ersten 3 Kapitel fertig.

arbeit/ma.md

@@ -180,14 +180,18 @@ 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.
 
-\improvement[inline]{Weitere Eigenschaften erwähnen:
-\newline Convex hull
-\newline $\sum_i N_i = 1$?
-\newline Bilder von Basisfunktionen zur Visualisierung.
+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.
+
+\improvement[inline]{
+Bilder von Basisfunktionen zur Visualisierung.
 }
 
 For a given number of points $v_1,\dots,v_n$ we can then calculate
-the contributions $n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
+the contributions \linebreak[4]$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the
 projection from the control--point--space into the object--space:
 $$
 v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
@@ -264,16 +268,21 @@ however, is very generic and we introduce it here in a broader sense.
 The general shape of an evolutionary algorithm (adapted from
 \cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here, $P(t)$ 
 denotes the population of parameters in step $t$ of the algorithm. The
-population contains $\mu$ individuals $a_i$ that fit the shape of the parameters
-we are looking for. Typically these are initialized by a random guess or just
-zero. Further on we need a so--called *fitness--function* $\Phi : I \mapsto M$\improvement{Was ist $I,M$?\newline Bezug Genotyp/Phenotyp} that can take
-each parameter to a measurable space along a convergence--function $c : I \mapsto
-\mathbb{B}$ that terminates the optimization.
+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
+*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 *Genotypes*
+while $M$ represents the possible observable *Phenotypes*.
 
 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.  
+  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.
 - **Mutate** with a mutation--function $m : I^{\lambda} \mapsto I^{\lambda}$ to
   introduce new effects that cannot be produced by mere recombination of the
@@ -332,9 +341,23 @@ least get suboptimal solutions fast, which then refine over time.
 ## Criteria for the evolvability of linear deformations
 \label{sec:intro:rvi}
 
-\improvement[inline]{Nomenklatur. Was ist $\vec{U}$? Kurz Matrix--Darstellung
-des Problems & Rückgriff auf FFD-Kapitel.}
+As we have established in chapter \ref{sec:back:ffd}, we can describe a
+deformation by the formula
+$$
+V = UP
+$$
+where $V$ is a $n \times d$ matrix of vertices, $U$ 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
+$$
+\Delta V = U \cdot \Delta P
+$$
+which is isomorphic to the former due to the linear correlation in the
+deformation. One can see in this way, that the way the deformation behaves lies
+solely in the entries of $U$, which is why the three criteria focus on this.
 
 ### Variability
 
@@ -550,7 +573,8 @@ entfernen kann?}
 
 For our tests we chose different uniformly sized grids and added gaussian noise
 onto each control-point^[For the special case of the outer layer we only applied
-noise away from the object] to simulate different starting-conditions.
+noise away from the object, so the object is still confined in the convex hull
+of the control--points.] to simulate different starting-conditions.
 
 \unsure[inline]{verweis auf DM--FFD?}
 

arbeit/ma.tex

@@ -148,7 +148,8 @@ 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 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
@@ -169,15 +170,15 @@ Unless otherwise noted the following holds:
 \improvement[inline]{Mehr Bilder}
 
 Many modern industrial design processes require advanced optimization
-methods due 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.
+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 aerodynamic (i.e.~drag),
+fluid dynamics (i.e.~throughput of liquid) --- where the complexity
+increases with the temporal and spatial resolution of the simulation ---
+or known hard algorithmic problems in informatics (i.e.~layouting of
+circuit boards or stacking of 3D--objects). Moreover these are typically
+not static environments but requirements shift over time or from case to
+case.
 
 Evolutionary algorithms cope especially well with these problem domains
 while addressing all the issues at hand\cite{minai2006complex}. One of
@@ -228,7 +229,7 @@ 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 potential of such optimization.
+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
@@ -332,16 +333,20 @@ 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.
 
-\improvement[inline]{Weitere Eigenschaften erwähnen:
-\newline Convex hull
-\newline $\sum_i N_i = 1$?
-\newline Bilder von Basisfunktionen zur Visualisierung.
+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.
+
+\improvement[inline]{
+Bilder von Basisfunktionen zur Visualisierung.
 }
 
 For a given number of points \(v_1,\dots,v_n\) we can then calculate the
-contributions \(n_{i,j}~:=~N_{j,d,\tau}\) of each control point \(p_j\)
-to get the projection from the control--point--space into the
-object--space: \[
+contributions \linebreak[4]\(n_{i,j}~:=~N_{j,d,\tau}\) of each control
+point \(p_j\) to get the projection from the control--point--space into
+the object--space: \[
 v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
 \] or written for all points at the same time: \[
 \vec{v} = \vec{N} \vec{p}
@@ -420,14 +425,17 @@ broader sense.
 The general shape of an evolutionary algorithm (adapted from
 \cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here,
 \(P(t)\) denotes the population of parameters in step \(t\) of the
-algorithm. The population contains \(\mu\) individuals \(a_i\) that fit
-the shape of the parameters we are looking for. Typically these are
-initialized by a random guess or just zero. Further on we need a
-so--called \emph{fitness--function}
-\(\Phi : I \mapsto M\)\improvement{Was ist $I,M$?\newline Bezug Genotyp/Phenotyp}
-that can take each parameter to a measurable space along a
-convergence--function \(c : I \mapsto \mathbb{B}\) that terminates the
-optimization.
+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}.
 
 The main algorithm just repeats the following steps:
 
@@ -435,8 +443,8 @@ The main algorithm just repeats the following steps:
 \tightlist
 \item
   \textbf{Recombine} with a recombination--function
-  \(r : I^{\mu} \mapsto I^{\lambda}\) to generate new individuals based
-  on the parents characteristics.\\
+  \(r : I^{\mu} \mapsto I^{\lambda}\) 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
@@ -507,8 +515,21 @@ deformations}\label{criteria-for-the-evolvability-of-linear-deformations}
 
 \label{sec:intro:rvi}
 
-\improvement[inline]{Nomenklatur. Was ist $\vec{U}$? Kurz Matrix--Darstellung
-des Problems & Rückgriff auf FFD-Kapitel.}
+As we have established in chapter \ref{sec:back:ffd}, we can describe a
+deformation by the formula \[
+V = UP
+\] where \(V\) is a \(n \times d\) matrix of vertices, \(U\) 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 \[
+\Delta V = U \cdot \Delta P
+\] which is isomorphic to the former due to the linear correlation in
+the deformation. One can see in this way, that the way the deformation
+behaves lies solely in the entries of \(U\), which is why the three
+criteria focus on this.
 
 \subsection{Variability}\label{variability}
 
@@ -730,8 +751,9 @@ entfernen kann?}
 
 For our tests we chose different uniformly sized grids and added
 gaussian noise onto each control-point\footnote{For the special case of
-  the outer layer we only applied noise away from the object} to
-simulate different starting-conditions.
+  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.
 
 \unsure[inline]{verweis auf DM--FFD?}
 

arbeit/settings/packages.tex

@@ -16,7 +16,7 @@
 \usepackage{color}						%\colorbox
 \usepackage{dsfont}						%\mathds
 \usepackage{draftwatermark}
-\SetWatermarkLightness{0.9} % default: 0.8
+\SetWatermarkLightness{0.95} % default: 0.8
 \usepackage{epigraph}
 % \usepackage{euler}  					% euler: uni, eucal: baake, ohne: standard
 \usepackage{eucal}						% euler calligraphy