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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}

\improvement[inline]{Mehr Bilder}

Many modern industrial design processes require advanced optimization
methods do to the increased complexity. These designs have to adhere to
more and more degrees of freedom as methods refine and/or other methods
are used. Examples for this are physical domains like aerodynamic
(i.e.~drag), fluid dynamics (i.e.~throughput of liquid) --- where the
complexity increases with the temporal and spatial resolution of the
simulation --- or known hard algorithmic problems in informatics
(i.e.~layouting of circuit boards or stacking of 3D--objects). Moreover
these are typically not static environments but requirements shift over
time or from case to case.

Evolutionary algorithms cope especially well with these problem domains
while addressing all the issues at hand\cite{minai2006complex}. One of
the main concerns in these algorithms is the formulation of the problems
in terms of a genome and a fitness function. While one can typically use
an arbitrary cost--function for the fitness--functions (i.e.~amount of
drag, amount of space, etc.), the translation of the problem--domain
into a simple parametric representation can be challenging.

The quality of such a representation in biological evolution is called
\emph{evolvability}\cite{wagner1996complex} and is at the core of this
thesis, as the parametrization of the problem has serious implications
on the convergence speed and the quality of the
solution\cite{Rothlauf2006}. However, there is no consensus on how
\emph{evolvability} is defined and the meaning varies from context to
context\cite{richter2015evolvability}.

As we transfer the results of Richter et al.\cite{anrichterEvol} from
using \acf{RBF} as a representation to manipulate a geometric mesh to
the use of \acf{FFD} we will use the same definition for evolvability
the original author used, namely \emph{regularity}, \emph{variability},
and \emph{improvement potential}. We introduce these term in detail in
Chapter \ref{sec:intro:rvi}.

In the original publication the author used random sampled points
weighted with \acf{RBF} to deform the mesh and showed that the mentioned
criteria of \emph{regularity}, \emph{variability}, and \emph{improvement
potential} correlate with 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 local deformation near the control
points and evaluate if the evolution--criteria still work as a predictor
given the different deformation scheme, as suspected in
\cite{anrichterEvol}.

\section{Outline of this thesis}\label{outline-of-this-thesis}

First we introduce different topics in isolation in Chapter
\ref{sec:back}. We take an abstract look at the definition of \ac{FFD}
for a one--dimensional line (in \ref{sec:back:ffd}) and discuss why this
is a sensible deformation function (in \ref{sec:back:ffdgood}). Then we
establish some background--knowledge of evolutionary algorithms (in
\ref{sec:back:evo}) and why this is useful in our domain (in
\ref{sec:back:evogood}). In a third step we take a look at the
definition of the different evolvability criteria established in
\cite{anrichterEvol}.

In Chapter \ref{sec:impl} we take a look at our implementation of
\ac{FFD} and the adaptation for 3D--meshes.

Next, in Chapter \ref{sec:eval}, we describe the different scenarios we
use to evaluate the different evolvability--criteria incorporating all
aspects introduced in Chapter \ref{sec:back}. Following that, we
evaluate the results in Chapter \ref{sec:res} with further on discussion
in Chapter \ref{sec:dis}.

\chapter{Background}\label{background}

\label{sec:back}

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

\label{sec:back:ffd}

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}

\label{sec:back:ffdgood}

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.

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/hsu_fig7.png}
\caption{Figure 7 from \cite{hsu1991dmffd}.}
\label{fig:hsu_fig7}
\end{figure}

But this approach also has downsides as can be seen in figure
\ref{fig:hsu_fig7}, 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 evolutionary
optimization?}\label{what-is-evolutionary-optimization}

\label{sec:back:evo}

In this thesis we are using an evolutionary optimization strategy to
solve the problem of finding the best parameters for our deformation.
This approach, however, is very generic and we introduce it here in a
broader sense.

\begin{algorithm}
\caption{An outline of evolutionary algorithms}
\label{alg:evo}
\begin{algorithmic}
\STATE t := 0;
\STATE initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$;
\STATE evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$;
\WHILE{$c(F(t)) \neq$ \TRUE}
    \STATE recombine: $P’(t) := r(P(t))$;
    \STATE mutate: $P''(t) := m(P’(t))$;
    \STATE evaluate $F''(t) : \{\Phi(x) | x \in P''(t)\}$
    \STATE select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$;
    \STATE t := t + 1;
\ENDWHILE
\end{algorithmic}
\end{algorithm}

The general shape of an evolutionary algorithm (adapted from
\cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here,
\(P(t)\) denotes the population of parameters in step \(t\) of the
algorithm. The population contains \(\mu\) individuals \(a_i\) that fit
the shape of the parameters we are looking for. Typically these are
initialized by a random guess or just zero. Further on we need a
so--called \emph{fitness--function} \(\Phi : I \mapsto M\) that can take
each parameter to a measurable space along a convergence--function
\(c : I \mapsto \mathbb{B}\) that terminates the optimization.

The main algorithm just repeats the following steps:

\begin{itemize}
\tightlist
\item
  \textbf{Recombine} with a recombination--function
  \(r : I^{\mu} \mapsto I^{\lambda}\) to generate new individuals based
  on the parents characteristics.\\
  This makes sure that the next guess is close to the old guess.
\item
  \textbf{Mutate} with a mutation--function
  \(m : I^{\lambda} \mapsto I^{\lambda}\) to introduce new effects that
  cannot be produced by mere recombination of the parents.\\
  Typically this just adds minor defects to individual members of the
  population like adding a random gaussian noise or amplifying/dampening
  random parts.
\item
  \textbf{Selection} takes a selection--function
  \(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
  selects from the previously generated \(I^\lambda\) children and
  optionally also the parents (denoted by the set \(Q\) in the
  algorithm) using the fitness--function \(\Phi\). The result of this
  operation is the next Population of \(\mu\) individuals.
\end{itemize}

All these functions can (and mostly do) have a lot of hidden parameters
that can be changed over time. One can for example start off with a high
mutation--rate that cools off over time (i.e.~by lowering the variance
of a gaussian noise).

\section{Advantages of evolutionary
algorithms}\label{advantages-of-evolutionary-algorithms}

\label{sec:back:evogood}

The main advantage of evolutionary algorithms is the ability to find
optima of general functions just with the help of a given
fitness--function. With this most problems of simple gradient--based
procedures, which often target the same error--function which measures
the fitness, as an evolutionary algorithm, but can easily get stuck in
local optima.

Components and techniques for evolutionary algorithms are specifically
known to help with different problems arising in the domain of
optimization\cite{weise2012evolutionary}. An overview of the typical
problems are shown in figure \ref{fig:probhard}.

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/weise_fig3.png}
\caption{Fig.~3. taken from \cite{weise2012evolutionary}}
\label{fig:probhard}
\end{figure}

Most of the advantages stem from the fact that a gradient--based
procedure has only one point of observation from where it evaluates the
next steps, whereas an evolutionary strategy starts with a population of
guessed solutions. Because an evolutionary 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 and is easily computable
(i.e.~because the error--function is convex) an evolutionary 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 either not convex or there are so many parameters that an
analytic solution (mostly meaning the equivalence to an exhaustive
search) is computationally not feasible. Here evolutionary optimization
has one more advantage as you can at least get a suboptimal solutions
fast, which then refine over time.

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

\label{sec:intro:rvi}

\subsection{Variability}\label{variability}

In \cite{anrichterEvol} \emph{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}

\emph{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
evolutionary 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 \emph{variability} and
\emph{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 \emph{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{\texorpdfstring{Implementation of
\acf{FFD}}{Implementation of }}\label{implementation-of}

\label{sec:impl}

The general formulation of B--Splines has two free parameters \(d\) and
\(\tau\) which must be chosen beforehand.

As we usually work with regular grids in our \ac{FFD} we define \(\tau\)
statically as \(\tau_i = \nicefrac{i}{n}\) whereby \(n\) is the number
of control--points in that direction.

\(d\) defines the \emph{degree} of the B--Spline--Function (the number
of times this function is differentiable) and for our purposes we fix
\(d\) to \(3\), but give the formulas for the general case so it can be
adapted quite freely.

\section{\texorpdfstring{Adaption of
\ac{FFD}}{Adaption of }}\label{adaption-of}

As we have established in Chapter \ref{sec:back:ffd} we can define an
\ac{FFD}--displacement as

\begin{equation}
\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
\end{equation}

Note that we only sum up the \(\Delta\)--displacements in the control
points \(c_i\) to get the change in position of the point we are
interested in.

In this way every deformed vertex is defined by \[
\textrm{Deform}(v_x) = v_x + \Delta_x(u)
\] with \(u \in [0..1[\) being the variable that connects the
high--detailed vertex--mesh to the low--detailed control--grid. To
actually calculate the new position of the vertex we first have to
calculate the \(u\)--value for each vertex. This is achieved by finding
out the parametrization of \(v\) in terms of \(c_i\) \[
v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
\] so we can minimize the error between those two: \[
\underset{u}{\argmin}\,Err(u,v_x) = \underset{u}{\argmin}\,2 \cdot \|v_x - \sum_i N_{i,d,\tau_i}(u) c_i\|^2_2
\] As this error--term is quadratic we just derive by \(u\) yielding \[
\begin{array}{rl}
\frac{\partial}{\partial u} & v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
= & - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
\end{array}
\] and do a gradient--descend to approximate the value of \(u\) up to an
\(\epsilon\) of \(0.0001\).

For this we use the Gauss--Newton algorithm\cite{gaussNewton} as the
solution to this problem may not be deterministic, because we usually
have way more vertices than control points (\(\#v~\gg~\#c\)).

\section{\texorpdfstring{Adaption of \ac{FFD} for a
3D--Mesh}{Adaption of  for a 3D--Mesh}}\label{adaption-of-for-a-3dmesh}

\label{3dffd}

This is a straightforward extension of the 1D--method presented in the
last chapter. But this time things get a bit more complicated. As we
have a 3--dimensional grid we may have a different amount of
control--points in each direction.

Given \(n,m,o\) control points in \(x,y,z\)--direction each Point on the
curve is defined by
\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]

In this case we have three different B--Splines (one for each dimension)
and also 3 variables \(u,v,w\) for each vertex we want to approximate.

Given a target vertex \(\vec{p}^*\) and an initial guess
\(\vec{p}=V(u,v,w)\) we define the error--function for the
gradient--descent as:

\[Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)\]

And the partial version for just one direction as

\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \]

To solve this we derive partially, like before:

\[
\begin{array}{rl}
    \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
  = & \displaystyle - \sum_i \sum_j \sum_k N'_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
\end{array}
\]

The other partial derivatives follow the same pattern yielding the
Jacobian:

\[
J(Err(u,v,w)) = 
\left(
\begin{array}{ccc}
\frac{\partial Err_x}{\partial u} & \frac{\partial Err_x}{\partial v} & \frac{\partial Err_x}{\partial w} \\
\frac{\partial Err_y}{\partial u} & \frac{\partial Err_y}{\partial v} & \frac{\partial Err_y}{\partial w} \\
\frac{\partial Err_z}{\partial u} & \frac{\partial Err_z}{\partial v} & \frac{\partial Err_z}{\partial w}
\end{array}
\right)
\] \[
\scriptsize
=
\left(
\begin{array}{ccc}
- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_x \\
- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_y \\
- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_z
\end{array}
\right)
\]

With the Gauss--Newton algorithm we iterate via the formula
\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]
and use Cramers rule for inverting the small Jacobian and solving this
system of linear equations.

\section{Deformation Grid}\label{deformation-grid}

As mentioned in chapter \ref{sec:back:evo}, the way of choosing the
representation to map the general problem (mesh--fitting/optimization in
our case) into a parameter-space it very important for the quality and
runtime of evolutionary algorithms\cite{Rothlauf2006}.

Because our control--points are arranged in a grid, we can accurately
represent each vertex--point inside the grids volume with proper
B--Spline--coefficients between \([0,1[\) and --- as a consequence ---
we have to embed our object into it (or create constant ``dummy''-points
outside).

The great advantage of B--Splines is the locality, direct impact of each
control point without having a \(1:1\)--correlation, and a smooth
deformation. While the advantages are great, the issues arise from the
problem to decide where to place the control--points and how many.

One would normally think, that the more control--points you add, the
better the result will be, but this is not the case for our B--Splines.
Given any point \(p\) only the \(2 \cdot (d-1)\) control--points
contribute to the parametrization of that point\footnote{Normally these
  are \(d-1\) to each side, but at the boundaries the number gets
  increased to the inside to meet the required smoothness}. This means,
that a high resolution can have many control-points that are not
contributing to any point on the surface and are thus completely
irrelevant to the solution.

\begin{figure}[!ht]
\begin{center}
\includegraphics{img/enoughCP.png}
\end{center}
\caption{A high resolution ($10 \times 10$) of control--points over a circle.
yellow/green points contribute to the parametrization, red points don't.\newline
An Example--point (blue) is solely determined by the position of the green
control--points.}
\label{fig:enoughCP}
\end{figure}

We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
central points are not relevant for the parametrization of the circle.

\unsure[inline]{erwähnen, dass man aus $\vec{D}$ einfach die Null--Spalten
entfernen kann?}

For our tests we chose different uniformly sized grids and added
gaussian noise onto each control-point\footnote{For the special case of
  the outer layer we only applied noise away from the object} to
simulate different starting-conditions.

\begin{itemize}
\tightlist
\item
  Nachteile von Parametrisierung
\item
  wie in kap. \ref{sec:back:evo} zu sehen, ist Parametrisierung
  wichtig\cite{Rothlauf2006}.
\item
  Parametrisierung zwar lokal, aber nicht 1:1
\item
  Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
\item
  =\textgreater{} DM--FFD kann abhelfen, further study.
\item
  Schlechte Parametrisierung sorgt dafür, dass CP u.U. nicht zur
  Parametrisierung verwendet werden.
\end{itemize}

\chapter{\texorpdfstring{Scenarios for testing evolvability criteria
using
\acf{FFD}}{Scenarios for testing evolvability criteria using }}\label{scenarios-for-testing-evolvability-criteria-using}

\label{sec:eval}

\section{Test Scenario: 1D Function
Approximation}\label{test-scenario-1d-function-approximation}

\subsection{Optimierungszenario}\label{optimierungszenario}

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

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

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

\subsection{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}

\section{Test Scenario: 3D Function
Approximation}\label{test-scenario-3d-function-approximation}

\subsection{Optimierungsszenario}\label{optimierungsszenario}

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

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

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

\subsection{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 of Scenarios}\label{evaluation-of-scenarios}

\label{sec:res}

\section{Spearman/Pearson--Metriken}\label{spearmanpearsonmetriken}

\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}

\section{Results of 1D Function
Approximation}\label{results-of-1d-function-approximation}

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/evolution1d/20171005-all_appended.png}
\caption{Results 1D}

\end{figure}

\section{Results of 3D Function
Approximation}\label{results-of-3d-function-approximation}

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/evolution3d/20171007_3dFit_all_append.png}
\caption{Results 3D}

\end{figure}

\chapter{Schluss}\label{schluss}

\label{sec:dis}

HAHA .. als ob -.-

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