added evo-section

arbeit/bibma.bib

@@ -115,3 +115,14 @@ eprint = {https://doi.org/10.1137/0111030}
   organization={IEEE},
   url={http://www.graphics.uni-bielefeld.de/publications/cec15.pdf}
 }
+@article{back1993overview,
+  title={An overview of evolutionary algorithms for parameter optimization},
+  author={B{\"a}ck, Thomas and Schwefel, Hans-Paul},
+  journal={Evolutionary computation},
+  volume={1},
+  number={1},
+  pages={1--23},
+  year={1993},
+  publisher={MIT Press},
+  url={https://www.researchgate.net/profile/Hans-Paul_Schwefel/publication/220375001_An_Overview_of_Evolutionary_Algorithms_for_Parameter_Optimization/links/543663d00cf2dc341db30452.pdf}
+}

arbeit/ma.md

@@ -174,7 +174,56 @@ mesh albeit the downsides.
 ## What is evolutional optimization?
 \label{sec:back:evo}
 
-\change[inline]{Write this section}
+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}

arbeit/ma.tex

@@ -148,20 +148,20 @@ Unless otherwise noted the following holds:
 \tightlist
 \item
   lowercase letters \(x,y,z\)\\
-   refer to real variables and represent a point in 3D--Space.
+  refer to real variables and represent a point in 3D--Space.
 \item
   lowercase letters \(u,v,w\)\\
-   refer to real variables between \(0\) and \(1\) used as coefficients
+  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.
+  refer to other scalar (real) variables.
 \item
   lowercase \textbf{bold} letters (e.g. \(\vec{x},\vec{y}\))\\
-   refer to 3D coordinates
+  refer to 3D coordinates
 \item
   uppercase \textbf{BOLD} letters (e.g. \(\vec{D}, \vec{M}\))\\
-   refer to Matrices
+  refer to Matrices
 \end{itemize}
 
 \chapter{Introduction}\label{introduction}
@@ -331,7 +331,68 @@ optimization?}\label{what-is-evolutional-optimization}
 
 \label{sec:back:evo}
 
-\change[inline]{Write this section}
+\change[inline]{Write this section} 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 \emph{fitness-function} \(\Phi : I \mapsto M\) that can take
+each parameter to a measurable space along a convergence-function
+\(c : I \mapsto \mathbb{B}\) that terminates the optimization.
+
+The main algorithm just repeats the following steps:
+
+\begin{itemize}
+\tightlist
+\item
+  \textbf{Recombine} with a recombination-function
+  \(r : I^{\mu} \mapsto I^{\lambda}\) to generate new individuals based
+  on the parents characteristics.\\
+  This makes sure that the next guess is close to the old guess.
+\item
+  \textbf{Mutate} with a mutation-function
+  \(m : I^{\lambda} \mapsto I^{\lambda}\) to introduce new effects that
+  cannot be produced by mere recombination of the parents.\\
+  Typically this just adds minor defects to individual members of the
+  population like adding a random gaussian noise or amplifying/dampening
+  random parts.
+\item
+  \textbf{Selection} takes a selection-function
+  \(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\) that
+  selects from the previously generated \(I^\lambda\) children and
+  optionally also the parents (denoted by the set \(Q\) in the
+  algorithm) using the fitness-function \(\Phi\). The result of this
+  operation is the next Population of \(\mu\) individuals.
+\end{itemize}
+
+All these functions can (and mostly do) have a lot of hidden parameters
+that can be changed over time. One can for example start off with a high
+mutation--rate that cools off over time (i.e.~by lowering the variance
+of a gaussian noise). As the recombination and selection-steps are
+usually pure
 
 \section{Advantages of evolutional
 algorithms}\label{advantages-of-evolutional-algorithms}