---
fontsize: 12pt
---

\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:

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

# 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.

Evolutional 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
*evolvability*\cite{wagner1996complex} and is at the core of this thesis.
However, there is no consensus on how *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 *regularity*, *variability*, and *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
*regularity*, *variability*, and *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}.

## 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 evolutional 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}.


# Background
\label{sec:back}

## What is \acf{FFD}?
\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 $i$^[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
$N$^[*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.

### Why is \ac{FFD} 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 *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.

## What is evolutional optimization?
\label{sec:back:evo}

In this thesis we are using an evolutional 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 evolutional 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 evolutional 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 *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:

- **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.
- **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.
- **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.

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). As the recombination and selection-steps are usually pure 

## Advantages of evolutional algorithms
\label{sec:back:evogood}

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

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

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

## Criteria for the evolvability of linear deformations
\label{sec:intro:rvi}

### Variability

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

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

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

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

### Regularity

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

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

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

### Improvement Potential

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

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

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

# Implementation of \acf{FFD}
\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 *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.


## Adaption of \ac{FFD}

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 ($\#v~\gg~\#c$).

## Adaption of \ac{FFD} for a 3D--Mesh
\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.


## Parametrisierung sinnvoll?

- Nachteile von Parametrisierung
  - Deformation ist um einen Kontrollpunkt viel direkter zu steuern.
  - => DM--FFD?


# Scenarios for testing evolvability criteria using \acf{FFD}
\label{sec:eval}

## Test Scenario: 1D Function Approximation

### Optimierungszenario

- Ebene -> Template--Fit

### Matching in 1D

- Trivial

### Besonderheiten der Auswertung

- Analytische Lösung einzig beste
  - Ergebnis auch bei Rauschen konstant?
  - normierter 1--Vektor auf den Gradienten addieren
    - Kegel entsteht

## Test Scenario: 3D Function Approximation

### Optimierungsszenario

- Ball zu Mario

### Matching in 3D

- alternierende Optimierung

### Besonderheiten der Optimierung

- Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig
- Kriterien trotzdem gut

# Evaluation of Scenarios
\label{sec:res}

## Spearman/Pearson--Metriken

- Was ist das?
- Wieso sollte uns das interessieren?
- Wieso reicht Monotonie?
- Haben wir das gezeigt?
- Statistik, Bilder, blah!

## Results of 1D Function Approximation

\begin{figure}[!ht]
\includegraphics[width=\textwidth]{img/evolution1d/20170830-evolution1D_5x5_100Times-all_appended.png}
\caption{Results 1D}

\end{figure}

<!-- ![Improvement potential vs steps](img/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-steps.png) -->
<!--  -->
<!-- ![Improvement potential vs evolutional error](img/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-evo-error.png) -->
<!--  -->
<!-- ![Regularity vs steps](img/evolution1d/20170830-evolution1D_5x5_100Times-all_regularity-vs-steps.png) -->

## Results of 3D Function Approximation


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

\end{figure}

<!-- ![Improvement potential vs steps](img/evolution3d/20170926_3dFit_both_improvement-vs-steps.png) -->
<!--  -->
<!-- ![Improvement potential vs evolutional -->
<!-- error](img/evolution3d/20170926_3dFit_both_improvement-vs-evo-error.png) -->
<!--  -->
<!-- ![Regularity vs steps](img/evolution3d/20170926_3dFit_both_regularity-vs-steps.png) -->

# Schluss
\label{sec:dis}

HAHA .. als ob -.-
