done?

arbeit/files/erklaerung.tex

@@ -2,10 +2,9 @@
 \thispagestyle{empty}
 \vspace*{\stretch{1}}
 \noindent
-{\huge Declaration of own work(?)}\\[1cm]
+{\huge Declaration of own work}\\[1cm]
 I hereby declare that this thesis is my own work and effort. Where other sources of information have been used, they have been acknowledged.
-\improvement[inline]{write proper declaration..}
+\\[2cm]
-%\\[2cm]
+Bielefeld, \today\hspace{\fill}
-Bielefeld, den \today\hspace{\fill}
 \parbox[t]{5cm}{\dotfill\\ \centering Stefan Dresselhaus}
 \vspace*{\stretch{3}}

arbeit/ma.md

@@ -125,8 +125,8 @@ Chapter \ref{sec:dis}.
 \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 (esp. meshes in our case) in the first
+tool for deforming geometric objects (especially meshes in our case) in the
-place. For simplicity we only summarize the 1D--case from
+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}.
 
@@ -150,7 +150,7 @@ corresponding deformation to generate a deformed objet}
 \end{figure}
 
 In the 1--dimensional example in figure \ref{fig:bspline}, the control--points
-are indicated as red dots and the color-gradient should hint at the $u$--values
+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:  
@@ -169,7 +169,7 @@ N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+
 \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
+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 $i$^[one more for each recursive step.], which gives
 us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local
@@ -216,18 +216,18 @@ where $\vec{N}$ is the $n \times m$ transformation--matrix (later on called
 \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
+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.
+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
+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--splines--basis--functions form a partition of unity for all, but
+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 $\sum_j n_{i,j} p_j = p_i$
 for these points.
@@ -308,9 +308,29 @@ initialized by a random guess or just zero. Further on we need a so--called
 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 *Genotypes*
+Biologically speaking the set $I$ corresponds to the set of possible *genotypes*
-while $M$ represents the possible observable *Phenotypes*.
+while $M$ represents the possible observable *phenotypes*. *Genotypes* define
-\improvement[inline]{Erklären, was das ist. Quellen!}
+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 *phenotypes* make certain behaviour observable (algorithmically through our
+*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 *fitness--function*
+and a so called *genotype--phenotype--mapping*. The first one directly applies
+the *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 *genotype* can be an arbitrary vector (the genes), the
+*phenotype* is then a deformed object, and the performance can be a single
+measurement like an air--drag--coefficient. The *genotype--phenotype--mapping*
+would then just be the generation of different objects from that
+starting--vector, whereas the *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:
 
@@ -335,15 +355,16 @@ can be changed over time. A good overview of this is given in
 
 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 part. An even more sophisticated example would be the \glqq 1/5
+$\sigma$ to every component of those parameters. An even more sophisticated
-success rule\grqq \ from \cite{rechenberg1973evolutionsstrategie}.
+example would be the \glqq 1/5 success rule\grqq \ from
+\cite{rechenberg1973evolutionsstrategie}.
 
-Also in selection it may not be wise to only take the best--performing
+Also in the selection--function it may not be wise to only take the
-individuals, because it may be that the optimization has to overcome a barrier
+best--performing individuals, because it may be that the optimization has to
-of bad fitness to achieve a better local optimum.
+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 grouping into account.
+also take ancestry, distance of genes or groups of individuals into account.
 
 ## Advantages of evolutionary algorithms
 \label{sec:back:evogood}
@@ -364,13 +385,11 @@ are shown in figure \ref{fig:probhard}.
 Most of the advantages stem from the fact that a gradient--based procedure has
 only one point of observation from where it evaluates the next steps, whereas an
 evolutionary strategy starts with a population of guessed solutions. Because an
-evolutionary strategy modifies the solution randomly, keeping some solutions
+evolutionary strategy can be modified according to the problem--domain (i.e. by
-and purging others, it can also target multiple different hypothesis at the
+the ideas given above) it can also approximate very difficult problems in an
-same time where the local optima die out in the face of other, better
+efficient manner and even self--tune parameters depending on the ancestry at
-candidates.
+runtime^[Some examples of this are explained in detail in
-
+\cite{eiben1999parameter}].
-\improvement[inline]{Verweis auf MO-CMA etc. Vielleicht auch etwas
-ausführlicher.}
 
 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.
@@ -381,7 +400,7 @@ 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 the same solution.
+converge to a decent solution much faster than an exhaustive search.
 
 ## Criteria for the evolvability of linear deformations
 \label{sec:intro:rvi}
@@ -445,8 +464,8 @@ 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 
+convergence speed of evolutionary algorithms\cite{anrichterEvol} as it is tied
-the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}.
+to the notion of locality\cite{weise2012evolutionary,thorhauer2014locality}.
 
 ### Improvement Potential
 
@@ -582,7 +601,7 @@ $$
 
 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
+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
@@ -818,10 +837,9 @@ 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 CMA--ES (covariance matrix
+For the evolutionary optimization we employ the \afc{CMA--ES} of the shark3.1
-adaptation evolution strategy) of the shark3.1 library \cite{shark08}, as this
+library \cite{shark08}, as this algorithm was used by \cite{anrichterEvol} as
-algorithm was used by \cite{anrichterEvol} as well. We leave the parameters at
+well. We leave the parameters at their sensible defaults as further explained in
-their sensible defaults as further explained in
 \cite[Appendix~A: Table~1]{hansen2016cma}.
 
 ## Procedure: 1D Function Approximation
@@ -911,7 +929,7 @@ 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 boxplot in figure
+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
@@ -1114,7 +1132,7 @@ in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
 
 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 boxplot in figure \ref{fig:3dvar}.
+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
@@ -1133,7 +1151,8 @@ deformation--matrix.
 \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.
+control--points each showing in this case how many control points are actually
+used in the calculations.
 }
 \label{fig:histrank3d}
 \end{figure}
@@ -1306,7 +1325,7 @@ before with the 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
+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 *weak* to
 *moderate* correlation between the improvement potential and the fitting error,
 but all findings (except for $7 \times 4 \times 4$ and $6 \times 6 \times 6$)
@@ -1319,9 +1338,28 @@ quality is naturally tied to the number of control--points.
 All in all the improvement potential seems to be a good and sensible measure of
 quality, even given gradients of varying quality.
 
-\improvement[inline]{improvement--potential vs. steps ist anders als in 1d! Plot
+Lastly, a small note on the behaviour of improvement potential and convergence
-und zeigen!}
+speed, as we used this in the 1D case to argue, why the *regularity* defied our
+expectations. As a contrast we wanted to show, that improvement potential cannot
+serve for good predictions of the convergence speed. In figure
+\ref{fig:imp1d3d} we show improvement potential against number of iterations
+for both scenarios. As one can see, in the 1D scenario we have a *strong*
+and *significant* correlation (with $-r_S = -0.72$, $p = 0$), whereas in the 3D
+scenario we have the opposite *significant* and *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 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}
 
 # Discussion and outlook
 \label{sec:dis}
@@ -1355,23 +1393,27 @@ between $0.34$ to $0.87$.
 
 Taking these results into consideration, one can say, that *variability* and
 *improvement potential* are very good estimates for the quality of a fit using
-\acf{FFD} as a deformation function.
+\acf{FFD} as a deformation function, while we could not reproduce similar
+compelling results as Richter et al. for *regularity and convergence speed*.
 
 One reason for the bad or erratic behaviour of the *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.
+that are only contributing to the whole parametrization in negligible amounts,
-This results in very small right singular values of the deformation--matrix
+resulting in very small right singular values of the deformation--matrix
 $\vec{U}$ that influence the condition--number and thus the *regularity* in a
 significant way. Further research is needed to refine *regularity* so that these
-problems get addressed.
+problems get addressed, like taking all singular values into account when
+capturing the notion of *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 *regularity* as
-\improvement[inline]{write more outlook/further research}
+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.
 
 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
-Direktlinks des Autors.\newline
+Direktlinks des Autors.}
-Außerdem bricht url über Seitengrenzen den Seitenspiegel.}

arbeit/ma.tex

@@ -278,10 +278,10 @@ outlook in Chapter \ref{sec:dis}.
 \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 (esp. meshes in our case) in
+good tool for deforming geometric objects (especially meshes in our
-the first place. For simplicity we only summarize the 1D--case from
+case) in the first place. For simplicity we only summarize the 1D--case
-\cite{spitzmuller1996bezier} here and go into the extension to the 3D
+from \cite{spitzmuller1996bezier} here and go into the extension to the
-case in chapter~\ref{3dffd}.
+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
@@ -302,8 +302,8 @@ corresponding deformation to generate a deformed objet}
 \end{figure}
 
 In the 1--dimensional example in figure~\ref{fig:bspline}, the
-control--points are indicated as red dots and the color-gradient should
+control--points are indicated as red dots and the colour--gradient
-hint at the \(u\)--values ranging from \(0\) to \(1\).
+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
@@ -324,7 +324,7 @@ N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+
 
 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
+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,
@@ -368,18 +368,18 @@ and \(m\) control--points.
 \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
+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.
+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
+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--splines--basis--functions form a partition of unity for
+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 \(\sum_j n_{i,j} p_j = p_i\)
@@ -469,8 +469,32 @@ 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{genotypes} while \(M\) represents the possible observable
-\emph{Phenotypes}. \improvement[inline]{Erklären, was das ist. Quellen!}
+\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:
 
@@ -503,16 +527,18 @@ that can be changed over time. A good overview of this is given in
 
 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 part. An even more
+or giving a different \(\sigma\) to every component of those parameters.
-sophisticated example would be the \glqq 1/5 success rule\grqq ~from
+An even more sophisticated example would be the \glqq 1/5 success
-\cite{rechenberg1973evolutionsstrategie}.
+rule\grqq ~from \cite{rechenberg1973evolutionsstrategie}.
 
-Also in selection it may not be wise to only take the best--performing
+Also in the selection--function it may not be wise to only take the
-individuals, because it may be that the optimization has to overcome a
+best--performing individuals, because it may be that the optimization
-barrier of bad fitness to achieve a better local optimum.
+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 grouping into account.
+but can also take ancestry, distance of genes or groups of individuals
+into account.
 
 \section{Advantages of evolutionary
 algorithms}\label{advantages-of-evolutionary-algorithms}
@@ -535,13 +561,11 @@ typical problems are shown in figure \ref{fig:probhard}.
 Most of the advantages stem from the fact that a gradient--based
 procedure has only one point of observation from where it evaluates the
 next steps, whereas an evolutionary strategy starts with a population of
-guessed solutions. Because an evolutionary strategy modifies the
+guessed solutions. Because an evolutionary strategy can be modified
-solution randomly, keeping some solutions and purging others, it can
+according to the problem--domain (i.e.~by the ideas given above) it can
-also target multiple different hypothesis at the same time where the
+also approximate very difficult problems in an efficient manner and even
-local optima die out in the face of other, better candidates.
+self--tune parameters depending on the ancestry at runtime\footnote{Some
-
+  examples of this are explained in detail in \cite{eiben1999parameter}}.
-\improvement[inline]{Verweis auf MO-CMA etc. Vielleicht auch etwas
-ausführlicher.}
 
 If an analytic best solution exists and is easily computable
 (i.e.~because the error--function is convex) an evolutionary algorithm
@@ -553,8 +577,8 @@ 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 the same
+fast, which then refine over time and still converge to a decent
-solution.
+solution much faster than an exhaustive search.
 
 \section{Criteria for the evolvability of linear
 deformations}\label{criteria-for-the-evolvability-of-linear-deformations}
@@ -763,7 +787,7 @@ J(Err(u,v,w)) =
 
 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
+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
@@ -1036,11 +1060,11 @@ 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 CMA--ES (covariance
+For the evolutionary optimization we employ the \afc{CMA--ES} of the
-matrix adaptation evolution strategy) of the shark3.1 library
+shark3.1 library \cite{shark08}, as this algorithm was used by
-\cite{shark08}, as this algorithm was used by \cite{anrichterEvol} as
+\cite{anrichterEvol} as well. We leave the parameters at their sensible
-well. We leave the parameters at their sensible defaults as further
+defaults as further explained in
-explained in \cite[Appendix~A: Table~1]{hansen2016cma}.
+\cite[Appendix~A: Table~1]{hansen2016cma}.
 
 \section{Procedure: 1D Function
 Approximation}\label{procedure-1d-function-approximation}
@@ -1139,7 +1163,7 @@ 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 boxplot in figure \ref{fig:1dvar}
+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 variability, they perform not better
@@ -1361,7 +1385,7 @@ in brackets for three cases of increasing variability ($\mathrm{X} \in [4,5,7],
 
 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 boxplot in figure
+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
@@ -1382,7 +1406,8 @@ the variability via the rank of the deformation--matrix.
 \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.
+control--points each showing in this case how many control points are actually
+used in the calculations.
 }
 \label{fig:histrank3d}
 \end{figure}
@@ -1562,7 +1587,7 @@ we did before with the 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
+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 improvement
 potential and the fitting error, but all findings (except for
@@ -1576,8 +1601,30 @@ control--points.
 All in all the improvement potential seems to be a good and sensible
 measure of quality, even given gradients of varying quality.
 
-\improvement[inline]{improvement--potential vs. steps ist anders als in 1d! Plot
+Lastly, a small note on the behaviour of improvement potential and
-und zeigen!}
+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 improvement potential cannot serve for good predictions of
+the convergence speed. In figure \ref{fig:imp1d3d} we show 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 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}
 
@@ -1617,28 +1664,32 @@ 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.
+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. This results in very
+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.
+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.
+better with the criteria we examined. This can also solve the problem of
-
+bad singular values for the \emph{regularity} as the incorporation of
-\improvement[inline]{write more outlook/further research}
+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.
 
 \improvement[inline]{Bibliotheksverzeichnis links anpassen. DOI überschreibt
-Direktlinks des Autors.\newline
+Direktlinks des Autors.}
-Außerdem bricht url über Seitengrenzen den Seitenspiegel.}
 
 % \backmatter
 \cleardoublepage

arbeit/settings/abkuerzungen.tex

@@ -10,8 +10,9 @@
 %
 %\acro{GPL}{GNU General Public License} --
 % License for free software, see \url{http://www.gnu.org/copyleft/gpl.html}.
-\acro{FFD}{Freeform--Deformation}
+\acro{CMA--ES}{Covariance Matrix Adaption Evolution Strategy}
 \acro{DM--FFD}{Direct Manipulation Freeform--Deformation}
+\acro{FFD}{Freeform--Deformation}
 \acro{RBF}{Radial Basis Function}
 
 %