\) and \(1\) used as coefficients
  in a 3D B-Spline grid.
\item
  other lowercase letters\\
   refer to other scalar (real) variables.
\item
  lowercase \textbf{bold} letters (e.g. \(\vec{x},\vec{y}\))\\
   refer to 3D coordinates
\item
  uppercase \textbf{BOLD} letters (e.g. \(\vec{D}, \vec{M}\))\\
   refer to Matrices
\end{itemize}

\chapter{Introduction}\label{introduction}

In this Master Thesis we try to extend a previously proposed concept of
predicting the evolvability of \acf{FFD} given a
Deformation-Matrix\cite{anrichterEvol}. In the original publication the
author used random sampled points weighted with \acf{RBF} to deform the
mesh and defined three different criteria that can be calculated prior
to using an evolutional optimization algorithm to asses the quality and
potential of such optimization.

We will replicate the same setup on the same meshes but use \acf{FFD}
instead of \acf{RBF} to create a deformation and evaluate if the
evolution-criteria still work as a predictor given the different
deformation scheme.

\section{\texorpdfstring{What is \acf{FFD}?}{What is ?}}\label{what-is}

First of all we have to establish how a \ac{FFD} works and why this is a
good tool for deforming meshes in the first place. For simplicity we
only summarize the 1D-case from \cite{spitzmuller1996bezier} here and go
into the extension to the 3D case in chapter \ref{3dffd}.

Given an arbitrary number of points \(p_i\) alongside a line, we map a
scalar value \(\tau_i \in [0,1[\) to each point with
\(\tau_i < \tau_{i+1} \forall i\). Given a degree of the target
polynomial \(d\) we define the curve \(N_{i,d,\tau_i}(u)\) as follows:

\begin{equation} \label{eqn:ffd1d1}
N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
\end{equation}

and

\begin{equation} \label{eqn:ffd1d2}
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)
\end{equation}

If we now multiply every \(p_i\) with the corresponding
\(N_{i,d,\tau_i}(u)\) we get the contribution of each point \(p_i\) to
the final curve-point parameterized only by \(u \in [0,1[\). As can be
seen from \eqref{eqn:ffd1d2} we only access points \([i..i+d]\) for any
given \(i\)\footnote{one more for each recursive step.}, which gives us,
in combination with choosing \(p_i\) and \(\tau_i\) in order, only a
local interference of \(d+1\) points.

We can even derive this equation straightforward for an arbitrary
\(N\)\footnote{\emph{Warning:} in the case of \(d=1\) the
  recursion-formula yields a \(0\) denominator, but \(N\) is also \(0\).
  The right solution for this case is a derivative of \(0\)}:

\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]

For a B-Spline \[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\] these
derivations yield \(\frac{\partial^d}{\partial u} s(u) = 0\).

Another interesting property of these recursive polynomials is that they
are continuous (given \(d \ge 1\)) as every \(p_i\) gets blended in
linearly between \(\tau_i\) and \(\tau_{i+d}\) and out linearly between
\(\tau_{i+1}\) and \(\tau_{i+d+1}\) as can bee seen from the two
coefficients in every step of the recursion.

\subsection{\texorpdfstring{Why is \ac{FFD} a good deformation
function?}{Why is  a good deformation function?}}\label{why-is-a-good-deformation-function}

The usage of \ac{FFD} as a tool for manipulating follows directly from
the properties of the polynomials and the correspondence to the control
points. Having only a few control points gives the user a nicer
high-level-interface, as she only needs to move these points and the
model follows in an intuitive manner. The deformation is smooth as the
underlying polygon is smooth as well and affects as many vertices of the
model as needed. Moreover the changes are always local so one risks not
any change that a user cannot immediately see.

But there are also disadvantages of this approach. The user loses the
ability to directly influence vertices and even seemingly simple tasks
as creating a plateau can be difficult to
achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}.

This disadvantages led to the formulation of
\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly
interacts with the surface-mesh. All interactions will be applied
proportionally to the control-points that make up the parametrization of
the interaction-point itself yielding a smooth deformation of the
surface \emph{at} the surface without seemingly arbitrary scattered
control-points. Moreover this increases the efficiency of an
evolutionary optimization\cite{Menzel2006}, which we will use later on.

But this approach also has downsides as can be seen in
\cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the
tessellation of the invisible grid has a major impact on the deformation
itself.

All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a
high-polygon mesh albeit the downsides.

\section{What is evolutional
optimization?}\label{what-is-evolutional-optimization}

\section{Wieso ist evo-Opt so cool?}\label{wieso-ist-evo-opt-so-cool}

The main advantage of evolutional algorithms is the ability to find
optima of general functions just with the help of a given error-function
(or fitness-function in this domain). This avoids the general pitfalls
of gradient-based procedures, which often target the same error-function
as an evolutional algorithm, but can get stuck in local optima.

This is mostly due to the fact that a gradient-based procedure has only
one point of observation from where it evaluates the next steps, whereas
an evolutional strategy starts with a population of guessed solutions.
Because an evolutional strategy modifies the solution randomly, keeps
the best solutions and purges the worst, it can also target multiple
different hypothesis at the same time where the local optima die out in
the face of other, better candidates.

If an analytic best solution exists (i.e.~because the error-function is
convex) an evolutional algorithm is not the right choice. Although both
converge to the same solution, the analytic one is usually faster. But
in reality many problems have no analytic solution, because the problem
is not convex. Here evolutional optimization has one more advantage as
you get bad solutions fast, which refine over time.

\section{Criteria for the evolvability of linear
deformations}\label{criteria-for-the-evolvability-of-linear-deformations}

\subsection{Variability}\label{variability}

In \cite{anrichterEvol} variability is defined as
\[V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},\] whereby \(\vec{U}\)
is the \(m \times n\) deformation-Matrix used to map the \(m\) control
points onto the \(n\) vertices.

Given \(n = m\), an identical number of control-points and vertices,
this quotient will be \(=1\) if all control points are independent of
each other and the solution is to trivially move every control-point
onto a target-point.

In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because as
there are only few control-points for many vertices, so \(m \ll n\).

Additionally in our setup we connect neighbouring control-points in a
grid so each control point is not independent, but typically depends on
\(4^d\) control-points for an \(d\)-dimensional control mesh.

\subsection{Regularity}\label{regularity}

Regularity is defined\cite{anrichterEvol} as
\[R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\]
where \(\sigma_{min}\) and \(\sigma_{max}\) are the smallest and
greatest right singular value of the deformation-matrix \(\vec{U}\).

As we deform the given Object only based on the parameters as
\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\) this makes sure that
\(\|\vec{Up}\| \propto \|\vec{p}\|\) when \(\kappa(\vec{U}) \approx 1\).
The inversion of \(\kappa(\vec{U})\) is only performed to map the
criterion-range to \([0..1]\), whereas \(1\) is the optimal value and
\(0\) is the worst value.

This criterion should be characteristic for numeric stability on the on
hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of
evolutional algorithms on the other hand\cite{anrichterEvol} as it is
tied to the notion of
locality\cite{weise2012evolutionary,thorhauer2014locality}.

\subsection{Improvement Potential}\label{improvement-potential}

In contrast to the general nature of variability and regularity, which
are agnostic of the fitness-function at hand the third criterion should
reflect a notion of potential.

As during optimization some kind of gradient \(g\) is available to
suggest a direction worth pursuing we use this to guess how much change
can be achieved in the given direction.

The definition for an improvement potential \(P\)
is\cite{anrichterEvol}: \[
P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F
\] given some approximate \(n \times d\) fitness-gradient \(\vec{G}\),
normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the
Frobenius-Norm.

\chapter{Hauptteil}\label{hauptteil}

\section{Was ist FFD?}\label{was-ist-ffd}

\label{3dffd}

\begin{itemize}
\tightlist
\item
  Definition
\item
  Wieso Newton-Optimierung?
\item
  Was folgt daraus?
\end{itemize}

\section{Szenarien vorstellen}\label{szenarien-vorstellen}

\subsection{1D}\label{d}

\subsubsection{Optimierungszenario}\label{optimierungszenario}

\begin{itemize}
\tightlist
\item
  Ebene -\textgreater{} Template-Fit
\end{itemize}

\subsubsection{Matching in 1D}\label{matching-in-1d}

\begin{itemize}
\tightlist
\item
  Trivial
\end{itemize}

\subsubsection{Besonderheiten der
Auswertung}\label{besonderheiten-der-auswertung}

\begin{itemize}
\tightlist
\item
  Analytische Lösung einzig beste
\item
  Ergebnis auch bei Rauschen konstant?
\item
  normierter 1-Vektor auf den Gradienten addieren

  \begin{itemize}
  \tightlist
  \item
    Kegel entsteht
  \end{itemize}
\end{itemize}

\subsection{3D}\label{d-1}

\subsubsection{Optimierungsszenario}\label{optimierungsszenario}

\begin{itemize}
\tightlist
\item
  Ball zu Mario
\end{itemize}

\subsubsection{Matching in 3D}\label{matching-in-3d}

\begin{itemize}
\tightlist
\item
  alternierende Optimierung
\end{itemize}

\subsubsection{Besonderheiten der
Optimierung}\label{besonderheiten-der-optimierung}

\begin{itemize}
\tightlist
\item
  Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
\item
  Kriterien trotzdem gut
\end{itemize}

\chapter{Evaluation}\label{evaluation}

\section{Spearman/Pearson-Metriken}\label{spearmanpearson-metriken}

\begin{itemize}
\tightlist
\item
  Was ist das?
\item
  Wieso sollte uns das interessieren?
\item
  Wieso reicht Monotonie?
\item
  Haben wir das gezeigt?
\item
  Statistik, Bilder, blah!
\end{itemize}

\chapter{Schluss}\label{schluss}

HAHA .. als ob -.-

\backmatter
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\chapter*{Appendix}
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%%%%%%%%%%%%%%% Literaturverzeichnis %%%%%%%%%%%%%%%
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