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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 the coordinates of 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}

Many modern industrial design processes require advanced optimization
methods due to the increased complexity resulting from more and more
degrees of freedom as methods refine and/or other methods are used.
Examples for this are physical domains like aerodynamics (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.

\begin{figure}[hbt]
\centering
\includegraphics[width=\textwidth]{img/Evo_overview.png}
\caption{Example of the use of evolutionary algorithms in automotive design
(from \cite{anrichterEvol}).}
\end{figure}

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 \emph{genome} and \emph{fitness--function}. While one can
typically use an arbitrary cost--function for the
\emph{fitness--functions} (i.e.~amount of drag, amount of space, etc.),
the translation of the problem--domain into a simple parametric
representation (the \emph{genome}) can be challenging.

This translation is often necessary as the target of the optimization
may have too many degrees of freedom for a reasonable computation. In
the example of an aerodynamic simulation of drag onto an object, those
object--designs tend to have a high number of vertices to adhere to
various requirements (visual, practical, physical, etc.). A simpler
representation of the same object in only a few parameters that
manipulate the whole in a sensible matter are desirable, as this often
decreases the computation time significantly.

Additionally one can exploit the fact, that drag in this case is
especially sensitive to non--smooth surfaces, so that a smooth local
manipulation of the surface as a whole is more advantageous than merely
random manipulation of the vertices.

The quality of such a low--dimensional representation in biological
evolution is strongly tied to the notion of
\emph{evolvability}\cite{wagner1996complex}, 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 a consequence
there is need for some criteria we can measure, so that we are able to
compare different representations to learn and improve upon these.

\begin{figure}[hbt]
\centering
\includegraphics[width=\textwidth]{img/deformations.png}
\caption{Example of RBF--based deformation and FFD targeting the same mesh.}
\end{figure}

One example of such a general representation of an object is to generate
random points and represent vertices of an object as distances to these
points --- for example via \acf{RBF}. If one (or the algorithm) would
move such a point the object will get deformed only locally (due to the
\ac{RBF}). As this results in a simple mapping from the parameter--space
onto the object one can try out different representations of the same
object and evaluate which criteria may be suited to describe this notion
of \emph{evolvability}. This is exactly what Richter et
al.\cite{anrichterEvol} have done.

As we transfer the results of Richter et al.\cite{anrichterEvol} from
using \acf{RBF} as a representation to manipulate geometric objects to
the use of \acf{FFD} we will use the same definition for
\emph{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 could show a correlation between these
evolvability--criteria with the quality and convergence speed of such
optimization.

We will replicate the same setup on the same objects 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 for \emph{evolvability} of the representation given the
different deformation scheme, as suspected in \cite{anrichterEvol}.

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}) followed by the definition of the different
evolvability--criteria established in \cite{anrichterEvol} (in
\ref {sec:intro:rvi}).

In Chapter \ref{sec:impl} we take a look at our implementation of
\ac{FFD} and the adaptation for 3D--meshes that were used. 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, summary and
outlook 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 geometric objects (especially meshes in our
case) 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}.

The main idea of \ac{FFD} is to create a function
\(s : [0,1[^d \mapsto \mathbb{R}^d\) that spans a certain part of a
vector--space and is only linearly parametrized by some special
control--points \(p_i\) and an constant attribution--function
\(a_i(u)\), so \[
s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
\] can be thought of a representation of the inside of the convex hull
generated by the control--points where each position inside can be
accessed by the right \(u \in [0,1[^d\).

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{img/B-Splines.png}
\end{center}
\caption[Example of B--Splines]{Example of a parametrization of a line with
corresponding deformation to generate a deformed objet}
\label{fig:bspline}
\end{figure}

In the 1--dimensional example in figure~\ref{fig:bspline}, the
control--points are indicated as red dots and the colour--gradient
should hint at the \(u\)--values ranging from \(0\) to \(1\).

We now define a \acf{FFD} by the following:\\
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
\(\tau_i < \tau_{i+1} \forall i\) according to the position of \(p_i\)
on said line. Additionally, 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 parametrized only by \(u \in [0,1[\). As can be
seen from \eqref{eqn:ffd1d2} we only access points \([p_i..p_{i+d}]\)
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
\(\left(\frac{\partial}{\partial u}\right)^d s(u) = 0\).

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

This means that all changes are only a local linear combination between
the control--point \(p_i\) to \(p_{i+d+1}\) and consequently this yields
to the convex--hull--property of B--Splines --- meaning, that no matter
how we choose our coefficients, the resulting points all have to lie
inside convex--hull of the control--points.

For a given point \(s_i\) we can then calculate the contributions
\(u_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\) to get the
projection from the control--point--space into the object--space: \[
s_i = \sum_j u_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
\] or written for all points at the same time: \[
\vec{s} = \vec{U} \vec{p}
\] where \(\vec{U}\) is the \(n \times m\) transformation--matrix (later
on called \textbf{deformation matrix}) for \(n\) object--space--points
and \(m\) control--points.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{img/unity.png}
\end{center}
\caption[B--spline--basis--function as partition of unity]{From \cite[Figure 2.13]{brunet2010contributions}:\newline
\glqq Some interesting properties of the B--splines. On the natural definition domain
of the B--spline ($[k_0,k_4]$ on this figure), the B--Spline basis functions sum
up to one (partition of unity). In this example, we use B--Splines of degree 2.
The horizontal segment below the abscissa axis represents the domain of
influence of the B--splines basis function, i.e. the interval on which they are
not null. At a given point, there are at most $ d+1$ non--zero B--Spline basis
functions (compact support).\grqq \newline
Note, that Brunet starts his index at $-d$ opposed to our definition, where we
start at $0$.}
\label{fig:partition_unity}
\end{figure}

Furthermore B--Spline--basis--functions form a partition of unity for
all, but the first and last \(d\)
control--points\cite{brunet2010contributions}. Therefore we later on use
the border--points \(d+1\) times, such that \(\sum_j u_{i,j} p_j = p_i\)
for these points.

The locality of the influence of each control--point and the partition
of unity was beautifully pictured by Brunet, which we included here as
figure \ref{fig:partition_unity}.

\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 \(a_i\) from the
possible individual--set \(I\) 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 \(M\) (usually \(M = \mathbb{R}\)) along a convergence--function
\(c : I \mapsto \mathbb{B}\) that terminates the optimization.

Biologically speaking the set \(I\) corresponds to the set of possible
\emph{genotypes} while \(M\) represents the possible observable
\emph{phenotypes}. \emph{Genotypes} define all initial properties of an
individual, but their properties are not directly observable. It is the
genes, that evolve over time (and thus correspond to the parameters we
are tweaking in our algorithms or the genes in nature), but only the
\emph{phenotypes} make certain behaviour observable (algorithmically
through our \emph{fitness--function}, biologically by the ability to
survive and produce offspring). Any individual in our algorithm thus
experience a biologically motivated life cycle of inheriting genes from
the parents, modified by mutations occurring, performing according to a
fitness--metric, and generating offspring based on this. Therefore each
iteration in the while--loop above is also often named generation.

One should note that there is a subtle difference between
\emph{fitness--function} and a so called
\emph{genotype--phenotype--mapping}. The first one directly applies the
\emph{genotype--phenotype--mapping} and evaluates the performance of an
individual, thus going directly from genes/parameters to
reproduction--probability/score. In a concrete example the
\emph{genotype} can be an arbitrary vector (the genes), the
\emph{phenotype} is then a deformed object, and the performance can be a
single measurement like an air--drag--coefficient. The
\emph{genotype--phenotype--mapping} would then just be the generation of
different objects from that starting--vector, whereas the
\emph{fitness--function} would go directly from such a starting--vector
to the coefficient that we want to optimize.

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 \(\lambda\) new
  individuals based on the characteristics of the \(\mu\) parents.\\
   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 \emph{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. A good overview of this is given in
\cite{eiben1999parameter}, so we only give a small excerpt here.

For example the mutation can consist of merely a single \(\sigma\)
determining the strength of the gaussian defects in every parameter ---
or giving a different \(\sigma\) to every component of those parameters.
An even more sophisticated example would be the \glqq 1/5 success
rule\grqq ~from \cite{rechenberg1973evolutionsstrategie}.

Also in the selection--function it may not be wise to only take the
best--performing individuals, because it may be that the optimization
has to overcome a barrier of bad fitness to achieve a better local
optimum.

Recombination also does not have to be mere random choosing of parents,
but can also take ancestry, distance of genes or groups of individuals
into account.

\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
\emph{fitness--function}. 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 usually 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 can be
modified according to the problem--domain (i.e.~by the ideas given
above) it can also approximate very difficult problems in an efficient
manner and even self--tune parameters depending on the ancestry at
runtime\footnote{Some examples of this are explained in detail in
  \cite{eiben1999parameter}}.

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 one can at least get suboptimal solutions
fast, which then refine over time and still converge to a decent
solution much faster than an exhaustive search.

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

\label{sec:intro:rvi}

As we have established in chapter \ref{sec:back:ffd}, we can describe a
deformation by the formula \[
\vec{S} = \vec{U}\vec{P}
\] where \(\vec{S}\) is a \(n \times d\) matrix of vertices\footnote{We
  use \(\vec{S}\) in this notation, as we will use this parametrization
  of a source--mesh to manipulate \(\vec{S}\) into a target--mesh
  \(\vec{T}\) via \(\vec{P}\)}, \(\vec{U}\) are the (during
parametrization) calculated deformation--coefficients and \(P\) is a
\(m \times d\) matrix of control--points that we interact with during
deformation.

We can also think of the deformation in terms of differences from the
original coordinates \[
\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
\] which is isomorphic to the former due to the linearity of the
deformation. One can see in this way, that the way the deformation
behaves lies solely in the entries of \(\vec{U}\), which is why the
three criteria focus on this.

\subsection{Variability}\label{variability}

In \cite{anrichterEvol} \emph{variability} is defined as
\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\]
whereby \(\vec{U}\) is the \(n \times m\) 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
there are only few control--points for many vertices, so \(m \ll n\).

This criterion should correlate to the degrees of freedom the given
parametrization has. This can be seen from the fact, that
\(\mathrm{rank}(\vec{U})\) is limited by \(\min(m,n)\) and --- as \(n\)
is constant --- can never exceed \(n\).

The rank itself is also interesting, as control--points could
theoretically be placed on top of each other or be linear dependent in
another way --- but will in both cases lower the rank below the number
of control--points \(m\) and are thus measurable by the
\emph{variability}.

\subsection{Regularity}\label{regularity}

\emph{Regularity} is defined\cite{anrichterEvol} as
\[\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\]
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 \([0..1]\), where \(1\) is the optimal value and
\(0\) is the worst value.

On the one hand this criterion should be characteristic for numeric
stability\cite[chapter 2.7]{golub2012matrix} and on the other hand for
the convergence speed of evolutionary algorithms\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 \emph{fitness--function} at
hand, the third criterion should reflect a notion of the potential for
optimization, taking a guess into account.

Most of the times some kind of gradient \(g\) is available to suggest a
direction worth pursuing; either from a previous iteration or by
educated guessing. 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}: \[
\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
\] 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}

\label{sec:ffd:adapt}

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 employ the Gauss--Newton algorithm\cite{gaussNewton}, which
converges into the least--squares solution. An exact solution of this
problem is impossible most of the time, 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 Cramer's rule for inverting the small Jacobian and solving this
system of linear equations.

As there is no strict upper bound of the number of iterations for this
algorithm, we just iterate it long enough to be within the given
\(\epsilon\)--error above. This takes --- depending on the shape of the
object and the grid --- about \(3\) to \(5\) iterations that we observed
in practice.

Another issue that we observed in our implementation is, that multiple
local optima may exist on self--intersecting grids. We solve this
problem by defining self--intersecting grids to be \emph{invalid} and do
not test any of them.

This is not such a big problem as it sounds at first, as
self--intersections mean, that control--points being further away from a
given vertex have more influence over the deformation than
control--points closer to this vertex. Also this contradicts the notion
of locality that we want to achieve and deemed beneficial for a good
behaviour of the evolutionary algorithm.

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

\label{sec:impl: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 is 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 local, 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 to
place at all.

\begin{figure}[!tbh]
\centering
\includegraphics{img/enoughCP.png}
\caption[Example of a high resolution control--grid]{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}

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 \(\vec{p}\) 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 border points get used
  multiple times to meet the number of points required}. 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.

We illustrate this phenomenon in figure \ref{fig:enoughCP}, where the
red central points are not relevant for the parametrization of the
circle. This leads to artefacts in the deformation--matrix \(\vec{U}\),
as the columns corresponding to those control--points are \(0\).

This also leads to useless increased complexity, as the parameters
corresponding to those points will never have any effect, but a naive
algorithm will still try to optimize them yielding numeric artefacts in
the best and non--terminating or ill--defined solutions\footnote{One
  example would be, when parts of an algorithm depend on the inverse of
  the minimal right singular value leading to a division by \(0\).} at
worst.

One can of course neglect those columns and their corresponding
control--points, but this raises the question why they were introduced
in the first place. We will address this in a special scenario in
\ref{sec:res:3d:var}.

For our tests we chose different uniformly sized grids and added noise
onto each control--point\footnote{For the special case of the outer
  layer we only applied noise away from the object, so the object is
  still confined in the convex hull of the control--points.} to simulate
different starting--conditions.

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

\label{sec:eval}

In our experiments we use the same two testing--scenarios, that were
also used by Richter et al.\cite{anrichterEvol} The first scenario
deforms a plane into a shape originally defined by Giannelli et
al.\cite{giannelli2012thb}, where we setup control--points in a
2--dimensional manner and merely deform in the height--coordinate to get
the resulting shape.

In the second scenario we increase the degrees of freedom significantly
by using a 3--dimensional control--grid to deform a sphere into a face,
so each control point has three degrees of freedom in contrast to first
scenario.

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

In this scenario we used the shape defined by Giannelli et
al.\cite{giannelli2012thb}, which is also used by Richter et
al.\cite{anrichterEvol} using the same discretization to
\(150 \times 150\) points for a total of \(n = 22\,500\) vertices. The
shape is given by the following definition

\begin{equation}
t(x,y) =
\begin{cases}
0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 & q(x,y) < \frac{1}{16},\\
2(y-x) & 0 < y-x < 0.5,\\
1 & 0.5 < y - x
\end{cases}
\end{equation}

with \((x,y) \in [0,2] \times [0,1]\) and
\(q(x,y)=(x-1.5)^2 + (y-0.5)^2\), which we have visualized in figure
\ref{fig:1dtarget}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{img/1dtarget.png}
\end{center}
\caption[The 1D--target--shape]{The target--shape for our 1--dimensional optimization--scenario
including a wireframe--overlay of the vertices.}
\label{fig:1dtarget}
\end{figure}

As the starting--plane we used the same shape, but set all
\(z\)--coordinates to \(0\), yielding a flat plane, which is partially
already correct.

Regarding the \emph{fitness--function} \(\mathrm{f}(\vec{p})\), we use
the very simple approach of calculating the squared distances for each
corresponding vertex

\begin{equation}
\mathrm{f}(\vec{p}) = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
\end{equation}

where \(t_i\) are the respective target--vertices to the parametrized
source--vertices\footnote{The parametrization is encoded in \(\vec{U}\)
  and the initial position of the control--points. See
  \ref{sec:ffd:adapt}} with the current deformation--parameters
\(\vec{p} = (p_1,\dots, p_m)\). We can do this
one--to--one--correspondence because we have exactly the same number of
source and target--vertices do to our setup of just flattening the
object.

This formula is also the least--squares approximation error for which we
can compute the analytic solution \(\vec{p^{*}} = \vec{U^+}\vec{t}\),
yielding us the correct gradient in which the evolutionary optimizer
should move.

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

\label{sec:test:3dfa} Opposed to the 1--dimensional scenario before, the
3--dimensional scenario is much more complex --- not only because we
have more degrees of freedom on each control point, but also, because
the \emph{fitness--function} we will use has no known analytic solution
and multiple local minima.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{img/3dtarget.png}
\end{center}
\caption[3D source and target meshes]{\newline
Left: The sphere we start from with 10\,807 vertices\newline
Right: The face we want to deform the sphere into with 12\,024 vertices.}
\label{fig:3dtarget}
\end{figure}

First of all we introduce the set up: We have given a triangulated model
of a sphere consisting of \(10\,807\) vertices, that we want to deform
into a the target--model of a face with a total of \(12\,024\) vertices.
Both of these Models can be seen in figure \ref{fig:3dtarget}.

Opposed to the 1D--case we cannot map the source and target--vertices in
a one--to--one--correspondence, which we especially need for the
approximation of the fitting--error. Hence we state that the error of
one vertex is the distance to the closest vertex of the respective other
model and sum up the error from the source and target.

We therefore define the \emph{fitness--function} to be:

\begin{equation}
\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
\vec{s_i}\|_2^2}_{\textrm{source--to--target--distance}}
+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
+ \lambda \cdot \textrm{regularization}(\vec{P})
\label{eq:fit3d}
\end{equation}

where \(\vec{c_T(s_i)}\) denotes the target--vertex that is
corresponding to the source--vertex \(\vec{s_i}\) and \(\vec{c_S(t_i)}\)
denotes the source--vertex that corresponds to the target--vertex
\(\vec{t_i}\). Note that the target--vertices are given and fixed by the
target--model of the face we want to deform into, whereas the
source--vertices vary depending on the chosen parameters \(\vec{P}\), as
those get calculated by the previously introduces formula
\(\vec{S} = \vec{UP}\) with \(\vec{S}\) being the \(n \times 3\)--matrix
of source--vertices, \(\vec{U}\) the \(n \times m\)--matrix of
calculated coefficients for the \ac{FFD} --- analog to the 1D case ---
and finally \(\vec{P}\) being the \(m \times 3\)--matrix of the
control--grid defining the whole deformation.

As regularization--term we add a weighted Laplacian of the deformation
that has been used before by Aschenbach et
al.\cite[Section 3.2]{aschenbach2015} on similar models and was shown to
lead to a more precise fit. The Laplacian

\begin{equation}
\mathrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s}_j \in \mathcal{N}(\vec{s}_i)} w_j \cdot \|\Delta \vec{s}_j - \Delta \vec{s}_i\|^2 \right)
\label{eq:reg3d}
\end{equation}

is determined by the cotangent weighted displacement \(w_j\) of the to
\(s_i\) connected vertices \(\mathcal{N}(s_i)\) and \(A_i\) is the
Voronoi--area of the corresponding vertex \(\vec{s_i}\). We leave out
the \(\vec{R}_i\)--term from the original paper as our deformation is
merely linear.

This regularization--weight gives us a measure of stiffness for the
material that we will influence via the \(\lambda\)--coefficient to
start out with a stiff material that will get more flexible per
iteration. As a side--effect this also limits the effects of
overagressive movement of the control--points in the beginning of the
fitting process and thus should limit the generation of ill--defined
grids mentioned in section \ref{sec:impl:grid}.

\chapter{Evaluation of Scenarios}\label{evaluation-of-scenarios}

\label{sec:res}

To compare our results to the ones given by Richter et
al.\cite{anrichterEvol}, we also use Spearman's rank correlation
coefficient. Opposed to other popular coefficients, like the Pearson
correlation coefficient, which measures a linear relationship between
variables, the Spearman's coefficient assesses \glqq how well an
arbitrary monotonic function can describe the relationship between two
variables, without making any assumptions about the frequency
distribution of the variables\grqq\cite{hauke2011comparison}.

As we don't have any prior knowledge if any of the criteria is linear
and we are just interested in a monotonic relation between the criteria
and their predictive power, the Spearman's coefficient seems to fit out
scenario best and was also used before by Richter et
al.\cite{anrichterEvol}

For interpretation of these values we follow the same interpretation
used in \cite{anrichterEvol}, based on \cite{weir2015spearman}: The
coefficient intervals \(r_S \in [0,0.2[\), \([0.2,0.4[\), \([0.4,0.6[\),
\([0.6,0.8[\), and \([0.8,1]\) are classified as \emph{very weak},
\emph{weak}, \emph{moderate}, \emph{strong} and \emph{very strong}. We
interpret p--values smaller than \(0.01\) as \emph{significant} and cut
off the precision of p--values after four decimal digits (thus often
having a p--value of \(0\) given for p--values \(< 10^{-4}\)).

As we are looking for anti--correlation (i.e.~our criterion should be
maximized indicating a minimal result in --- for example --- the
reconstruction--error) instead of correlation we flip the sign of the
correlation--coefficient for readability and to have the
correlation--coefficients be in the classification--range given above.

For the evolutionary optimization we employ the \afc{CMA--ES} of the
shark3.1 library \cite{shark08}, as this algorithm was used by
\cite{anrichterEvol} as well. We leave the parameters at their sensible
defaults as further explained in
\cite[Appendix~A: Table~1]{hansen2016cma}.

\section{Procedure: 1D Function
Approximation}\label{procedure-1d-function-approximation}

\label{sec:proc:1d}

For our setup we first compute the coefficients of the
deformation--matrix and use the formulas for \emph{variability} and
\emph{regularity} to get our predictions. Afterwards we solve the
problem analytically to get the (normalized) correct gradient that we
use as guess for the \emph{improvement potential}. To further test the
\emph{improvement potential} we also consider a distorted gradient
\(\vec{g}_{\mathrm{d}}\): \[
\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
\] where \(\mathbb{1}\) is the vector consisting of \(1\) in every
dimension, \(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\) is the
calculated correct gradient, and \(\mu\) is used to blend between
\(\vec{g}_\mathrm{c}\) and \(\mathbb{1}\). As we always start with a
gradient of \(p = \mathbb{0}\) this means we can shorten the definition
of \(\vec{g}_\mathrm{c}\) to \(\vec{g}_\mathrm{c} = \vec{p^{*}}\).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{img/example1d_grid.png}
\end{center}
\caption[Example of a 1D--grid]{\newline Left: A regular $7 \times 4$--grid\newline Right: The same grid after a
random distortion to generate a testcase.}
\label{fig:example1d_grid}
\end{figure}

We then set up a regular 2--dimensional grid around the object with the
desired grid resolutions. To generate a testcase we then move the
grid--vertices randomly inside the x--y--plane. As self--intersecting
grids get tricky to solve with our implemented newtons--method (see
section \ref{3dffd}) we avoid the generation of such self--intersecting
grids for our testcases.

To achieve that we generated a gaussian distributed number with
\(\mu = 0, \sigma=0.25\) and clamped it to the range \([-0.25,0.25]\).
We chose such an \(r \in [-0.25,0.25]\) per dimension and moved the
control--points by that factor towards their respective
neighbours\footnote{Note: On the Edges this displacement is only applied
  outwards by flipping the sign of \(r\), if appropriate.}.

In other words we set

\begin{equation*}
p_i =
\begin{cases}
  p_i + (p_i - p_{i-1}) \cdot r, & \textrm{if } r \textrm{ negative} \\
  p_i + (p_{i+1} - p_i) \cdot r, & \textrm{if } r \textrm{ positive}
\end{cases}
\end{equation*}

in each dimension separately.

An Example of such a testcase can be seen for a \(7 \times 4\)--grid in
figure \ref{fig:example1d_grid}.

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

In the case of our 1D--Optimization--problem, we have the luxury of
knowing the analytical solution to the given problem--set. We use this
to experimentally evaluate the quality criteria we introduced before. As
an evolutional optimization is partially a random process, we use the
analytical solution as a stopping--criteria. We measure the convergence
speed as number of iterations the evolutional algorithm needed to get
within \(1.05 \times\) of the optimal solution.

We used different regular grids that we manipulated as explained in
Section \ref{sec:proc:1d} with a different number of control--points. As
our grids have to be the product of two integers, we compared a
\(5 \times 5\)--grid with \(25\) control--points to a \(4 \times 7\) and
\(7 \times 4\)--grid with \(28\) control--points. This was done to
measure the impact an \glqq improper\grqq ~ setup could have and how
well this is displayed in the criteria we are examining.

Additionally we also measured the effect of increasing the total
resolution of the grid by taking a closer look at \(5 \times 5\),
\(7 \times 7\) and \(10 \times 10\) grids.

\subsection{Variability}\label{variability-1}

\begin{figure}[tbh]
\centering
\includegraphics[width=0.7\textwidth]{img/evolution1d/variability_boxplot.png}
\caption[1D Fitting Errors for various grids]{The squared error for the various
grids we examined.\newline
Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.}
\label{fig:1dvar}
\end{figure}

\emph{Variability} should characterize the potential for design space
exploration and is defined in terms of the normalized rank of the
deformation matrix \(\vec{U}\):
\(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\), whereby \(n\) is the
number of vertices. As all our tested matrices had a constant rank
(being \(m = x \cdot y\) for a \(x \times y\) grid), we have merely
plotted the errors in the box plot in figure \ref{fig:1dvar}

It is also noticeable, that although the \(7 \times 4\) and
\(4 \times 7\) grids have a higher \emph{variability}, they perform not
better than the \(5 \times 5\) grid. Also the \(7 \times 4\) and
\(4 \times 7\) grids differ distinctly from each other with a
mean\(\pm\)sigma of \(233.09 \pm 12.32\) for the former and
\(286.32 \pm 22.36\) for the latter, although they have the same number
of control--points. This is an indication of an impact a proper or
improper grid--setup can have. We do not draw scientific conclusions
from these findings, as more research on non--squared grids seem
necessary.

Leaving the issue of the grid--layout aside we focused on grids having
the same number of prototypes in every dimension. For the
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) grids we found a
\emph{very strong} correlation (\(-r_S = 0.94, p = 0\)) between the
\emph{variability} and the evolutionary error.

\subsection{Regularity}\label{regularity-1}

\begin{figure}[tbh]
\centering
\includegraphics[width=\textwidth]{img/evolution1d/55_to_1010_steps.png}
\caption[Improvement potential and regularity against iterations]{\newline
Left: *Improvement potential* against number of iterations until convergence\newline
Right: *Regularity* against number of iterations until convergence\newline
Coloured by their grid--resolution, both with a linear fit over the whole
dataset.}
\label{fig:1dreg}
\end{figure}

\begin{table}[b]
\centering
\begin{tabular}{c|c|c|c|c}
$5 \times 5$ & $7 \times 4$ & $4 \times 7$ & $7 \times 7$ & $10 \times 10$\\
\hline
$0.28$ ($0.0045$) & \textcolor{red}{$0.21$} ($0.0396$) & \textcolor{red}{$0.1$} ($0.3019$) & \textcolor{red}{$0.01$} ($0.9216$) & \textcolor{red}{$0.01$} ($0.9185$)
\end{tabular}
\caption[Correlation 1D *regularity* against iterations]{Inverted Spearman's correlation (and p--values)
between *regularity* and number of iterations for the 1D function approximation
problem.
\newline Note: Not significant results are marked in \textcolor{red}{red}.
}
\label{tab:1dreg}
\end{table}

\emph{Regularity} should correspond to the convergence speed (measured
in iteration--steps of the evolutionary algorithm), and is computed as
inverse condition number \(\kappa(\vec{U})\) of the deformation--matrix.

As can be seen from table \ref{tab:1dreg}, we could only show a
\emph{weak} correlation in the case of a \(5 \times 5\) grid. As we
increment the number of control--points the correlation gets worse until
it is completely random in a single dataset. Taking all presented
datasets into account we even get a \emph{strong} correlation of
\(- r_S = -0.72, p = 0\), that is opposed to our expectations.

To explain this discrepancy we took a closer look at what caused these
high number of iterations. In figure \ref{fig:1dreg} we also plotted the
\emph{improvement potential} against the steps next to the
\emph{regularity}--plot. Our theory is that the \emph{very strong}
correlation (\(-r_S = -0.82, p=0\)) between \emph{improvement potential}
and number of iterations hints that the employed algorithm simply takes
longer to converge on a better solution (as seen in figure
\ref{fig:1dvar} and \ref{fig:1dimp}) offsetting any gain the
regularity--measurement could achieve.

\subsection{Improvement Potential}\label{improvement-potential-1}

\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution1d/55_to_1010_improvement-vs-evo-error.png}
\caption[Correlation 1D Improvement vs. Error]{*Improvement potential* plotted
against the error yielded by the evolutionary optimization for different
grid--resolutions}
\label{fig:1dimp}
\end{figure}

The \emph{improvement potential} should correlate to the quality of the
fitting--result. We plotted the results for the tested grid--sizes
\(5 \times 5\), \(7 \times 7\) and \(10 \times 10\) in figure
\ref{fig:1dimp}. We tested the \(4 \times 7\) and \(7 \times 4\) grids
as well, but omitted them from the plot.

Additionally we tested the results for a distorted gradient described in
\ref{sec:proc:1d} with a \(\mu\)--value of \(0.25\), \(0.5\), \(0,75\),
and \(1.0\) for the \(5 \times 5\) grid and with a \(\mu\)--value of
\(0.5\) for all other cases.

All results show the identical \emph{very strong} and \emph{significant}
correlation with a Spearman--coefficient of \(- r_S = 1.0\) and p--value
of \(0\).

These results indicate, that \(\|\mathbb{1} - \vec{U}\vec{U}^{+}\|_F\)
is close to \(0\), reducing the impacts of any kind of gradient.
Nevertheless, the improvement potential seems to be suited to make
estimated guesses about the quality of a fit, even lacking an exact
gradient.

\section{Procedure: 3D Function
Approximation}\label{procedure-3d-function-approximation}

\label{sec:proc:3dfa}

As explained in section \ref{sec:test:3dfa} in detail, we do not know
the analytical solution to the global optimum. Additionally we have the
problem of finding the right correspondences between the original
sphere--model and the target--model, as they consist of \(10\,807\) and
\(12\,024\) vertices respectively, so we cannot make a
one--to--one--correspondence between them as we did in the
one--dimensional case.

Initially we set up the correspondences \(\vec{c_T(\dots)}\) and
\(\vec{c_S(\dots)}\) to be the respectively closest vertices of the
other model. We then calculate the analytical solution given these
correspondences via \(\vec{P^{*}} = \vec{U^+}\vec{T}\), and also use the
first solution as guessed gradient for the calculation of the
\emph{improvement potential}, as the optimal solution is not known. We
then let the evolutionary algorithm run up within \(1.05\) times the
error of this solution and afterwards recalculate the correspondences
\(\vec{c_T(\dots)}\) and \(\vec{c_S(\dots)}\).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{img/example3d_grid.png}
\end{center}
\caption[Example of a 3D--grid]{\newline Left: The 3D--setup with a $4\times
4\times 4$--grid.\newline Right: The same grid after added noise to the
control--points.}
\label{fig:setup3d}
\end{figure}

For the next step we then halve the regularization--impact \(\lambda\)
(starting at \(1\)) of our \emph{fitness--function} (\ref{eq:fit3d}) and
calculate the next incremental solution
\(\vec{P^{*}} = \vec{U^+}\vec{T}\) with the updated correspondences
(again, mapping each vertex to its closest neighbour in the respective
other model) to get our next target--error. We repeat this process as
long as the target--error keeps decreasing and use the number of these
iterations as measure of the convergence speed. As the resulting
evolutional error without regularization is in the numeric range of
\(\approx 100\), whereas the regularization is numerically
\(\approx 7000\) we need at least \(10\) to \(15\) iterations until the
regularization--effect wears off.

The grid we use for our experiments is just very coarse due to
computational limitations. We are not interested in a good
reconstruction, but an estimate if the mentioned evolvability--criteria
are good.

In figure \ref{fig:setup3d} we show an example setup of the scene with a
\(4\times 4\times 4\)--grid. Identical to the 1--dimensional scenario
before, we create a regular grid and move the control--points in the
exact same random manner between their neighbours as described in
section \ref{sec:proc:1d}, but in three instead of two
dimensions\footnote{Again, we flip the signs for the edges, if necessary
  to have the object still in the convex hull.}.

\begin{figure}[!htb]
\includegraphics[width=\textwidth]{img/3d_grid_resolution.png}
\caption[Different resolution of 3D grids]{\newline
Left: A $7 \times 4 \times 4$ grid suited to better deform into facial
features.\newline
Right: A $4 \times 4 \times 7$ grid that we expect to perform worse.}
\label{fig:3dgridres}
\end{figure}

As is clearly visible from figure \ref{fig:3dgridres}, the target--model
has many vertices in the facial area, at the ears and in the
neck--region. Therefore we chose to increase the grid--resolutions for
our tests in two different dimensions and see how well the criteria
predict a suboptimal placement of these control--points.

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

In the 3D--Approximation we tried to evaluate further on the impact of
the grid--layout to the overall criteria. As the target--model has many
vertices in concentrated in the facial area we start from a
\(4 \times 4 \times 4\) grid and only increase the number of
control--points in one dimension, yielding a resolution of
\(7 \times 4 \times 4\) and \(4 \times 4 \times 7\) respectively. We
visualized those two grids in figure \ref{fig:3dgridres}.

To evaluate the performance of the evolvability--criteria we also tested
a more neutral resolution of \(4 \times 4 \times 4\),
\(5 \times 5 \times 5\), and \(6 \times 6 \times 6\) --- similar to the
1D--setup.

\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{img/evolution3d/variability_boxplot.png}
\caption[3D Fitting Errors for various grids]{The fitting error for the various
grids we examined.\newline
Note that the number of control--points is a product of the resolution, so $X
\times 4 \times 4$ and $4 \times 4 \times X$ have the same number of
control--points.}
\label{fig:3dvar}
\end{figure}

\subsection{Variability}\label{variability-2}

\label{sec:res:3d:var}

\begin{table}[tbh]
\centering
\begin{tabular}{c|c|c|c}
$4 \times 4 \times \mathrm{X}$ & $\mathrm{X} \times 4 \times 4$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ & all \\
\hline
0.89 (0) & 0.9 (0) & 0.91 (0) & 0.94 (0)
\end{tabular}
\caption[Correlation between *variability* and fitting error for 3D]{Correlation
between *variability* and fitting error for the 3D fitting scenario.\newline
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for three cases of increasing *variability* ($\mathrm{X} \in [4,5,7],
\mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dvar}
\end{table}

Similar to the 1D case all our tested matrices had a constant rank
(being \(m = x \cdot y \cdot z\) for a \(x \times y \times z\) grid), so
we again have merely plotted the errors in the box plot in figure
\ref{fig:3dvar}.

As expected the \(\mathrm{X} \times 4 \times 4\) grids performed
slightly better than their \(4 \times 4 \times \mathrm{X}\) counterparts
with a mean\(\pm\)sigma of \(101.25 \pm 7.45\) to \(102.89 \pm 6.74\)
for \(\mathrm{X} = 5\) and \(85.37 \pm 7.12\) to \(89.22 \pm 6.49\) for
\(\mathrm{X} = 7\).

Interestingly both variants end up closer in terms of fitting error than
we anticipated, which shows that the evolutionary algorithm we employed
is capable of correcting a purposefully created \glqq bad\grqq ~grid.
Also this confirms, that in our cases the number of control--points is
more important for quality than their placement, which is captured by
the \emph{variability} via the rank of the deformation--matrix.

Overall the correlation between \emph{variability} and fitness--error
were \emph{significant} and showed a \emph{very strong} correlation in
all our tests. The detailed correlation--coefficients are given in table
\ref{tab:3dvar} alongside their p--values.

As introduces in section \ref{sec:impl:grid} and visualized in figure
\ref{fig:enoughCP}, we know, that not all control--points have to
necessarily contribute to the parametrization of our 3D--model. Because
we are starting from a sphere, some control--points are too far away
from the surface to contribute to the deformation at all.

One can already see in 2D in figure \ref{fig:enoughCP}, that this effect
starts with a regular \(9 \times 9\) grid on a perfect circle. To make
sure we observe this, we evaluated the \emph{variability} for 100
randomly moved \(10 \times 10 \times 10\) grids on the sphere we start
out with.

\begin{figure}[hbt]
\centering
\includegraphics[width=0.8\textwidth]{img/evolution3d/variability2_boxplot.png}
\caption[Histogram of ranks of high--resolution deformation--matrices]{
Histogram of ranks of various $10 \times 10 \times 10$ grids with $1000$
control--points each showing in this case how many control--points are actually
used in the calculations.
}
\label{fig:histrank3d}
\end{figure}

As the \emph{variability} is defined by
\(\frac{\mathrm{rank}(\vec{U})}{n}\) we can easily recover the rank of
the deformation--matrix \(\vec{U}\). The results are shown in the
histogram in figure \ref{fig:histrank3d}. Especially in the centre of
the sphere and in the corners of our grid we effectively loose
control--points for our parametrization.

This of course yields a worse error as when those control--points would
be put to use and one should expect a loss in quality evident by a
higher reconstruction--error opposed to a grid where they are used.
Sadly we could not run a in--depth test on this due to computational
limitations.

Nevertheless this hints at the notion, that \emph{variability} is a good
measure for the overall quality of a fit.

\subsection{Regularity}\label{regularity-2}

\begin{table}[tbh]
\centering
\begin{tabular}{c|c|c|c}
 & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\
\cline{2-4}
 & \textcolor{red}{0.15} (0.147) & \textcolor{red}{0.09} (0.37) & 0.46 (0) \B \\
\cline{2-4}
\multicolumn{4}{c}{} \\[-1.4em]
\hline
$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\
\hline
0.38 (0) & \textcolor{red}{0.17} (0.09) & 0.40 (0) & 0.46 (0) \B \\
\hline
\multicolumn{4}{c}{} \\[-1.4em]
\cline{2-4}
 & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\
\cline{2-4}
 & \textcolor{red}{-0.18} (0.0775) & \textcolor{red}{-0.13} (0.1715) & -0.25 (0) \B \\
\cline{2-4}
\multicolumn{4}{c}{} \\[-1.4em]
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.15 (0) \T
\end{tabular}
\caption[Correlation between *regularity* and iterations for 3D]{Correlation
between *regularity* and number of iterations for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dreg}
\end{table}

Opposed to the predictions of \emph{variability} our test on
\emph{regularity} gave a mixed result --- similar to the 1D--case.

In roughly half of the scenarios we have a \emph{significant}, but
\emph{weak} to \emph{moderate} correlation between \emph{regularity} and
number of iterations. On the other hand in the scenarios where we
increased the number of control--points, namely \(125\) for the
\(5 \times 5 \times 5\) grid and \(216\) for the \(6 \times 6 \times 6\)
grid we found a \emph{significant}, but \emph{weak}
\textbf{anti}--correlation when taking all three tests into
account\footnote{Displayed as \(Y \times Y \times Y\)}, which seem to
contradict the findings/trends for the sets with \(64\), \(80\), and
\(112\) control--points (first two rows of table \ref{tab:3dreg}).

Taking all results together we only find a \emph{very weak}, but
\emph{significant} link between \emph{regularity} and the number of
iterations needed for the algorithm to converge.

\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{img/evolution3d/regularity_montage.png}
\caption[Regularity for different 3D--grids]{
Plots of *regularity* against number of iterations for various scenarios together
with a linear fit to indicate trends.}
\label{fig:resreg3d}
\end{figure}

As can be seen from figure \ref{fig:resreg3d}, we can observe that
increasing the number of control--points helps the convergence--speeds.
The regularity--criterion first behaves as we would like to, but then
switches to behave exactly opposite to our expectations, as can be seen
in the first three plots. While the number of control--points increases
from red to green to blue and the number of iterations decreases, the
\emph{regularity} seems to increase at first, but then decreases again
on higher grid--resolutions.

This can be an artefact of the definition of \emph{regularity}, as it is
defined by the inverse condition--number of the deformation--matrix
\(\vec{U}\), being the fraction
\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\) between the
least and greatest right singular value.

As we observed in the previous section, we cannot guarantee that each
control--point has an effect (see figure \ref{fig:histrank3d}) and so a
small minimal right singular value occurring on higher grid--resolutions
seems likely the problem.

Adding to this we also noted, that in the case of the
\(10 \times 10 \times 10\)--grid the \emph{regularity} was always \(0\),
as a non--contributing control--point yields a \(0\)--column in the
deformation--matrix, thus letting \(\sigma_\mathrm{min} = 0\). A better
definition for \emph{regularity} (i.e.~using the smallest non--zero
right singular value) could solve this particular issue, but not fix the
trend we noticed above.

\subsection{Improvement Potential}\label{improvement-potential-2}

\begin{table}[tbh]
\centering
\begin{tabular}{c|c|c|c}
 & $5 \times 4 \times 4$ & $7 \times 4 \times 4$ & $\mathrm{X} \times 4 \times 4$ \\
\cline{2-4}
 & 0.3 (0.0023) & \textcolor{red}{0.23} (0.0233) & 0.89 (0) \B \\
\cline{2-4}
\multicolumn{4}{c}{} \\[-1.4em]
\hline
$4 \times 4 \times 4$ & $4 \times 4 \times 5$ & $4 \times 4 \times 7$ & $4 \times 4 \times \mathrm{X}$ \T \\
\hline
0.5 (0) & 0.38 (0) & 0.32 (0.0012) & 0.9 (0) \B \\
\hline
\multicolumn{4}{c}{} \\[-1.4em]
\cline{2-4}
 & $5 \times 5 \times 5$ & $6 \times 6 \times 6$ & $\mathrm{Y} \times \mathrm{Y} \times \mathrm{Y}$ \T \\
\cline{2-4}
 & 0.47 (0) & \textcolor{red}{-0.01} (0.8803) & 0.89 (0) \B \\
\cline{2-4}
\multicolumn{4}{c}{} \\[-1.4em]
\cline{2-4}
\multicolumn{3}{c}{} & all: 0.95 (0) \T
\end{tabular}
\caption[Correlation between *improvement potential* and fitting--error for 3D]{Correlation
between *improvement potential* and fitting--error for the 3D fitting scenario.
Displayed are the negated Spearman coefficients with the corresponding p--values
in brackets for various given grids ($\mathrm{X} \in [4,5,7], \mathrm{Y} \in [4,5,6]$).
\newline Note: Not significant results are marked in \textcolor{red}{red}.}
\label{tab:3dimp}
\end{table}

Comparing to the 1D--scenario, we do not know the optimal solution to
the given problem and for the calculation we only use the initial
gradient produced by the initial correlation between both objects. This
gradient changes with every iteration and will be off our first guess
very quickly. This is the reason we are not trying to create
artificially bad gradients, as we have a broad range in quality of such
gradients anyway.

\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{img/evolution3d/improvement_montage.png}
\caption[Improvement potential for different 3D--grids]{
Plots of *improvement potential* against error given by our *fitness--function*
after convergence together with a linear fit of each of the plotted data to
indicate trends.}
\label{fig:resimp3d}
\end{figure}

We plotted our findings on the \emph{improvement potential} in a similar
way as we did before with the \emph{regularity}. In figure
\ref{fig:resimp3d} one can clearly see the correlation and the spread
within each setup and the behaviour when we increase the number of
control--points.

Along with this we also give the Spearman--coefficients along with their
p--values in table \ref{tab:3dimp}. Within one scenario we only find a
\emph{weak} to \emph{moderate} correlation between the \emph{improvement
potential} and the fitting error, but all findings (except for
\(7 \times 4 \times 4\) and \(6 \times 6 \times 6\)) are significant.

If we take multiple datasets into account the correlation is \emph{very
strong} and \emph{significant}, which is good, as this functions as a
litmus--test, because the quality is naturally tied to the number of
control--points.

All in all the \emph{improvement potential} seems to be a good and
sensible measure of quality, even given gradients of varying quality.

Lastly, a small note on the behaviour of \emph{improvement potential}
and convergence speed, as we used this in the 1D case to argue, why the
\emph{regularity} defied our expectations. As a contrast we wanted to
show, that \emph{improvement potential} cannot serve for good
predictions of the convergence speed. In figure \ref{fig:imp1d3d} we
show \emph{improvement potential} against number of iterations for both
scenarios. As one can see, in the 1D scenario we have a \emph{strong}
and \emph{significant} correlation (with \(-r_S = -0.72\), \(p = 0\)),
whereas in the 3D scenario we have the opposite \emph{significant} and
\emph{strong} effect (with \(-r_S = 0.69\), \(p=0\)), so these
correlations clearly seem to be dependent on the scenario and are not
suited for generalization.

\begin{figure}[hbt]
\centering
\includegraphics[width=\textwidth]{img/imp1d3d.png}
\caption[Improvement potential and convergence speed\newline for 1D and 3D--scenarios]{
\newline
Left: *Improvement potential* against convergence speed for the
1D--scenario\newline
Right: *Improvement potential* against convergence speed for the 3D--scnario
}
\label{fig:imp1d3d}
\end{figure}

\chapter{Discussion and outlook}\label{discussion-and-outlook}

\label{sec:dis}

In this thesis we took a look at the different criteria for
\emph{evolvability} as introduced by Richter et al.\cite{anrichterEvol},
namely \emph{variability}, \emph{regularity} and \emph{improvement
potential} under different setup--conditions. Where Richter et al. used
\acf{RBF}, we employed \acf{FFD} to set up a low--complexity
parametrization of a more complex vertex--mesh.

In our findings we could show in the 1D--scenario, that there were
statistically \emph{significant} \emph{very strong} correlations between
\emph{variability and fitting error} (\(0.94\)) and \emph{improvement
potential and fitting error} (\(1.0\)) with comparable results than
Richter et al. (with \(0.31\) to \(0.88\) for the former and \(0.75\) to
\(0.99\) for the latter), whereas we found only \emph{weak} correlations
for \emph{regularity and convergence--speed} (\(0.28\)) opposed to
Richter et al. with \(0.39\) to \(0.91\).\footnote{We only took
  statistically \emph{significant} results into consideration when
  compiling these numbers. Details are given in the respective chapters.}

For the 3D--scenario our results show a \emph{very strong},
\emph{significant} correlation between \emph{variability and fitting
error} with \(0.89\) to \(0.94\), which are pretty much in line with the
findings of Richter et al. (\(0.65\) to \(0.95\)). The correlation
between \emph{improvement potential and fitting error} behave similar,
with our findings having a significant coefficient of \(0.3\) to
\(0.95\) depending on the grid--resolution compared to the \(0.61\) to
\(0.93\) from Richter et al. In the case of the correlation of
\emph{regularity and convergence speed} we found very different (and
often not significant) correlations and anti--correlations ranging from
\(-0.25\) to \(0.46\), whereas Richter et al. reported correlations
between \(0.34\) to \(0.87\).

Taking these results into consideration, one can say, that
\emph{variability} and \emph{improvement potential} are very good
estimates for the quality of a fit using \acf{FFD} as a deformation
function, while we could not reproduce similar compelling results as
Richter et al. for \emph{regularity and convergence speed}.

One reason for the bad or erratic behaviour of the
\emph{regularity}--criterion could be that in an \ac{FFD}--setting we
have a likelihood of having control--points that are only contributing
to the whole parametrization in negligible amounts, resulting in very
small right singular values of the deformation--matrix \(\vec{U}\) that
influence the condition--number and thus the \emph{regularity} in a
significant way. Further research is needed to refine \emph{regularity}
so that these problems get addressed, like taking all singular values
into account when capturing the notion of \emph{regularity}.

Richter et al. also compared the behaviour of direct and indirect
manipulation in \cite{anrichterEvol}, whereas we merely used an indirect
\ac{FFD}--approach. As direct manipulations tend to perform better than
indirect manipulations, the usage of \acf{DM--FFD} could also work
better with the criteria we examined. This can also solve the problem of
bad singular values for the \emph{regularity} as the incorporation of
the parametrization of the points on the surface --- which are the
essential part of a direct--manipulation --- could cancel out a bad
control--grid as the bad control--points are never or negligibly used to
parametrize those surface--points.

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