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 <section id="introduction" class="slide level1">
 <h1>Introduction</h1>
-<p>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.</p>
-
-<p>Evolutionary algorithms cope especially well with these problem domains while addressing all the issues at hand. One of the main concerns in these algorithms is the formulation of the problems in terms of a <em>genome</em> and <em>fitness–function</em>. While one can typically use an arbitrary cost–function for the <em>fitness–functions</em> (i.e. amount of drag, amount of space, etc.), the translation of the problem–domain into a simple parametric representation (the <em>genome</em>) can be challenging.</p>
-<p>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.</p>
-<p>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.</p>
-<p>The quality of such a low–dimensional representation in biological evolution is strongly tied to the notion of <em>evolvability</em>, as the parametrization of the problem has serious implications on the convergence speed and the quality of the solution. However, there is no consensus on how <em>evolvability</em> is defined and the meaning varies from context to context. 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.</p>
-
-<p>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 . If one (or the algorithm) would move such a point the object will get deformed only locally (due to the ). 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 <em>evolvability</em>. This is exactly what Richter et al. have done.</p>
-<p>As we transfer the results of Richter et al. from using  as a representation to manipulate geometric objects to the use of  we will use the same definition for <em>evolvability</em> the original author used, namely <em>regularity</em>, <em>variability</em>, and <em>improvement potential</em>. We introduce these term in detail in Chapter . In the original publication the author could show a correlation between these evolvability–criteria with the quality and convergence speed of such optimization.</p>
-<p>We will replicate the same setup on the same objects but use  instead of  to create a local deformation near the control–points and evaluate if the evolution–criteria still work as a predictor for <em>evolvability</em> of the representation given the different deformation scheme, as suspected in .</p>
-<p>First we introduce different topics in isolation in Chapter . We take an abstract look at the definition of  for a one–dimensional line (in ) and discuss why this is a sensible deformation function (in ). Then we establish some background–knowledge of evolutionary algorithms (in ) and why this is useful in our domain (in ) followed by the definition of the different evolvability–criteria established in  (in ).</p>
-<p>In Chapter  we take a look at our implementation of  and the adaptation for 3D–meshes that were used. Next, in Chapter , we describe the different scenarios we use to evaluate the different evolvability–criteria incorporating all aspects introduced in Chapter . Following that, we evaluate the results in Chapter  with further on discussion, summary and outlook in Chapter .</p>
-</section>
-<section id="background" class="slide level1">
-<h1>Background</h1>
-
-<section id="what-is" class="level2">
-<h2>What is ?</h2>
-
-<p>First of all we have to establish how a  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  here and go into the extension to the 3D case in chapter .</p>
-<p>The main idea of  is to create a function <span class="math inline">\(s : [0,1[^d \mapsto \mathbb{R}^d\)</span> that spans a certain part of a vector–space and is only linearly parametrized by some special control–points <span class="math inline">\(p_i\)</span> and an constant attribution–function <span class="math inline">\(a_i(u)\)</span>, so <span class="math display">\[
-s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
-\]</span> 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 <span class="math inline">\(u \in [0,1[^d\)</span>.</p>
-
-<p>In the 1–dimensional example in figure , the control–points are indicated as red dots and the colour–gradient should hint at the <span class="math inline">\(u\)</span>–values ranging from <span class="math inline">\(0\)</span> to <span class="math inline">\(1\)</span>.</p>
-<p>We now define a  by the following:<br />
-Given an arbitrary number of points <span class="math inline">\(p_i\)</span> alongside a line, we map a scalar value <span class="math inline">\(\tau_i \in [0,1[\)</span> to each point with <span class="math inline">\(\tau_i &lt; \tau_{i+1} \forall i\)</span> according to the position of <span class="math inline">\(p_i\)</span> on said line. Additionally, given a degree of the target polynomial <span class="math inline">\(d\)</span> we define the curve <span class="math inline">\(N_{i,d,\tau_i}(u)\)</span> as follows:</p>
-<span class="math display">\[\begin{equation} \label{eqn:ffd1d1}
-N_{i,0,\tau}(u) = \begin{cases} 1, &amp; u \in [\tau_i, \tau_{i+1}[ \\ 0, &amp; \mbox{otherwise} \end{cases}
-\end{equation}\]</span>
-<p>and <span class="math display">\[\begin{equation} \label{eqn:ffd1d2}
-N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
-\end{equation}\]</span></p>
-<p>If we now multiply every <span class="math inline">\(p_i\)</span> with the corresponding <span class="math inline">\(N_{i,d,\tau_i}(u)\)</span> we get the contribution of each point <span class="math inline">\(p_i\)</span> to the final curve–point parametrized only by <span class="math inline">\(u \in [0,1[\)</span>. As can be seen from  we only access points <span class="math inline">\([p_i..p_{i+d}]\)</span> for any given <span class="math inline">\(i\)</span><a href="#/fn1" class="footnote-ref" id="fnref1"><sup>1</sup></a>, which gives us, in combination with choosing <span class="math inline">\(p_i\)</span> and <span class="math inline">\(\tau_i\)</span> in order, only a local interference of <span class="math inline">\(d+1\)</span> points.</p>
-<p>We can even derive this equation straightforward for an arbitrary <span class="math inline">\(N\)</span><a href="#/fn2" class="footnote-ref" id="fnref2"><sup>2</sup></a>:</p>
-<p><span class="math display">\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]</span></p>
-<p>For a B–Spline <span class="math display">\[s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i\]</span> these derivations yield <span class="math inline">\(\left(\frac{\partial}{\partial u}\right)^d s(u) = 0\)</span>.</p>
-<p>Another interesting property of these recursive polynomials is that they are continuous (given <span class="math inline">\(d \ge 1\)</span>) as every <span class="math inline">\(p_i\)</span> gets blended in between <span class="math inline">\(\tau_i\)</span> and <span class="math inline">\(\tau_{i+d}\)</span> and out between <span class="math inline">\(\tau_{i+1}\)</span>, and <span class="math inline">\(\tau_{i+d+1}\)</span> as can bee seen from the two coefficients in every step of the recursion.</p>
-<p>This means that all changes are only a local linear combination between the control–point <span class="math inline">\(p_i\)</span> to <span class="math inline">\(p_{i+d+1}\)</span> 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.</p>
-<p>For a given point <span class="math inline">\(s_i\)</span> we can then calculate the contributions <span class="math inline">\(u_{i,j}~:=~N_{j,d,\tau}\)</span> of each control point <span class="math inline">\(p_j\)</span> to get the projection from the control–point–space into the object–space: <span class="math display">\[
-s_i = \sum_j u_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p}
-\]</span> or written for all points at the same time: <span class="math display">\[
-\vec{s} = \vec{U} \vec{p}
-\]</span> where <span class="math inline">\(\vec{U}\)</span> is the <span class="math inline">\(n \times m\)</span> transformation–matrix (later on called <strong>deformation matrix</strong>) for <span class="math inline">\(n\)</span> object–space–points and <span class="math inline">\(m\)</span> control–points.</p>
-
-<p>Furthermore B–Spline–basis–functions form a partition of unity for all, but the first and last <span class="math inline">\(d\)</span> control–points. Therefore we later on use the border–points <span class="math inline">\(d+1\)</span> times, such that <span class="math inline">\(\sum_j u_{i,j} p_j = p_i\)</span> for these points.</p>
-<p>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 .</p>
-<section id="why-is-a-good-deformation-function" class="level3">
-<h3>Why is  a good deformation function?</h3>
-
-<p>The usage of  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.</p>
-<p>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.</p>
-<p>This disadvantages led to the formulation of  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 <em>at</em> the surface without seemingly arbitrary scattered control–points. Moreover this increases the efficiency of an evolutionary optimization, which we will use later on.</p>
-
-<p>But this approach also has downsides as can be seen in figure , as the tessellation of the invisible grid has a major impact on the deformation itself.</p>
-<p>All in all  and  are still good ways to deform a high–polygon mesh albeit the downsides.</p>
-</section>
-</section>
-<section id="what-is-evolutionary-optimization" class="level2">
-<h2>What is evolutionary optimization?</h2>
-
-<p>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.</p>
-
-<p>The general shape of an evolutionary algorithm (adapted from ) is outlined in Algorithm . Here, <span class="math inline">\(P(t)\)</span> denotes the population of parameters in step <span class="math inline">\(t\)</span> of the algorithm. The population contains <span class="math inline">\(\mu\)</span> individuals <span class="math inline">\(a_i\)</span> from the possible individual–set <span class="math inline">\(I\)</span> 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 <em>fitness–function</em> <span class="math inline">\(\Phi : I \mapsto M\)</span> that can take each parameter to a measurable space <span class="math inline">\(M\)</span> (usually <span class="math inline">\(M = \mathbb{R}\)</span>) along a convergence–function <span class="math inline">\(c : I \mapsto \mathbb{B}\)</span> that terminates the optimization.</p>
-<p>Biologically speaking the set <span class="math inline">\(I\)</span> corresponds to the set of possible <em>genotypes</em> while <span class="math inline">\(M\)</span> represents the possible observable <em>phenotypes</em>. <em>Genotypes</em> 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 <em>phenotypes</em> make certain behaviour observable (algorithmically through our <em>fitness–function</em>, 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.</p>
-<p>One should note that there is a subtle difference between <em>fitness–function</em> and a so called <em>genotype–phenotype–mapping</em>. The first one directly applies the <em>genotype–phenotype–mapping</em> and evaluates the performance of an individual, thus going directly from genes/parameters to reproduction–probability/score. In a concrete example the <em>genotype</em> can be an arbitrary vector (the genes), the <em>phenotype</em> is then a deformed object, and the performance can be a single measurement like an air–drag–coefficient. The <em>genotype–phenotype–mapping</em> would then just be the generation of different objects from that starting–vector, whereas the <em>fitness–function</em> would go directly from such a starting–vector to the coefficient that we want to optimize.</p>
-<p>The main algorithm just repeats the following steps:</p>
 <ul>
-<li><strong>Recombine</strong> with a recombination–function <span class="math inline">\(r : I^{\mu} \mapsto I^{\lambda}\)</span> to generate <span class="math inline">\(\lambda\)</span> new individuals based on the characteristics of the <span class="math inline">\(\mu\)</span> parents.<br />
-This makes sure that the next guess is close to the old guess.</li>
-<li><strong>Mutate</strong> with a mutation–function <span class="math inline">\(m : I^{\lambda} \mapsto I^{\lambda}\)</span> to introduce new effects that cannot be produced by mere recombination of the parents.<br />
-Typically this just adds minor defects to individual members of the population like adding a random gaussian noise or amplifying/dampening random parts.</li>
-<li><strong>Selection</strong> takes a selection–function <span class="math inline">\(s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu\)</span> that selects from the previously generated <span class="math inline">\(I^\lambda\)</span> children and optionally also the parents (denoted by the set <span class="math inline">\(Q\)</span> in the algorithm) using the <em>fitness–function</em> <span class="math inline">\(\Phi\)</span>. The result of this operation is the next Population of <span class="math inline">\(\mu\)</span> individuals.</li>
+<li>Many modern industrial design processes require advanced optimization methods due to increased complexity</li>
+<li>Examples are
+<ul>
+<li>physical domains
+<ul>
+<li>aerodynamics (i.e. drag)</li>
+<li>fluid dynamics (i.e. throughput of liquid)</li>
+</ul></li>
+<li>NP-hard problems
+<ul>
+<li>layouting of circuit boards</li>
+<li>stacking of 3D–objects</li>
+</ul></li>
+</ul></li>
 </ul>
-<p>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 , so we only give a small excerpt here.</p>
-<p>For example the mutation can consist of merely a single <span class="math inline">\(\sigma\)</span> determining the strength of the gaussian defects in every parameter — or giving a different <span class="math inline">\(\sigma\)</span> to every component of those parameters. An even more sophisticated example would be the 1/5 success rule from .</p>
-<p>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.</p>
-<p>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.</p>
 </section>
-<section id="advantages-of-evolutionary-algorithms" class="level2">
-<h2>Advantages of evolutionary algorithms</h2>
-
-<p>The main advantage of evolutionary algorithms is the ability to find optima of general functions just with the help of a given <em>fitness–function</em>. Components and techniques for evolutionary algorithms are specifically known to help with different problems arising in the domain of optimization. An overview of the typical problems are shown in figure .</p>
-
-<p>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<a href="#/fn3" class="footnote-ref" id="fnref3"><sup>3</sup></a>.</p>
-<p>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.</p>
-<p>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.</p>
+<section id="motivation" class="slide level1">
+<h1>Motivation</h1>
+<ul>
+<li>Evolutionary algorithms cope especially well with these problem domains <figure class="" style=""><img  src="../arbeit/img/Evo_overview.png" style=""></img><figcaption>Example of the use of evolutionary algorithms in automotive design</figcaption></figure></li>
+<li>But formulation can be tricky</li>
+</ul>
 </section>
-<section id="criteria-for-the-evolvability-of-linear-deformations" class="level2">
-<h2>Criteria for the evolvability of linear deformations</h2>
-
-<p>As we have established in chapter , we can describe a deformation by the formula <span class="math display">\[
-\vec{S} = \vec{U}\vec{P}
-\]</span> where <span class="math inline">\(\vec{S}\)</span> is a <span class="math inline">\(n \times d\)</span> matrix of vertices<a href="#/fn4" class="footnote-ref" id="fnref4"><sup>4</sup></a>, <span class="math inline">\(\vec{U}\)</span> are the (during parametrization) calculated deformation–coefficients and <span class="math inline">\(P\)</span> is a <span class="math inline">\(m \times d\)</span> matrix of control–points that we interact with during deformation.</p>
-<p>We can also think of the deformation in terms of differences from the original coordinates <span class="math display">\[
-\Delta \vec{S} = \vec{U} \cdot \Delta \vec{P}
-\]</span> 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 <span class="math inline">\(\vec{U}\)</span>, which is why the three criteria focus on this.</p>
-<section id="variability" class="level3">
-<h3>Variability</h3>
-<p>In  <em>variability</em> is defined as <span class="math display">\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n},\]</span> whereby <span class="math inline">\(\vec{U}\)</span> is the <span class="math inline">\(n \times m\)</span> deformation–Matrix used to map the <span class="math inline">\(m\)</span> control–points onto the <span class="math inline">\(n\)</span> vertices.</p>
-<p>Given <span class="math inline">\(n = m\)</span>, an identical number of control–points and vertices, this quotient will be <span class="math inline">\(=1\)</span> if all control–points are independent of each other and the solution is to trivially move every control–point onto a target–point.</p>
-<p>In praxis the value of <span class="math inline">\(V(\vec{U})\)</span> is typically <span class="math inline">\(\ll 1\)</span>, because there are only few control–points for many vertices, so <span class="math inline">\(m \ll n\)</span>.</p>
-<p>This criterion should correlate to the degrees of freedom the given parametrization has. This can be seen from the fact, that <span class="math inline">\(\mathrm{rank}(\vec{U})\)</span> is limited by <span class="math inline">\(\min(m,n)\)</span> and — as <span class="math inline">\(n\)</span> is constant — can never exceed <span class="math inline">\(n\)</span>.</p>
-<p>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 <span class="math inline">\(m\)</span> and are thus measurable by the <em>variability</em>.</p>
+<section id="motivation-1" class="slide level1">
+<h1>Motivation</h1>
+<ul>
+<li>Problems tend to be very complex
+<ul>
+<li>i.e. a surface with <span class="math inline">\(n\)</span> vertices has <span class="math inline">\(3\cdot n\)</span> Degrees of Freedom (DoF).</li>
+</ul></li>
+<li>Need for a small-dimensional representation that manipulates the high-dimensional problem-space.</li>
+<li>We concentrate on smooth deformations (<span class="math inline">\(C^3\)</span>-continuous)</li>
+<li>But what representation is good?</li>
+</ul>
 </section>
-<section id="regularity" class="level3">
-<h3>Regularity</h3>
-<p><em>Regularity</em> is defined as <span class="math display">\[\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\]</span> where <span class="math inline">\(\sigma_{min}\)</span> and <span class="math inline">\(\sigma_{max}\)</span> are the smallest and greatest right singular value of the deformation–matrix <span class="math inline">\(\vec{U}\)</span>.</p>
-<p>As we deform the given Object only based on the parameters as <span class="math inline">\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\)</span> this makes sure that <span class="math inline">\(\|\vec{Up}\| \propto \|\vec{p}\|\)</span> when <span class="math inline">\(\kappa(\vec{U}) \approx 1\)</span>. The inversion of <span class="math inline">\(\kappa(\vec{U})\)</span> is only performed to map the criterion–range to <span class="math inline">\([0..1]\)</span>, where <span class="math inline">\(1\)</span> is the optimal value and <span class="math inline">\(0\)</span> is the worst value.</p>
-<p>On the one hand this criterion should be characteristic for numeric stability and on the other hand for the convergence speed of evolutionary algorithms as it is tied to the notion of locality.</p>
+<section id="what-representation-is-good" class="slide level1">
+<h1>What representation is good?</h1>
+<ul>
+<li>In biological evolution this measure is called <em>evolvability</em>.
+<ul>
+<li>no consensus on definition</li>
+<li>meaning varies from context to context</li>
+<li>measurable?</li>
+</ul></li>
+<li>Measure depends on representation as well.</li>
+</ul>
 </section>
-<section id="improvement-potential" class="level3">
-<h3>Improvement Potential</h3>
-<p>In contrast to the general nature of <em>variability</em> and <em>regularity</em>, which are agnostic of the <em>fitness–function</em> at hand, the third criterion should reflect a notion of the potential for optimization, taking a guess into account.</p>
-<p>Most of the times some kind of gradient <span class="math inline">\(g\)</span> 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.</p>
-<p>The definition for an <em>improvement potential</em> <span class="math inline">\(P\)</span> is: <span class="math display">\[
-\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F
-\]</span> given some approximate <span class="math inline">\(n \times d\)</span> fitness–gradient <span class="math inline">\(\vec{G}\)</span>, normalized to <span class="math inline">\(\|\vec{G}\|_F = 1\)</span>, whereby <span class="math inline">\(\|\cdot\|_F\)</span> denotes the Frobenius–Norm.</p>
+<section id="rbf-and-ffd" class="slide level1">
+<h1>RBF and FFD</h1>
+<ul>
+<li>Andreas Richter uses Radial Basis Functions (RBF) to smoothly deform meshes</li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/deformations.png" style=""></img><figcaption>Example of RBF–based deformation and FFD targeting the same mesh.</figcaption></figure></p>
 </section>
+<section id="rbf-and-ffd-1" class="slide level1">
+<h1>RBF and FFD</h1>
+<ul>
+<li>My master thesis transferred his idea to Freeform-Deformation (FFD)
+<ul>
+<li>same setup</li>
+<li>same measurements</li>
+<li>same results?</li>
+</ul></li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/deformations.png" style=""></img><figcaption>Example of RBF–based deformation and FFD targeting the same mesh.</figcaption></figure></p>
 </section>
+<section id="outline" class="slide level1">
+<h1>Outline</h1>
+<ul>
+<li><strong>What is FFD?</strong></li>
+<li>What is evolutionary optimization?</li>
+<li>How to measure evolvability?</li>
+<li>Scenarios</li>
+<li>Results</li>
+</ul>
 </section>
-<section id="implementation-of" class="slide level1">
-<h1>Implementation of </h1>
-
-<p>The general formulation of B–Splines has two free parameters <span class="math inline">\(d\)</span> and <span class="math inline">\(\tau\)</span> which must be chosen beforehand.</p>
-<p>As we usually work with regular grids in our  we define <span class="math inline">\(\tau\)</span> statically as <span class="math inline">\(\tau_i = \nicefrac{i}{n}\)</span> whereby <span class="math inline">\(n\)</span> is the number of control–points in that direction.</p>
-<p><span class="math inline">\(d\)</span> defines the <em>degree</em> of the B–Spline–Function (the number of times this function is differentiable) and for our purposes we fix <span class="math inline">\(d\)</span> to <span class="math inline">\(3\)</span>, but give the formulas for the general case so it can be adapted quite freely.</p>
-<section id="adaption-of" class="level2">
-<h2>Adaption of </h2>
-
-<p>As we have established in Chapter  we can define an –displacement as <span class="math display">\[\begin{equation}
-\Delta_x(u) = \sum_i N_{i,d,\tau_i}(u) \Delta_x c_i
-\end{equation}\]</span></p>
-<p>Note that we only sum up the <span class="math inline">\(\Delta\)</span>–displacements in the control–points <span class="math inline">\(c_i\)</span> to get the change in position of the point we are interested in.</p>
-<p>In this way every deformed vertex is defined by <span class="math display">\[
-\textrm{Deform}(v_x) = v_x + \Delta_x(u)
-\]</span> with <span class="math inline">\(u \in [0..1[\)</span> 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 <span class="math inline">\(u\)</span>–value for each vertex. This is achieved by finding out the parametrization of <span class="math inline">\(v\)</span> in terms of <span class="math inline">\(c_i\)</span> <span class="math display">\[
-v_x \overset{!}{=} \sum_i N_{i,d,\tau_i}(u) c_i
-\]</span> so we can minimize the error between those two: <span class="math display">\[
-\underset{u}{\argmin}\,Err(u,v_x) = \underset{u}{\argmin}\,2 \cdot \|v_x - \sum_i N_{i,d,\tau_i}(u) c_i\|^2_2
-\]</span> As this error–term is quadratic we just derive by <span class="math inline">\(u\)</span> yielding <span class="math display">\[
-\begin{array}{rl}
-\frac{\partial}{\partial u} &amp; v_x - \sum_i N_{i,d,\tau_i}(u) c_i \\
-= &amp; - \sum_i \left( \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u) \right) c_i
-\end{array}
-\]</span> and do a gradient–descend to approximate the value of <span class="math inline">\(u\)</span> up to an <span class="math inline">\(\epsilon\)</span> of <span class="math inline">\(0.0001\)</span>.</p>
-<p>For this we employ the Gauss–Newton algorithm, 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 (<span class="math inline">\(\#v~\gg~\#c\)</span>).</p>
+<section id="what-is-ffd" class="slide level1">
+<h1>What is FFD?</h1>
+<ul>
+<li>Create a function <span class="math inline">\(s : [0,1[^d \mapsto \mathbb{R}^d\)</span> that is parametrized by some special control–points <span class="math inline">\(p_i\)</span> with coefficient functions <span class="math inline">\(a_i(u)\)</span>: <span class="math display">\[
+  s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
+  \]</span></li>
+<li>All points inside the convex hull of <span class="math inline">\(\vec{p_i}\)</span> accessed by the right <span class="math inline">\(u \in [0,1[^d\)</span>.</li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/B-Splines.png" style=""></img><figcaption>Example of a parametrization of a line with corresponding deformation to generate a deformed objet</figcaption></figure></p>
 </section>
-<section id="adaption-of-for-a-3dmesh" class="level2">
-<h2>Adaption of  for a 3D–Mesh</h2>
-
-<p>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.</p>
-<p>Given <span class="math inline">\(n,m,o\)</span> control–points in <span class="math inline">\(x,y,z\)</span>–direction each Point on the curve is defined by <span class="math display">\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]</span></p>
-<p>In this case we have three different B–Splines (one for each dimension) and also 3 variables <span class="math inline">\(u,v,w\)</span> for each vertex we want to approximate.</p>
-<p>Given a target vertex <span class="math inline">\(\vec{p}^*\)</span> and an initial guess <span class="math inline">\(\vec{p}=V(u,v,w)\)</span> we define the error–function for the gradient–descent as:</p>
-<p><span class="math display">\[Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)\]</span></p>
-<p>And the partial version for just one direction as</p>
-<p><span class="math display">\[Err_x(u,v,w,\vec{p}^{*}) = p^{*}_x - \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \]</span></p>
-<p>To solve this we derive partially, like before:</p>
-<p><span class="math display">\[
-\begin{array}{rl}
-    \displaystyle \frac{\partial Err_x}{\partial u} &amp; p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
-  = &amp; \displaystyle - \sum_i \sum_j \sum_k N&#39;_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
-\end{array}
-\]</span></p>
-<p>The other partial derivatives follow the same pattern yielding the Jacobian:</p>
+<section id="definition-b-splines" class="slide level1">
+<h1>Definition B-Splines</h1>
+<ul>
+<li>The coefficient functions <span class="math inline">\(a_i(u)\)</span> in <span class="math inline">\(s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}\)</span> are different for each control-point</li>
+<li>Given a degree <span class="math inline">\(d\)</span> and position <span class="math inline">\(\tau_i\)</span> for the <span class="math inline">\(i\)</span>th control-point <span class="math inline">\(p_i\)</span> we define <span class="math display">\[\begin{equation}
+N_{i,0,\tau}(u) = \begin{cases} 1, &amp; u \in [\tau_i, \tau_{i+1}[ \\ 0, &amp; \mbox{otherwise} \end{cases}
+\end{equation}\]</span> and <span class="math display">\[\begin{equation} \label{eqn:ffd1d2}
+N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
+\end{equation}\]</span></li>
+<li>The derivatives of these coefficients are also easy to compute: <span class="math display">\[\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)\]</span></li>
+</ul>
+</section>
+<section id="properties-of-b-splines" class="slide level1">
+<h1>Properties of B-Splines</h1>
+<ul>
+<li>Coefficients vanish after <span class="math inline">\(d\)</span> differentiations</li>
+<li>Coefficients are continuous with respect to <span class="math inline">\(u\)</span></li>
+<li>A change in prototypes only deforms the mapping locally<br />
+(between <span class="math inline">\(p_i\)</span> to <span class="math inline">\(p_{i+d+1}\)</span>)</li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/unity.png" style=""></img><figcaption>Example of Basis-Functions for degree <span class="math inline">\(2\)</span>. [Brunet, 2010]<br /> Note, that Brunet starts his index at <span class="math inline">\(-d\)</span> opposed to our definition, where we start at <span class="math inline">\(0\)</span>.</figcaption></figure></p>
+</section>
+<section id="definition-ffd" class="slide level1">
+<h1>Definition FFD</h1>
+<ul>
+<li>FFD is a space-deformation resulting based on the underlying B-Splines</li>
+<li>Coefficients of space-mapping <span class="math inline">\(s(u) = \sum_j a_j(u) p_j\)</span> for an initial vertex <span class="math inline">\(v_i\)</span> are constant</li>
+<li>Set <span class="math inline">\(u_{i,j}~:=~N_{j,d,\tau}\)</span> for each <span class="math inline">\(v_i\)</span> and <span class="math inline">\(p_j\)</span> to get the projection: <span class="math display">\[
+  v_i = \sum_j u_{i,j} \cdot p_j = \vec{u}_i^{T} \vec{p}
+  \]</span> or written with matrices: <span class="math display">\[
+  \vec{v} = \vec{U} \vec{p}
+  \]</span></li>
+<li><span class="math inline">\(\vec{U}\)</span> is called <strong>deformation matrix</strong></li>
+</ul>
+</section>
+<section id="implementation-of-ffd" class="slide level1">
+<h1>Implementation of FFD</h1>
+<ul>
+<li>As we deal with 3D-Models we have to extend the introduced 1D-version</li>
+<li>We get one parameter for each dimension: <span class="math inline">\(u,v,w\)</span> instead of <span class="math inline">\(u\)</span></li>
+<li>Task: Find correct <span class="math inline">\(u,v,w\)</span> for each vertex in our model
+<ul>
+<li>We used a gradient-descent (via the gauss-newton algorithm)</li>
+</ul></li>
+</ul>
+</section>
+<section id="implementation-of-ffd-1" class="slide level1">
+<h1>Implementation of FFD</h1>
+<ul>
+<li>Given <span class="math inline">\(n,m,o\)</span> control-points in <span class="math inline">\(x,y,z\)</span>–direction each Point inside the convex hull is defined by <span class="math display">\[V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.\]</span></li>
+<li>Given a target vertex <span class="math inline">\(\vec{p}^*\)</span> and an initial guess <span class="math inline">\(\vec{p}=V(u,v,w)\)</span> we define the error–function for the gradient–descent as: <span class="math display">\[Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)\]</span></li>
+</ul>
+</section>
+<section id="implementation-of-ffd-2" class="slide level1">
+<h1>Implementation of FFD</h1>
+<ul>
+<li>Derivation is straightforward <span class="math display">\[
+  \scriptsize
+  \begin{array}{rl}
+      \displaystyle \frac{\partial Err_x}{\partial u} &amp; p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
+    = &amp; \displaystyle - \sum_i \sum_j \sum_k N&#39;_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
+  \end{array}
+  \]</span> yielding a Jacobian:</li>
+</ul>
 <p><span class="math display">\[
+\scriptsize
 J(Err(u,v,w)) = 
 \left(
 \begin{array}{ccc}
@@ -276,226 +260,290 @@ J(Err(u,v,w)) =
 \frac{\partial Err_z}{\partial u} &amp; \frac{\partial Err_z}{\partial v} &amp; \frac{\partial Err_z}{\partial w}
 \end{array}
 \right)
-\]</span> <span class="math display">\[
-\scriptsize
-=
-\left(
-\begin{array}{ccc}
-- \displaystyle \sum_{i,j,k} N&#39;_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x &amp;- \displaystyle \sum_{i,j,k} N_{i}(u) N&#39;_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x &amp; - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N&#39;_{k}(w) \cdot {c_{ijk}}_x \\
-- \displaystyle \sum_{i,j,k} N&#39;_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y &amp;- \displaystyle \sum_{i,j,k} N_{i}(u) N&#39;_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y &amp; - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N&#39;_{k}(w) \cdot {c_{ijk}}_y \\
-- \displaystyle \sum_{i,j,k} N&#39;_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z &amp;- \displaystyle \sum_{i,j,k} N_{i}(u) N&#39;_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z &amp; - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N&#39;_{k}(w) \cdot {c_{ijk}}_z
-\end{array}
-\right)
 \]</span></p>
-<p>With the Gauss–Newton algorithm we iterate via the formula <span class="math display">\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]</span> and use Cramer’s rule for inverting the small Jacobian and solving this system of linear equations.</p>
-<p>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 <span class="math inline">\(\epsilon\)</span>–error above. This takes — depending on the shape of the object and the grid — about <span class="math inline">\(3\)</span> to <span class="math inline">\(5\)</span> iterations that we observed in practice.</p>
-<p>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 <em>invalid</em> and do not test any of them.</p>
-<p>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.</p>
-<div id="deformation-grid">
+</section>
+<section id="implementation-of-ffd-3" class="slide level1">
+<h1>Implementation of FFD</h1>
+<ul>
+<li>Armed with this we iterate the formula <span class="math display">\[J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)\]</span> using Cramer’s rule for inverting the small Jacobian.</li>
+<li>Usually terminates after <span class="math inline">\(3\)</span> to <span class="math inline">\(5\)</span> iteration with an <span class="math inline">\(\epsilon := \vec{p^*} - V(u,v,w) &lt; 10^{-4}\)</span></li>
+<li>self-intersecting grids can invalidate the results
+<ul>
+<li>no problem, as these get not generated and contradict some properties we want (like locality)</li>
+</ul></li>
+</ul>
+</section>
+<section id="outline-1" class="slide level1">
+<h1>Outline</h1>
+<ul>
+<li>What is FFD?</li>
+<li><strong>What is evolutionary optimization?</strong></li>
+<li>How to measure evolvability?</li>
+<li>Scenarios</li>
+<li>Results</li>
+</ul>
+</section>
+<section id="what-is-evolutionary-optimization" class="slide level1">
+<h1>What is evolutionary optimization?</h1>
+<div id="section">
 <div style="width:50%;float:left">
-
+<pre><code data-noescape data-trim class="" style="">$t := 0$;
+initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$;
+evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$;
+while($c(F(t)) \neq$ true) {
+    recombine: $P’(t) := r(P(t))$;
+    mutate: $P''(t) := m(P’(t))$;
+    evaluate $F(t) : \{\Phi(x) | x \in P''(t)\}$
+    select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$;
+    $t := t + 1$;
+}</code></pre>
 </div>
 <div style="width:50%;float:left">
-<p>As mentioned in chapter , 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.</p>
+<pre><code data-noescape data-trim class="" style="">$t$: Iteration-step
+$I$: Set of possible Individuals
+$P$: Population of Individuals
+$F$: Fitness of Individuals
+$Q$: Either set of parents or $\emptyset$
+
+$r(..) : I^\mu \mapsto I^\lambda$
+$m(..) : I^\lambda \mapsto I^\lambda$
+$s(..) : I^{\lambda + \mu} \mapsto I^\mu$</code></pre>
 </div>
 <div style="clear: both">
 
 </div>
 </div>
-<p>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 <span class="math inline">\([0,1[\)</span> and — as a consequence — we have to embed our object into it (or create constant “dummy”–points outside).</p>
-<p>The great advantage of B–Splines is the local, direct impact of each control point without having a <span class="math inline">\(1:1\)</span>–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.</p>
-
-<p>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 <span class="math inline">\(\vec{p}\)</span> only the <span class="math inline">\(2 \cdot (d-1)\)</span> control–points contribute to the parametrization of that point<a href="#/fn5" class="footnote-ref" id="fnref5"><sup>5</sup></a>. 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.</p>
-<p>We illustrate this phenomenon in figure , where the red central points are not relevant for the parametrization of the circle. This leads to artefacts in the deformation–matrix <span class="math inline">\(\vec{U}\)</span>, as the columns corresponding to those control–points are <span class="math inline">\(0\)</span>.</p>
-<p>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<a href="#/fn6" class="footnote-ref" id="fnref6"><sup>6</sup></a> at worst.</p>
-<p>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 .</p>
-<p>For our tests we chose different uniformly sized grids and added noise onto each control–point<a href="#/fn7" class="footnote-ref" id="fnref7"><sup>7</sup></a> to simulate different starting–conditions.</p>
+<ul>
+<li>Algorithm to model simple inheritance</li>
+<li>Consists of three main steps
+<ul>
+<li>recombination</li>
+<li>mutation</li>
+<li>selection</li>
+</ul></li>
+<li>An “individual” in our case is the displacement of control-points</li>
+</ul>
 </section>
+<section id="evolutional-loop" class="slide level1">
+<h1>Evolutional loop</h1>
+<ul>
+<li><strong>Recombination</strong> generates <span class="math inline">\(\lambda\)</span> new individuals based on the characteristics of the <span class="math inline">\(\mu\)</span> parents.
+<ul>
+<li>This makes sure that the next guess is close to the old guess.</li>
+</ul></li>
+<li><strong>Mutation</strong> introduces new effects that cannot be produced by mere recombination of the parents.
+<ul>
+<li>Typically these are minor defects to individual members of the population i.e. through added noise</li>
+</ul></li>
+<li><strong>Selection</strong> selects <span class="math inline">\(\mu\)</span> individuals from the children (and optionally the parents) using a <em>fitness–function</em> <span class="math inline">\(\Phi\)</span>.
+<ul>
+<li>Fitness could mean low error, good improvement, etc.</li>
+<li>Fitness not solely determines who survives, there are many possibilities</li>
+</ul></li>
+</ul>
 </section>
-<section id="scenarios-for-testing-evolvabilitycriteria-using" class="slide level1">
-<h1>Scenarios for testing evolvability–criteria using </h1>
-
-<p>In our experiments we use the same two testing–scenarios, that were also used by Richter et al. The first scenario deforms a plane into a shape originally defined by Giannelli et al., where we setup control–points in a 2–dimensional manner and merely deform in the height–coordinate to get the resulting shape.</p>
-<p>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.</p>
-<section id="test-scenario-1d-function-approximation" class="level2">
-<h2>Test Scenario: 1D Function Approximation</h2>
-<p>In this scenario we used the shape defined by Giannelli et al., which is also used by Richter et al. using the same discretization to <span class="math inline">\(150 \times 150\)</span> points for a total of <span class="math inline">\(n = 22\,500\)</span> vertices. The shape is given by the following definition <span class="math display">\[\begin{equation}
-t(x,y) =
-\begin{cases}
-0.5 \cos(4\pi \cdot q^{0.5}) + 0.5 &amp; q(x,y) &lt; \frac{1}{16},\\
-2(y-x) &amp; 0 &lt; y-x &lt; 0.5,\\
-1 &amp; 0.5 &lt; y - x
-\end{cases}
-\end{equation}\]</span><!-- </> --> with <span class="math inline">\((x,y) \in [0,2] \times [0,1]\)</span> and <span class="math inline">\(q(x,y)=(x-1.5)^2 + (y-0.5)^2\)</span>, which we have visualized in figure .</p>
-
-<p>As the starting–plane we used the same shape, but set all <span class="math inline">\(z\)</span>–coordinates to <span class="math inline">\(0\)</span>, yielding a flat plane, which is partially already correct.</p>
-<p>Regarding the <em>fitness–function</em> <span class="math inline">\(\mathrm{f}(\vec{p})\)</span>, we use the very simple approach of calculating the squared distances for each corresponding vertex <span class="math display">\[\begin{equation}
-\mathrm{f}(\vec{p}) = \sum_{i=1}^{n} \|(\vec{Up})_i - t_i\|_2^2 = \|\vec{Up} - \vec{t}\|^2 \rightarrow \min
-\end{equation}\]</span> where <span class="math inline">\(t_i\)</span> are the respective target–vertices to the parametrized source–vertices<a href="#/fn8" class="footnote-ref" id="fnref8"><sup>8</sup></a> with the current deformation–parameters <span class="math inline">\(\vec{p} = (p_1,\dots, p_m)\)</span>. 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.</p>
-<p>This formula is also the least–squares approximation error for which we can compute the analytic solution <span class="math inline">\(\vec{p^{*}} = \vec{U^+}\vec{t}\)</span>, yielding us the correct gradient in which the evolutionary optimizer should move.</p>
+<section id="outline-2" class="slide level1">
+<h1>Outline</h1>
+<ul>
+<li>What is FFD?</li>
+<li>What is evolutionary optimization?</li>
+<li><strong>How to measure evolvability?</strong></li>
+<li>Scenarios</li>
+<li>Results</li>
+</ul>
 </section>
-<section id="test-scenario-3d-function-approximation" class="level2">
-<h2>Test Scenario: 3D Function Approximation</h2>
-<p> 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 <em>fitness–function</em> we will use has no known analytic solution and multiple local minima.</p>
-
-<p>First of all we introduce the set up: We have given a triangulated model of a sphere consisting of <span class="math inline">\(10\,807\)</span> vertices, that we want to deform into a the target–model of a face with a total of <span class="math inline">\(12\,024\)</span> vertices. Both of these Models can be seen in figure .</p>
-<p>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.</p>
-<p>We therefore define the <em>fitness–function</em> to be:</p>
-<span class="math display">\[\begin{equation}
-\mathrm{f}(\vec{P}) = \frac{1}{n} \underbrace{\sum_{i=1}^n \|\vec{c_T(s_i)} -
-\vec{s_i}\|_2^2}_{\textrm{source--to--target--distance}}
-+ \frac{1}{m} \underbrace{\sum_{i=1}^m \|\vec{c_S(t_i)} -
-\vec{t_i}\|_2^2}_{\textrm{target--to--source--distance}}
-+ \lambda \cdot \textrm{regularization}(\vec{P})
-\label{eq:fit3d}
-\end{equation}\]</span>
-<p>where <span class="math inline">\(\vec{c_T(s_i)}\)</span> denotes the target–vertex that is corresponding to the source–vertex <span class="math inline">\(\vec{s_i}\)</span> and <span class="math inline">\(\vec{c_S(t_i)}\)</span> denotes the source–vertex that corresponds to the target–vertex <span class="math inline">\(\vec{t_i}\)</span>. 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 <span class="math inline">\(\vec{P}\)</span>, as those get calculated by the previously introduces formula <span class="math inline">\(\vec{S} = \vec{UP}\)</span> with <span class="math inline">\(\vec{S}\)</span> being the <span class="math inline">\(n \times 3\)</span>–matrix of source–vertices, <span class="math inline">\(\vec{U}\)</span> the <span class="math inline">\(n \times m\)</span>–matrix of calculated coefficients for the  — analog to the 1D case — and finally <span class="math inline">\(\vec{P}\)</span> being the <span class="math inline">\(m \times 3\)</span>–matrix of the control–grid defining the whole deformation.</p>
-<p>As regularization–term we add a weighted Laplacian of the deformation that has been used before by Aschenbach et al. on similar models and was shown to lead to a more precise fit. The Laplacian <span class="math display">\[\begin{equation}
-\mathrm{regularization}(\vec{P}) = \frac{1}{\sum_i A_i} \sum_{i=1}^n A_i \cdot \left( \sum_{\vec{s}_j \in \mathcal{N}(\vec{s}_i)} w_j \cdot \|\Delta \vec{s}_j - \Delta \vec{s}_i\|^2 \right)
-\label{eq:reg3d}
-\end{equation}\]</span> is determined by the cotangent weighted displacement <span class="math inline">\(w_j\)</span> of the to <span class="math inline">\(s_i\)</span> connected vertices <span class="math inline">\(\mathcal{N}(s_i)\)</span> and <span class="math inline">\(A_i\)</span> is the Voronoi–area of the corresponding vertex <span class="math inline">\(\vec{s_i}\)</span>. We leave out the <span class="math inline">\(\vec{R}_i\)</span>–term from the original paper as our deformation is merely linear.</p>
-<p>This regularization–weight gives us a measure of stiffness for the material that we will influence via the <span class="math inline">\(\lambda\)</span>–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 .</p>
+<section id="how-to-measure-evolvability" class="slide level1">
+<h1>How to measure evolvability?</h1>
+<ul>
+<li>Different (conflicting) optimization targets
+<ul>
+<li>convergence speed?</li>
+<li>convergence quality?</li>
+</ul></li>
+<li>As <span class="math inline">\(\vec{v} = \vec{U}\vec{p}\)</span> is linear, we can also look at <span class="math inline">\(\Delta \vec{v} = \vec{U}\,  \Delta \vec{p}\)</span>
+<ul>
+<li>We only change <span class="math inline">\(\Delta \vec{p}\)</span>, so evolvability should only use <span class="math inline">\(\vec{U}\)</span> for predictions</li>
+</ul></li>
+</ul>
 </section>
+<section id="evolvability-criteria" class="slide level1">
+<h1>Evolvability criteria</h1>
+<ul>
+<li><strong>Variability</strong>
+<ul>
+<li>roughly: “How many actual Degrees of Freedom exist?”</li>
+<li>Defined by <span class="math display">\[\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n} \in [0..1]\]</span></li>
+<li>in FFD this is <span class="math inline">\(1/\#\textrm{CP}\)</span> for the number of control-points used for parametrization</li>
+</ul></li>
+</ul>
 </section>
-<section id="evaluation-of-scenarios" class="slide level1">
-<h1>Evaluation of Scenarios</h1>
-
-<p>To compare our results to the ones given by Richter et al., 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 how well an arbitrary monotonic function can describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables.</p>
-<p>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.</p>
-<p>For interpretation of these values we follow the same interpretation used in , based on : The coefficient intervals <span class="math inline">\(r_S \in [0,0.2[\)</span>, <span class="math inline">\([0.2,0.4[\)</span>, <span class="math inline">\([0.4,0.6[\)</span>, <span class="math inline">\([0.6,0.8[\)</span>, and <span class="math inline">\([0.8,1]\)</span> are classified as <em>very weak</em>, <em>weak</em>, <em>moderate</em>, <em>strong</em> and <em>very strong</em>. We interpret p–values smaller than <span class="math inline">\(0.01\)</span> as <em>significant</em> and cut off the precision of p–values after four decimal digits (thus often having a p–value of <span class="math inline">\(0\)</span> given for p–values <span class="math inline">\(&lt; 10^{-4}\)</span>). <!-- </> --></p>
-<p>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.</p>
-<p>For the evolutionary optimization we employ the  of the shark3.1 library , as this algorithm was used by  as well. We leave the parameters at their sensible defaults as further explained in .</p>
-<section id="procedure-1d-function-approximation" class="level2">
-<h2>Procedure: 1D Function Approximation</h2>
-
-<p>For our setup we first compute the coefficients of the deformation–matrix and use the formulas for <em>variability</em> and <em>regularity</em> to get our predictions. Afterwards we solve the problem analytically to get the (normalized) correct gradient that we use as guess for the <em>improvement potential</em>. To further test the <em>improvement potential</em> we also consider a distorted gradient <span class="math inline">\(\vec{g}_{\mathrm{d}}\)</span>: <span class="math display">\[
-\vec{g}_{\mathrm{d}} = \frac{\mu \vec{g}_{\mathrm{c}} + (1-\mu)\mathbb{1}}{\|\mu \vec{g}_{\mathrm{c}} + (1-\mu) \mathbb{1}\|}
-\]</span> where <span class="math inline">\(\mathbb{1}\)</span> is the vector consisting of <span class="math inline">\(1\)</span> in every dimension, <span class="math inline">\(\vec{g}_\mathrm{c} = \vec{p^{*}} - \vec{p}\)</span> is the calculated correct gradient, and <span class="math inline">\(\mu\)</span> is used to blend between <span class="math inline">\(\vec{g}_\mathrm{c}\)</span> and <span class="math inline">\(\mathbb{1}\)</span>. As we always start with a gradient of <span class="math inline">\(p = \mathbb{0}\)</span> this means we can shorten the definition of <span class="math inline">\(\vec{g}_\mathrm{c}\)</span> to <span class="math inline">\(\vec{g}_\mathrm{c} = \vec{p^{*}}\)</span>.</p>
-
-<p>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 ) we avoid the generation of such self–intersecting grids for our testcases.</p>
-<p>To achieve that we generated a gaussian distributed number with <span class="math inline">\(\mu = 0, \sigma=0.25\)</span> and clamped it to the range <span class="math inline">\([-0.25,0.25]\)</span>. We chose such an <span class="math inline">\(r \in [-0.25,0.25]\)</span> per dimension and moved the control–points by that factor towards their respective neighbours<a href="#/fn9" class="footnote-ref" id="fnref9"><sup>9</sup></a>.</p>
-<p>In other words we set <span class="math display">\[\begin{equation*}
-p_i =
-\begin{cases}
-  p_i + (p_i - p_{i-1}) \cdot r, &amp; \textrm{if } r \textrm{ negative} \\
-  p_i + (p_{i+1} - p_i) \cdot r, &amp; \textrm{if } r \textrm{ positive}
-\end{cases}
-\end{equation*}\]</span> in each dimension separately.</p>
-<p>An Example of such a testcase can be seen for a <span class="math inline">\(7 \times 4\)</span>–grid in figure .</p>
+<section id="evolvability-criteria-1" class="slide level1">
+<h1>Evolvability criteria</h1>
+<ul>
+<li><strong>Regularity</strong>
+<ul>
+<li>roughly: “How numerically stable is the optimization?”</li>
+<li>Defined by <span class="math display">\[\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}} \in [0..1]\]</span> with <span class="math inline">\(\sigma_{min/max}\)</span> being the least/greatest right singular value.</li>
+<li>high, when <span class="math inline">\(\|\vec{Up}\| \propto \|\vec{p}\|\)</span></li>
+</ul></li>
+</ul>
 </section>
-<section id="results-of-1d-function-approximation" class="level2">
-<h2>Results of 1D Function Approximation</h2>
-<p>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 <span class="math inline">\(1.05 \times\)</span> of the optimal solution.</p>
-<p>We used different regular grids that we manipulated as explained in Section  with a different number of control–points. As our grids have to be the product of two integers, we compared a <span class="math inline">\(5 \times 5\)</span>–grid with <span class="math inline">\(25\)</span> control–points to a <span class="math inline">\(4 \times 7\)</span> and <span class="math inline">\(7 \times 4\)</span>–grid with <span class="math inline">\(28\)</span> control–points. This was done to measure the impact an improper  setup could have and how well this is displayed in the criteria we are examining.</p>
-<p>Additionally we also measured the effect of increasing the total resolution of the grid by taking a closer look at <span class="math inline">\(5 \times 5\)</span>, <span class="math inline">\(7 \times 7\)</span> and <span class="math inline">\(10 \times 10\)</span> grids.</p>
-<section id="variability-1" class="level3">
-<h3>Variability</h3>
-
-<p><em>Variability</em> should characterize the potential for design space exploration and is defined in terms of the normalized rank of the deformation matrix <span class="math inline">\(\vec{U}\)</span>: <span class="math inline">\(V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n}\)</span>, whereby <span class="math inline">\(n\)</span> is the number of vertices. As all our tested matrices had a constant rank (being <span class="math inline">\(m = x \cdot y\)</span> for a <span class="math inline">\(x \times y\)</span> grid), we have merely plotted the errors in the box plot in figure </p>
-<p>It is also noticeable, that although the <span class="math inline">\(7 \times 4\)</span> and <span class="math inline">\(4 \times 7\)</span> grids have a higher <em>variability</em>, they perform not better than the <span class="math inline">\(5 \times 5\)</span> grid. Also the <span class="math inline">\(7 \times 4\)</span> and <span class="math inline">\(4 \times 7\)</span> grids differ distinctly from each other with a mean<span class="math inline">\(\pm\)</span>sigma of <span class="math inline">\(233.09 \pm 12.32\)</span> for the former and <span class="math inline">\(286.32 \pm 22.36\)</span> 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.</p>
-<p>Leaving the issue of the grid–layout aside we focused on grids having the same number of prototypes in every dimension. For the <span class="math inline">\(5 \times 5\)</span>, <span class="math inline">\(7 \times 7\)</span> and <span class="math inline">\(10 \times 10\)</span> grids we found a <em>very strong</em> correlation (<span class="math inline">\(-r_S = 0.94, p = 0\)</span>) between the <em>variability</em> and the evolutionary error.</p>
+<section id="evolvability-criteria-2" class="slide level1">
+<h1>Evolvability criteria</h1>
+<ul>
+<li><strong>Improvement Potential</strong>
+<ul>
+<li>roughly: “How good can the best fit become?”</li>
+<li>Defined by <span class="math display">\[\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F\]</span> with a unit-normed guessed gradient <span class="math inline">\(\vec{G}\)</span></li>
+</ul></li>
+</ul>
 </section>
-<section id="regularity-1" class="level3">
-<h3>Regularity</h3>
-
-
-<p><em>Regularity</em> should correspond to the convergence speed (measured in iteration–steps of the evolutionary algorithm), and is computed as inverse condition number <span class="math inline">\(\kappa(\vec{U})\)</span> of the deformation–matrix.</p>
-<p>As can be seen from table , we could only show a <em>weak</em> correlation in the case of a <span class="math inline">\(5 \times 5\)</span> 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 <em>strong</em> correlation of <span class="math inline">\(- r_S = -0.72, p = 0\)</span>, that is opposed to our expectations.</p>
-<p>To explain this discrepancy we took a closer look at what caused these high number of iterations. In figure  we also plotted the <em>improvement potential</em> against the steps next to the <em>regularity</em>–plot. Our theory is that the <em>very strong</em> correlation (<span class="math inline">\(-r_S = -0.82, p=0\)</span>) between <em>improvement potential</em> and number of iterations hints that the employed algorithm simply takes longer to converge on a better solution (as seen in figure  and ) offsetting any gain the regularity–measurement could achieve.</p>
+<section id="outline-3" class="slide level1">
+<h1>Outline</h1>
+<ul>
+<li>What is FFD?</li>
+<li>What is evolutionary optimization?</li>
+<li>How to measure evolvability?</li>
+<li><strong>Scenarios</strong></li>
+<li>Results</li>
+</ul>
 </section>
-<section id="improvement-potential-1" class="level3">
-<h3>Improvement Potential</h3>
-
-<p>The <em>improvement potential</em> should correlate to the quality of the fitting–result. We plotted the results for the tested grid–sizes <span class="math inline">\(5 \times 5\)</span>, <span class="math inline">\(7 \times 7\)</span> and <span class="math inline">\(10 \times 10\)</span> in figure . We tested the <span class="math inline">\(4 \times 7\)</span> and <span class="math inline">\(7 \times 4\)</span> grids as well, but omitted them from the plot.</p>
-<p>Additionally we tested the results for a distorted gradient described in  with a <span class="math inline">\(\mu\)</span>–value of <span class="math inline">\(0.25\)</span>, <span class="math inline">\(0.5\)</span>, <span class="math inline">\(0,75\)</span>, and <span class="math inline">\(1.0\)</span> for the <span class="math inline">\(5 \times 5\)</span> grid and with a <span class="math inline">\(\mu\)</span>–value of <span class="math inline">\(0.5\)</span> for all other cases.</p>
-<p>All results show the identical <em>very strong</em> and <em>significant</em> correlation with a Spearman–coefficient of <span class="math inline">\(- r_S = 1.0\)</span> and p–value of <span class="math inline">\(0\)</span>.</p>
-<p>These results indicate, that <span class="math inline">\(\|\mathbb{1} - \vec{U}\vec{U}^{+}\|_F\)</span> is close to <span class="math inline">\(0\)</span>, 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.</p>
+<section id="scenarios" class="slide level1">
+<h1>Scenarios</h1>
+<ul>
+<li>2 Testing Scenarios</li>
+<li>1-dimensional fit
+<ul>
+<li><span class="math inline">\(xy\)</span>-plane to <span class="math inline">\(xyz\)</span>-model, where only the <span class="math inline">\(z\)</span>-coordinate changes</li>
+<li>can be solved analytically with known global optimum</li>
+</ul></li>
+<li>3-dimensional fit
+<ul>
+<li>fit a parametrized sphere into a face</li>
+<li>cannot be solved analytically</li>
+<li>number of vertices differ between models</li>
+</ul></li>
+</ul>
 </section>
+<section id="d-scenario" class="slide level1">
+<h1>1D-Scenario</h1>
+<p><figure class="" style=""><img  src="../arbeit/img/example1d_grid.png" style=""></img><figcaption>Left: A regular <span class="math inline">\(7 \times 4\)</span>–grid<br />Right: The same grid after a random distortion to generate a testcase.</figcaption></figure></p>
+<p><figure class="" style="width:70%;"><img  src="../arbeit/img/1dtarget.png" style="width:70%;"></img><figcaption>The target–shape for our 1–dimensional optimization–scenario including a wireframe–overlay of the vertices.</figcaption></figure></p>
 </section>
-<section id="procedure-3d-function-approximation" class="level2">
-<h2>Procedure: 3D Function Approximation</h2>
-
-<p>As explained in section  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 <span class="math inline">\(10\,807\)</span> and <span class="math inline">\(12\,024\)</span> vertices respectively, so we cannot make a one–to–one–correspondence between them as we did in the one–dimensional case.</p>
-<p>Initially we set up the correspondences <span class="math inline">\(\vec{c_T(\dots)}\)</span> and <span class="math inline">\(\vec{c_S(\dots)}\)</span> to be the respectively closest vertices of the other model. We then calculate the analytical solution given these correspondences via <span class="math inline">\(\vec{P^{*}} = \vec{U^+}\vec{T}\)</span>, and also use the first solution as guessed gradient for the calculation of the <em>improvement potential</em>, as the optimal solution is not known. We then let the evolutionary algorithm run up within <span class="math inline">\(1.05\)</span> times the error of this solution and afterwards recalculate the correspondences <span class="math inline">\(\vec{c_T(\dots)}\)</span> and <span class="math inline">\(\vec{c_S(\dots)}\)</span>.</p>
-
-<p>For the next step we then halve the regularization–impact <span class="math inline">\(\lambda\)</span> (starting at <span class="math inline">\(1\)</span>) of our <em>fitness–function</em> () and calculate the next incremental solution <span class="math inline">\(\vec{P^{*}} = \vec{U^+}\vec{T}\)</span> 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 <span class="math inline">\(\approx 100\)</span>, whereas the regularization is numerically <span class="math inline">\(\approx 7000\)</span> we need at least <span class="math inline">\(10\)</span> to <span class="math inline">\(15\)</span> iterations until the regularization–effect wears off.</p>
-<p>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.</p>
-<p>In figure  we show an example setup of the scene with a <span class="math inline">\(4\times 4\times 4\)</span>–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 , but in three instead of two dimensions<a href="#/fn10" class="footnote-ref" id="fnref10"><sup>10</sup></a>.</p>
-
-<p>As is clearly visible from figure , 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.</p>
+<section id="d-scenarios" class="slide level1">
+<h1>3D-Scenarios</h1>
+<p><figure class="" style=""><img  src="../arbeit/img/3dtarget.png" style=""></img><figcaption>Left: The sphere we start from with 10 807 vertices<br />Right: The face we want to deform the sphere into with 12 024 vertices.</figcaption></figure></p>
 </section>
-<section id="results-of-3d-function-approximation" class="level2">
-<h2>Results of 3D Function Approximation</h2>
-<p>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 <span class="math inline">\(4 \times 4 \times 4\)</span> grid and only increase the number of control–points in one dimension, yielding a resolution of <span class="math inline">\(7 \times 4 \times 4\)</span> and <span class="math inline">\(4 \times 4 \times 7\)</span> respectively. We visualized those two grids in figure .</p>
-<p>To evaluate the performance of the evolvability–criteria we also tested a more neutral resolution of <span class="math inline">\(4 \times 4 \times 4\)</span>, <span class="math inline">\(5 \times 5 \times 5\)</span>, and <span class="math inline">\(6 \times 6 \times 6\)</span> — similar to the 1D–setup.</p>
-
-<section id="variability-2" class="level3">
-<h3>Variability</h3>
-
-
-<p>Similar to the 1D case all our tested matrices had a constant rank (being <span class="math inline">\(m = x \cdot y \cdot z\)</span> for a <span class="math inline">\(x \times y \times z\)</span> grid), so we again have merely plotted the errors in the box plot in figure .</p>
-<p>As expected the <span class="math inline">\(\mathrm{X} \times 4 \times 4\)</span> grids performed slightly better than their <span class="math inline">\(4 \times 4 \times \mathrm{X}\)</span> counterparts with a mean<span class="math inline">\(\pm\)</span>sigma of <span class="math inline">\(101.25 \pm 7.45\)</span> to <span class="math inline">\(102.89 \pm 6.74\)</span> for <span class="math inline">\(\mathrm{X} = 5\)</span> and <span class="math inline">\(85.37 \pm 7.12\)</span> to <span class="math inline">\(89.22 \pm 6.49\)</span> for <span class="math inline">\(\mathrm{X} = 7\)</span>.</p>
-<p>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 bad 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 <em>variability</em> via the rank of the deformation–matrix.</p>
-<p>Overall the correlation between <em>variability</em> and fitness–error were <em>significant</em> and showed a <em>very strong</em> correlation in all our tests. The detailed correlation–coefficients are given in table  alongside their p–values.</p>
-<p>As introduces in section  and visualized in figure , 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.</p>
-<p>One can already see in 2D in figure , that this effect starts with a regular <span class="math inline">\(9 \times 9\)</span> grid on a perfect circle. To make sure we observe this, we evaluated the <em>variability</em> for 100 randomly moved <span class="math inline">\(10 \times 10 \times 10\)</span> grids on the sphere we start out with.</p>
-
-<p>As the <em>variability</em> is defined by <span class="math inline">\(\frac{\mathrm{rank}(\vec{U})}{n}\)</span> we can easily recover the rank of the deformation–matrix <span class="math inline">\(\vec{U}\)</span>. The results are shown in the histogram in figure . Especially in the centre of the sphere and in the corners of our grid we effectively loose control–points for our parametrization.</p>
-<p>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.</p>
-<p>Nevertheless this hints at the notion, that <em>variability</em> is a good measure for the overall quality of a fit.</p>
+<section id="outline-4" class="slide level1">
+<h1>Outline</h1>
+<ul>
+<li>What is FFD?</li>
+<li>What is evolutionary optimization?</li>
+<li>How to measure evolvability?</li>
+<li>Scenarios</li>
+<li><strong>Results</strong></li>
+</ul>
 </section>
-<section id="regularity-2" class="level3">
-<h3>Regularity</h3>
-
-<p>Opposed to the predictions of <em>variability</em> our test on <em>regularity</em> gave a mixed result — similar to the 1D–case.</p>
-<p>In roughly half of the scenarios we have a <em>significant</em>, but <em>weak</em> to <em>moderate</em> correlation between <em>regularity</em> and number of iterations. On the other hand in the scenarios where we increased the number of control–points, namely <span class="math inline">\(125\)</span> for the <span class="math inline">\(5 \times 5 \times 5\)</span> grid and <span class="math inline">\(216\)</span> for the <span class="math inline">\(6 \times 6 \times 6\)</span> grid we found a <em>significant</em>, but <em>weak</em> <strong>anti</strong>–correlation when taking all three tests into account<a href="#/fn11" class="footnote-ref" id="fnref11"><sup>11</sup></a>, which seem to contradict the findings/trends for the sets with <span class="math inline">\(64\)</span>, <span class="math inline">\(80\)</span>, and <span class="math inline">\(112\)</span> control–points (first two rows of table ).</p>
-<p>Taking all results together we only find a <em>very weak</em>, but <em>significant</em> link between <em>regularity</em> and the number of iterations needed for the algorithm to converge.</p>
-
-<p>As can be seen from figure , 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 <em>regularity</em> seems to increase at first, but then decreases again on higher grid–resolutions.</p>
-<p>This can be an artefact of the definition of <em>regularity</em>, as it is defined by the inverse condition–number of the deformation–matrix <span class="math inline">\(\vec{U}\)</span>, being the fraction <span class="math inline">\(\frac{\sigma_{\mathrm{min}}}{\sigma_{\mathrm{max}}}\)</span> between the least and greatest right singular value.</p>
-<p>As we observed in the previous section, we cannot guarantee that each control–point has an effect (see figure ) and so a small minimal right singular value occurring on higher grid–resolutions seems likely the problem.</p>
-<p>Adding to this we also noted, that in the case of the <span class="math inline">\(10 \times 10 \times 10\)</span>–grid the <em>regularity</em> was always <span class="math inline">\(0\)</span>, as a non–contributing control–point yields a <span class="math inline">\(0\)</span>–column in the deformation–matrix, thus letting <span class="math inline">\(\sigma_\mathrm{min} = 0\)</span>. A better definition for <em>regularity</em> (i.e. using the smallest non–zero right singular value) could solve this particular issue, but not fix the trend we noticed above.</p>
+<section id="variability-1d" class="slide level1">
+<h1>Variability 1D</h1>
+<ul>
+<li>Should measure Degrees of Freedom and thus quality</li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/evolution1d/variability_boxplot.png" style=""></img><figcaption>The squared error for the various grids we examined.<br /> Note that <span class="math inline">\(7 \times 4\)</span> and <span class="math inline">\(4 \times 7\)</span> have the same number of control–points.</figcaption></figure></p>
+<ul>
+<li><span class="math inline">\(5 \times 5\)</span>, <span class="math inline">\(7 \times 7\)</span> and <span class="math inline">\(10 \times 10\)</span> have <em>very strong</em> correlation (<span class="math inline">\(-r_S = 0.94, p = 0\)</span>) between the <em>variability</em> and the evolutionary error.</li>
+</ul>
 </section>
-<section id="improvement-potential-2" class="level3">
-<h3>Improvement Potential</h3>
-
-<p>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.</p>
-
-<p>We plotted our findings on the <em>improvement potential</em> in a similar way as we did before with the <em>regularity</em>. In figure  one can clearly see the correlation and the spread within each setup and the behaviour when we increase the number of control–points.</p>
-<p>Along with this we also give the Spearman–coefficients along with their p–values in table . Within one scenario we only find a <em>weak</em> to <em>moderate</em> correlation between the <em>improvement potential</em> and the fitting error, but all findings (except for <span class="math inline">\(7 \times 4 \times 4\)</span> and <span class="math inline">\(6 \times 6 \times 6\)</span>) are significant.</p>
-<p>If we take multiple datasets into account the correlation is <em>very strong</em> and <em>significant</em>, which is good, as this functions as a litmus–test, because the quality is naturally tied to the number of control–points.</p>
-<p>All in all the <em>improvement potential</em> seems to be a good and sensible measure of quality, even given gradients of varying quality.</p>
-<p>Lastly, a small note on the behaviour of <em>improvement potential</em> and convergence speed, as we used this in the 1D case to argue, why the <em>regularity</em> defied our expectations. As a contrast we wanted to show, that <em>improvement potential</em> cannot serve for good predictions of the convergence speed. In figure  we show <em>improvement potential</em> against number of iterations for both scenarios. As one can see, in the 1D scenario we have a <em>strong</em> and <em>significant</em> correlation (with <span class="math inline">\(-r_S = -0.72\)</span>, <span class="math inline">\(p = 0\)</span>), whereas in the 3D scenario we have the opposite <em>significant</em> and <em>strong</em> effect (with <span class="math inline">\(-r_S = 0.69\)</span>, <span class="math inline">\(p=0\)</span>), so these correlations clearly seem to be dependent on the scenario and are not suited for generalization.</p>
-
+<section id="variability-3d" class="slide level1">
+<h1>Variability 3D</h1>
+<ul>
+<li>Should measure Degrees of Freedom and thus quality</li>
+</ul>
+<p><figure class="" style=""><img  src="../arbeit/img/evolution3d/variability_boxplot.png" style=""></img><figcaption>The fitting error for the various grids we examined.<br />Note that the number of control–points is a product of the resolution, so <span class="math inline">\(X \times 4 \times 4\)</span> and <span class="math inline">\(4 \times 4 \times X\)</span> have the same number of control–points.</figcaption></figure></p>
+<ul>
+<li><span class="math inline">\(4 \times 4 \times 4\)</span>, <span class="math inline">\(5 \times 5 \times 5\)</span> and <span class="math inline">\(6 \times 6 \times 6\)</span> have <em>very strong</em> correlation (<span class="math inline">\(-r_S = 0.91, p = 0\)</span>) between the <em>variability</em> and the evolutionary error.</li>
+</ul>
 </section>
-</section>
-</section>
-<section id="discussion-and-outlook" class="slide level1">
-<h1>Discussion and outlook</h1>
+<section id="varying-variability" class="slide level1">
+<h1>Varying Variability</h1>
+<div id="section-1">
+<div style="width:50%;float:left">
+<p><figure class="" style=""><img  src="../arbeit/img/enoughCP.png" style=""></img><figcaption>A high resolution (<span class="math inline">\(10 \times 10\)</span>) of control–points over a circle. Yellow/green points contribute to the parametrization, red points don’t.<br />An Example–point (blue) is solely determined by the position of the green control–points.</figcaption></figure></p>
+</div>
+<div style="width:50%;float:left">
+<p><figure class="" style=""><img  src="../arbeit/img/evolution3d/variability2_boxplot.png" style=""></img><figcaption>Histogram of ranks of various <span class="math inline">\(10 \times 10 \times 10\)</span> grids with <span class="math inline">\(1000\)</span> control–points each showing in this case how many control–points are actually used in the calculations.</figcaption></figure></p>
+</div>
+<div style="clear: both">
 
-<p>In this thesis we took a look at the different criteria for <em>evolvability</em> as introduced by Richter et al., namely <em>variability</em>, <em>regularity</em> and <em>improvement potential</em> under different setup–conditions. Where Richter et al. used , we employed  to set up a low–complexity parametrization of a more complex vertex–mesh.</p>
-<p>In our findings we could show in the 1D–scenario, that there were statistically <em>significant</em> <em>very strong</em> correlations between <em>variability and fitting error</em> (<span class="math inline">\(0.94\)</span>) and <em>improvement potential and fitting error</em> (<span class="math inline">\(1.0\)</span>) with comparable results than Richter et al. (with <span class="math inline">\(0.31\)</span> to <span class="math inline">\(0.88\)</span> for the former and <span class="math inline">\(0.75\)</span> to <span class="math inline">\(0.99\)</span> for the latter), whereas we found only <em>weak</em> correlations for <em>regularity and convergence–speed</em> (<span class="math inline">\(0.28\)</span>) opposed to Richter et al. with <span class="math inline">\(0.39\)</span> to <span class="math inline">\(0.91\)</span>.<a href="#/fn12" class="footnote-ref" id="fnref12"><sup>12</sup></a></p>
-<p>For the 3D–scenario our results show a <em>very strong</em>, <em>significant</em> correlation between <em>variability and fitting error</em> with <span class="math inline">\(0.89\)</span> to <span class="math inline">\(0.94\)</span>, which are pretty much in line with the findings of Richter et al. (<span class="math inline">\(0.65\)</span> to <span class="math inline">\(0.95\)</span>). The correlation between <em>improvement potential and fitting error</em> behave similar, with our findings having a significant coefficient of <span class="math inline">\(0.3\)</span> to <span class="math inline">\(0.95\)</span> depending on the grid–resolution compared to the <span class="math inline">\(0.61\)</span> to <span class="math inline">\(0.93\)</span> from Richter et al. In the case of the correlation of <em>regularity and convergence speed</em> we found very different (and often not significant) correlations and anti–correlations ranging from <span class="math inline">\(-0.25\)</span> to <span class="math inline">\(0.46\)</span>, whereas Richter et al. reported correlations between <span class="math inline">\(0.34\)</span> to <span class="math inline">\(0.87\)</span>.</p>
-<p>Taking these results into consideration, one can say, that <em>variability</em> and <em>improvement potential</em> are very good estimates for the quality of a fit using  as a deformation function, while we could not reproduce similar compelling results as Richter et al. for <em>regularity and convergence speed</em>.</p>
-<p>One reason for the bad or erratic behaviour of the <em>regularity</em>–criterion could be that in an –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 <span class="math inline">\(\vec{U}\)</span> that influence the condition–number and thus the <em>regularity</em> in a significant way. Further research is needed to refine <em>regularity</em> so that these problems get addressed, like taking all singular values into account when capturing the notion of <em>regularity</em>.</p>
-<p>Richter et al. also compared the behaviour of direct and indirect manipulation in , whereas we merely used an indirect –approach. As direct manipulations tend to perform better than indirect manipulations, the usage of  could also work better with the criteria we examined. This can also solve the problem of bad singular values for the <em>regularity</em> 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.</p>
+</div>
+</div>
 </section>
-<section class="footnotes">
-<hr />
-<ol>
-<li id="fn1"><p>one more for each recursive step.<a href="#/fnref1" class="footnote-back">↩</a></p></li>
-<li id="fn2"><p><em>Warning:</em> in the case of <span class="math inline">\(d=1\)</span> the recursion–formula yields a <span class="math inline">\(0\)</span> denominator, but <span class="math inline">\(N\)</span> is also <span class="math inline">\(0\)</span>. The right solution for this case is a derivative of <span class="math inline">\(0\)</span><a href="#/fnref2" class="footnote-back">↩</a></p></li>
-<li id="fn3"><p>Some examples of this are explained in detail in <a href="#/fnref3" class="footnote-back">↩</a></p></li>
-<li id="fn4"><p>We use <span class="math inline">\(\vec{S}\)</span> in this notation, as we will use this parametrization of a source–mesh to manipulate <span class="math inline">\(\vec{S}\)</span> into a target–mesh <span class="math inline">\(\vec{T}\)</span> via <span class="math inline">\(\vec{P}\)</span><a href="#/fnref4" class="footnote-back">↩</a></p></li>
-<li id="fn5"><p>Normally these are <span class="math inline">\(d-1\)</span> to each side, but at the boundaries border points get used multiple times to meet the number of points required<a href="#/fnref5" class="footnote-back">↩</a></p></li>
-<li id="fn6"><p>One example would be, when parts of an algorithm depend on the inverse of the minimal right singular value leading to a division by <span class="math inline">\(0\)</span>.<a href="#/fnref6" class="footnote-back">↩</a></p></li>
-<li id="fn7"><p>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.<a href="#/fnref7" class="footnote-back">↩</a></p></li>
-<li id="fn8"><p>The parametrization is encoded in <span class="math inline">\(\vec{U}\)</span> and the initial position of the control–points. See <a href="#/fnref8" class="footnote-back">↩</a></p></li>
-<li id="fn9"><p>Note: On the Edges this displacement is only applied outwards by flipping the sign of <span class="math inline">\(r\)</span>, if appropriate.<a href="#/fnref9" class="footnote-back">↩</a></p></li>
-<li id="fn10"><p>Again, we flip the signs for the edges, if necessary to have the object still in the convex hull.<a href="#/fnref10" class="footnote-back">↩</a></p></li>
-<li id="fn11"><p>Displayed as <span class="math inline">\(Y \times Y \times Y\)</span><a href="#/fnref11" class="footnote-back">↩</a></p></li>
-<li id="fn12"><p>We only took statistically <em>significant</em> results into consideration when compiling these numbers. Details are given in the respective chapters.<a href="#/fnref12" class="footnote-back">↩</a></p></li>
-</ol>
+<section id="regularity-1d" class="slide level1">
+<h1>Regularity 1D</h1>
+<ul>
+<li>Should measure convergence speed</li>
+</ul>
+<p><figure class="" style="width:70%;"><img  src="../arbeit/img/evolution1d/55_to_1010_steps.png" style="width:70%;"></img><figcaption>Left: <em>Improvement potential</em> against number of iterations until convergence<br />Right: <em>Regularity</em> against number of iterations until convergence<br />Coloured by their grid–resolution, both with a linear fit over the whole dataset.</figcaption></figure></p>
+<ul>
+<li>Not in our scenarios - maybe due to the fact that a better solution simply takes longer to converge, thus dominating.</li>
+</ul>
+</section>
+<section id="regularity-3d" class="slide level1">
+<h1>Regularity 3D</h1>
+<ul>
+<li>Should measure convergence speed</li>
+</ul>
+<p><figure class="" style="width:70%;"><img  src="../arbeit/img/evolution3d/regularity_montage.png" style="width:70%;"></img><figcaption>Plots of <em>regularity</em> against number of iterations for various scenarios together with a linear fit to indicate trends.</figcaption></figure></p>
+<ul>
+<li>Only <em>very weak</em> correlation</li>
+<li>Point that contributes the worst dominates regularity by lowering the least right singular value towards 0.</li>
+</ul>
+</section>
+<section id="improvement-potential-in-1d" class="slide level1">
+<h1>Improvement Potential in 1D</h1>
+<ul>
+<li>Should measure expected quality given a gradient</li>
+</ul>
+<p><figure class="" style="width:70%;"><img  src="../arbeit/img/evolution1d/55_to_1010_improvement-vs-evo-error.png" style="width:70%;"></img><figcaption><em>Improvement potential</em> plotted against the error yielded by the evolutionary optimization for different grid–resolutions</figcaption></figure></p>
+<ul>
+<li><em>very strong</em> correlation of <span class="math inline">\(- r_S = 1.0, p = 0\)</span>.</li>
+<li>Even with a distorted gradient</li>
+</ul>
+</section>
+<section id="improvement-potential-in-3d" class="slide level1">
+<h1>Improvement Potential in 3D</h1>
+<ul>
+<li>Should measure expected quality given a gradient</li>
+</ul>
+<p><figure class="" style="width:70%;"><img  src="../arbeit/img/evolution3d/improvement_montage.png" style="width:70%;"></img><figcaption>Plots of <em>improvement potential</em> against error given by our <em>fitness–function</em> after convergence together with a linear fit of each of the plotted data to indicate trends.</figcaption></figure></p>
+<ul>
+<li><em>weak</em> to <em>moderate</em> correlation within each group.</li>
+</ul>
+</section>
+<section id="summary" class="slide level1">
+<h1>Summary</h1>
+<ul>
+<li><em>Variability</em> and <em>Improvement Potential</em> are good measurements in our cases</li>
+<li><em>Regularity</em> does not work well because of small singular right values
+<ul>
+<li>But optimizing for regularity <em>could</em> still lead to a better grid-setup (not shown, but likely)</li>
+<li>Effect can be dominated by other factors (i.e. better solutions just take longer)</li>
+</ul></li>
+</ul>
+</section>
+<section id="outlook-further-research" class="slide level1">
+<h1>Outlook / Further research</h1>
+<ul>
+<li>Only focused on FFD, but will DM-FFD perform better?
+<ul>
+<li>for RBF the indirect manipulation also performed worse than the direct one</li>
+</ul></li>
+<li>Do grids with high regularity indeed perform better?</li>
+</ul>
+</section>
+<section id="thank-you" class="slide level1">
+<h1>Thank you</h1>
+<p>Any questions?</p>
 </section>
 
         
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diff --git a/presentation/presentation.md b/presentation/presentation.md
index 1be7093..4247c84 100644
--- a/presentation/presentation.md
+++ b/presentation/presentation.md
@@ -7,562 +7,158 @@ affiliation: Graphics & Geometry Group
 
 # 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
-*genome* and  *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 (the *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 *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 *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 *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 *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 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 *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}.
-
-
-# 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 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 $i$^[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 $\left(\frac{\partial}{\partial u}\right)^d s(u) = 0$.
-
-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
-**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}.
-
-### 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 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
-*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*. *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 *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:
-
-- **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.
-- **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. 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.
-
-## 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 *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^[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.
-
-## 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^[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.
-
-### Variability
-
-In \cite{anrichterEvol} *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 *variability*.
-
-### Regularity
-
-*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}.
-
-### 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 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 *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.
-
-# 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}
-\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 ($\#v~\gg~\#c$).
-
-## 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:
+- Many modern industrial design processes require advanced optimization methods
+  due to increased complexity
+- Examples are
+    - physical domains
+        - aerodynamics (i.e. drag)
+        - fluid dynamics (i.e. throughput of liquid)
+    - NP-hard problems 
+        - layouting of circuit boards
+        - stacking of 3D--objects
+
+-----
+
+# Motivation
+
+- Evolutionary algorithms cope especially well with these problem domains
+    ![Example of the use of evolutionary algorithms in automotive design](../arbeit/img/Evo_overview.png)
+- But formulation can be tricky
+
+-----
+
+# Motivation
+
+- Problems tend to be very complex
+    - i.e. a surface with $n$ vertices has $3\cdot n$ Degrees of Freedom (DoF).
+- Need for a small-dimensional representation that manipulates the
+  high-dimensional problem-space.
+- We concentrate on smooth deformations ($C^3$-continuous)
+- But what representation is good?
+
+-----
+
+# What representation is good?
+
+- In biological evolution this measure is called *evolvability*.
+    - no consensus on definition
+    - meaning varies from context to context
+    - measurable?
+- Measure depends on representation as well.
+
+-----
+
+# RBF and FFD
+
+- Andreas Richter uses Radial Basis Functions (RBF) to smoothly deform meshes
+
+![Example of RBF--based deformation and FFD targeting the same mesh.](../arbeit/img/deformations.png)
+
+-----
+
+# RBF and FFD
+
+- My master thesis transferred his idea to Freeform-Deformation (FFD)
+    - same setup
+    - same measurements
+    - same results?
+
+![Example of RBF--based deformation and FFD targeting the same mesh.](../arbeit/img/deformations.png)
+
+-----
+
+# Outline
+
+- **What is FFD?**
+- What is evolutionary optimization?
+- How to measure evolvability?
+- Scenarios
+- Results
+
+-----
+
+# What is FFD?
+
+- Create a function $s : [0,1[^d \mapsto \mathbb{R}^d$ that is parametrized
+    by some special control--points $p_i$ with coefficient functions $a_i(u)$:
+    $$
+    s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
+    $$
+- All points inside the convex hull of $\vec{p_i}$ accessed by the right $u \in [0,1[^d$.
+
+![Example of a parametrization of a line with corresponding deformation to generate a deformed objet](../arbeit/img/B-Splines.png)
+
+-----
+
+# Definition B-Splines
+
+- The coefficient functions $a_i(u)$ in $s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}$ are different for each control-point
+- Given a degree $d$ and position $\tau_i$ for the $i$th control-point $p_i$ we
+  define
+  \begin{equation}
+  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}
+- The derivatives of these coefficients are also easy to compute:
+  $$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
+
+# Properties of B-Splines
+
+- Coefficients vanish after $d$ differentiations
+- Coefficients are continuous with respect to $u$
+- A change in prototypes only deforms the mapping locally  
+  (between $p_i$ to $p_{i+d+1}$)
+
+![Example of Basis-Functions for degree $2$. [Brunet, 2010]<br /> Note, that Brunet starts his index at $-d$ opposed to our definition, where we start at $0$.](../arbeit/img/unity.png)
+
+# Definition FFD
+
+- FFD is a space-deformation resulting based on the underlying B-Splines
+- Coefficients of space-mapping $s(u) = \sum_j a_j(u) p_j$ for an initial vertex
+  $v_i$ are constant
+- Set $u_{i,j}~:=~N_{j,d,\tau}$ for each $v_i$ and $p_j$ to get the projection:
+    $$
+    v_i = \sum_j u_{i,j} \cdot p_j = \vec{u}_i^{T} \vec{p}
+    $$
+    or written with matrices:
+    $$
+    \vec{v} = \vec{U} \vec{p}
+    $$
+- $\vec{U}$ is called **deformation matrix**
+
+# Implementation of FFD
+
+- As we deal with 3D-Models we have to extend the introduced 1D-version
+- We get one parameter for each dimension: $u,v,w$ instead of $u$
+- Task: Find correct $u,v,w$ for each vertex in our model
+    - We used a gradient-descent (via the gauss-newton algorithm)
+
+# Implementation of FFD
+
+- Given $n,m,o$ control-points in $x,y,z$--direction each Point inside the
+  convex hull is defined by
+    $$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
+- 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:
+  $$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$
+
+# Implementation of FFD
+
+- Derivation is straightforward
+    $$
+    \scriptsize
+    \begin{array}{rl}
+        \displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
+      = & \displaystyle - \sum_i \sum_j \sum_k N'_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
+    \end{array}
+    $$
+  yielding a Jacobian:
 
 $$
+\scriptsize
 J(Err(u,v,w)) = 
 \left(
 \begin{array}{ccc}
@@ -572,831 +168,240 @@ J(Err(u,v,w)) =
 \end{array}
 \right)
 $$
-$$
-\scriptsize
-=
-\left(
-\begin{array}{ccc}
-- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_x & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_x \\
-- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_y & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_y \\
-- \displaystyle \sum_{i,j,k} N'_{i}(u) N_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z &- \displaystyle \sum_{i,j,k} N_{i}(u) N'_{j}(v) N_{k}(w) \cdot {c_{ijk}}_z & - \displaystyle \sum_{i,j,k} N_{i}(u) N_{j}(v) N'_{k}(w) \cdot {c_{ijk}}_z
-\end{array}
-\right)
-$$
 
-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.
+# Implementation of FFD
 
-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.
+- Armed with this we iterate the formula
+  $$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
+  using Cramer's rule for inverting the small Jacobian.
+- Usually terminates after $3$ to $5$ iteration with an $\epsilon := \vec{p^*} -
+  V(u,v,w) < 10^{-4}$
+- self-intersecting grids can invalidate the results
+  - no problem, as these get not generated and contradict some properties we
+    want (like locality)
 
-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 *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.
+# Outline
 
-## Deformation Grid
-\label{sec:impl:grid}
+- What is FFD?
+- **What is evolutionary optimization?**
+- How to measure evolvability?
+- Scenarios
+- Results
 
-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).
+# What is evolutionary optimization?
 
-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.
+## 1 1
 
-\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^[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^[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^[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.
-
-# Scenarios for testing evolvability--criteria using \ac{FFD}
-\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.
-
-## 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 *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^[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.
-
-## 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 *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 *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}.
-
-# 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 *very weak*, *weak*, *moderate*, *strong* and *very strong*. We
-interpret p--values smaller than $0.01$ as *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}.
-
-## 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 *variability* and *regularity* to get our predictions.
-Afterwards we solve the problem analytically to get the (normalized) correct
-gradient that we use as guess for the *improvement potential*. To further test
-the *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^[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}.
-
-## 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.
-
-### Variability
-
-\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}
-
-*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 *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 *very strong* correlation ($-r_S = 0.94, p = 0$)
-between the *variability* and the evolutionary error.
-
-### Regularity
-
-\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]{Negated 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}.
+~~~
+$t := 0$;
+initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$;
+evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$;
+while($c(F(t)) \neq$ true) {
+    recombine: $P’(t) := r(P(t))$;
+    mutate: $P''(t) := m(P’(t))$;
+    evaluate $F(t) : \{\Phi(x) | x \in P''(t)\}$
+    select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$;
+    $t := t + 1$;
 }
-\label{tab:1dreg}
-\end{table}
+~~~
 
-*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.
+~~~
+$t$: Iteration-step
+$I$: Set of possible Individuals
+$P$: Population of Individuals
+$F$: Fitness of Individuals
+$Q$: Either set of parents or $\emptyset$
 
-As can be seen from table \ref{tab:1dreg}, we could only show a *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 *strong*
-correlation of $- r_S = -0.72, p = 0$, that is opposed to our expectations.
+$r(..) : I^\mu \mapsto I^\lambda$
+$m(..) : I^\lambda \mapsto I^\lambda$
+$s(..) : I^{\lambda + \mu} \mapsto I^\mu$
+~~~
 
-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
-*improvement potential* against the steps next to the *regularity*--plot. Our theory
-is that the *very strong* correlation ($-r_S = -0.82, p=0$) between
-*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.
+- Algorithm to model simple inheritance
+- Consists of three main steps
+    - recombination
+    - mutation
+    - selection
+- An "individual" in our case is the displacement of control-points
 
-### Improvement Potential
+# Evolutional loop
 
-\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}
+- **Recombination** generates $\lambda$ new individuals based on the characteristics of the $\mu$ parents.
+    - This makes sure that the next guess is close to the old guess.
+- **Mutation** introduces new effects that cannot be produced by mere recombination of the
+  parents.
+    - Typically these are minor defects to individual members of the population
+      i.e. through added noise
+- **Selection** selects $\mu$ individuals from the children (and optionally the
+    parents) using a *fitness--function* $\Phi$.
+    - Fitness could mean low error, good improvement, etc.
+    - Fitness not solely determines who survives, there are many possibilities
 
-The *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.
+# Outline
 
-All results show the identical *very strong* and *significant* correlation with
-a Spearman--coefficient of $- r_S = 1.0$ and p--value of $0$.
+- What is FFD?
+- What is evolutionary optimization?
+- **How to measure evolvability?**
+- Scenarios
+- Results
 
-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.
+# How to measure evolvability?
 
-## Procedure: 3D Function Approximation
-\label{sec:proc:3dfa}
+- Different (conflicting) optimization targets
+    - convergence speed?
+    - convergence quality?
+- As $\vec{v} = \vec{U}\vec{p}$ is linear, we can also look at $\Delta \vec{v} = \vec{U}\,
+    \Delta \vec{p}$
+    - We only change $\Delta \vec{p}$, so evolvability should only use $\vec{U}$
+      for predictions
 
-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.
+# Evolvability criteria
 
-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
-*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 $\vec{c_S(\dots)}$.
+- **Variability**
+    - roughly: "How many actual Degrees of Freedom exist?"
+    - Defined by
+        $$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n} \in [0..1]$$
+    - in FFD this is $1/\#\textrm{CP}$ for the number of control-points used for
+      parametrization
 
-\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}
+# Evolvability criteria
 
-For the next step we then halve the regularization--impact $\lambda$ (starting
-at $1$) of our *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.
+- **Regularity**
+    - roughly: "How numerically stable is the optimization?"
+    - Defined by
+        $$\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}} \in [0..1]$$
+        with $\sigma_{min/max}$ being the least/greatest right singular value.
+    - high, when $\|\vec{Up}\| \propto \|\vec{p}\|$
 
-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.
+# Evolvability criteria
 
-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^[Again, we flip the signs for the edges, if necessary to have
-the object still in the convex hull.].
+- **Improvement Potential**
+    - roughly: "How good can the best fit become?"
+    - Defined by
+        $$\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F$$
+        with a unit-normed guessed gradient $\vec{G}$
 
-\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}
+# Outline
 
-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.
+- What is FFD?
+- What is evolutionary optimization?
+- How to measure evolvability?
+- **Scenarios**
+- Results
 
-## Results of 3D Function Approximation
+# Scenarios
 
-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}.
+- 2 Testing Scenarios
+- 1-dimensional fit
+    - $xy$-plane to $xyz$-model, where only the $z$-coordinate changes
+    - can be solved analytically with known global optimum
+- 3-dimensional fit
+    - fit a parametrized sphere into a face
+    - cannot be solved analytically
+    - number of vertices differ between models
 
-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.
+# 1D-Scenario
 
-\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}
+![Left: A regular $7 \times 4$--grid<br />Right: The same grid after a
+random distortion to generate a testcase.](../arbeit/img/example1d_grid.png)
 
-### Variability
-\label{sec:res:3d:var}
+![The target--shape for our 1--dimensional optimization--scenario including a wireframe--overlay of the vertices.](../arbeit/img/1dtarget.png){width=70%}
 
-\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}
+# 3D-Scenarios
 
-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}.
+![\newline Left: The sphere we start from with 10 807 vertices<br />Right: The face we want to deform the sphere into with 12 024 vertices.](../arbeit/img/3dtarget.png)
 
-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 $\mathrm{X} = 7$.
+# Outline
 
-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 *variability* via the rank of the
-deformation--matrix.
+- What is FFD?
+- What is evolutionary optimization?
+- How to measure evolvability?
+- Scenarios
+- **Results**
 
-Overall the correlation between *variability* and fitness--error were
-*significant* and showed a *very strong* correlation in all our tests.
-The detailed correlation--coefficients are given in table \ref{tab:3dvar}
-alongside their p--values.
+# Variability 1D
 
-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.
+- Should measure Degrees of Freedom and thus quality
 
-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 *variability* for 100 randomly moved $10 \times 10 \times 10$
-grids on the sphere we start out with.
+![The squared error for the various grids we examined.<br /> Note that $7 \times 4$ and $4 \times 7$ have the same number of control--points.](../arbeit/img/evolution1d/variability_boxplot.png)
 
-\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}
+- $5 \times 5$, $7 \times 7$ and $10 \times 10$ have *very strong* correlation ($-r_S = 0.94, p = 0$) between the *variability* and the evolutionary error.
 
-As the *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.
+# Variability 3D
 
-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.
+- Should measure Degrees of Freedom and thus quality
 
-Nevertheless this hints at the notion, that *variability* is a good measure for
-the overall quality of a fit.
+![The fitting error for the various grids we examined.<br />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.](../arbeit/img/evolution3d/variability_boxplot.png)
 
-### Regularity
+- $4 \times 4 \times 4$, $5 \times 5 \times 5$ and $6 \times 6 \times 6$ have *very strong* correlation ($-r_S = 0.91, p = 0$) between the *variability* and the evolutionary error.
 
-\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}
+# Varying Variability
 
-Opposed to the predictions of *variability* our test on *regularity* gave a mixed
-result --- similar to the 1D--case.
+## 1 1
 
-In roughly half of the scenarios we have a *significant*, but *weak* to *moderate*
-correlation between *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 *significant*, but *weak* **anti**--correlation when taking all three tests into
-account^[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}).
+![A high resolution ($10 \times 10$) of control--points over a circle. Yellow/green points contribute to the parametrization, red points don't.<br />An Example--point (blue) is solely determined by the position of the green control--points.](../arbeit/img/enoughCP.png)
 
-Taking all results together we only find a *very weak*, but *significant* link
-between *regularity* and the number of iterations needed for the algorithm to
-converge.
+![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.](../arbeit/img/evolution3d/variability2_boxplot.png)
 
-\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}
+# Regularity 1D
 
-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 *regularity* seems to increase at
-first, but then decreases again on higher grid--resolutions.
+- Should measure convergence speed
 
-This can be an artefact of the definition of *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.
+![Left: *Improvement potential* against number of iterations until convergence<br />Right: *Regularity* against number of iterations until convergence<br />Coloured by their grid--resolution, both with a linear fit over the whole
+dataset.](../arbeit/img/evolution1d/55_to_1010_steps.png){width=70%}
 
-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.
+- Not in our scenarios - maybe due to the fact that a better solution simply
+  takes longer to converge, thus dominating.
 
-Adding to this we also noted, that in the case of the $10 \times 10 \times
-10$--grid the *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 *regularity* (i.e. using the
-smallest non--zero right singular value) could solve this particular issue, but
-not fix the trend we noticed above.
+# Regularity 3D
 
-### Improvement Potential
+- Should measure convergence speed
 
-\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}
+![Plots of *regularity* against number of iterations for various scenarios together
+with a linear fit to indicate trends.](../arbeit/img/evolution3d/regularity_montage.png){width=70%}
 
-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.
+- Only *very weak* correlation
+- Point that contributes the worst dominates regularity by lowering the least
+  right singular value towards 0.
 
-\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}
+# Improvement Potential in 1D
 
-We plotted our findings on the *improvement potential* in a similar way as 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.
+- Should measure expected quality given a gradient
 
-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 $6 \times 6 \times 6$)
-are significant.
+![*Improvement potential* plotted against the error yielded by the evolutionary optimization for different grid--resolutions](../arbeit/img/evolution1d/55_to_1010_improvement-vs-evo-error.png){width=70%}
 
-If we take multiple datasets into account the correlation is *very strong* and 
-*significant*, which is good, as this functions as a litmus--test, because the
-quality is naturally tied to the number of control--points.
+- *very strong* correlation of $- r_S = 1.0, p = 0$.
+- Even with a distorted gradient
 
-All in all the *improvement potential* seems to be a good and sensible measure of
-quality, even given gradients of varying quality.
+# Improvement Potential in 3D
 
-Lastly, a small note on the behaviour of *improvement potential* and convergence
-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.
+- Should measure expected quality given a gradient
 
-\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}
+![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.](../arbeit/img/evolution3d/improvement_montage.png){width=70%}
 
-# Discussion and outlook
-\label{sec:dis}
+- *weak* to *moderate* correlation within each group.
 
-In this thesis we took a look at the different criteria for *evolvability* as
-introduced by Richter et al.\cite{anrichterEvol}, namely *variability*,
-*regularity* and *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
-*significant* *very strong* correlations between *variability and fitting error*
-($0.94$) and *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 *weak* correlations for *regularity and convergence--speed* ($0.28$)
-opposed to Richter et al. with $0.39$ to $0.91$.^[We only took statistically
-*significant* results into consideration when compiling these numbers. Details
-are given in the respective chapters.]
+# Summary
 
-For the 3D--scenario our results show a *very strong*, *significant*  correlation
-between *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 *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 *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 $0.87$.
+- *Variability* and *Improvement Potential* are good measurements in our cases
+- *Regularity* does not work well because of small singular right values
+    - But optimizing for regularity *could* still lead to a better grid-setup
+      (not shown, but likely)
+    - Effect can be dominated by other factors (i.e. better solutions just take
+      longer)
 
-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, while we could not reproduce similar
-compelling results as Richter et al. for *regularity and convergence speed*.
+# Outlook / Further research
 
-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,
-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, like taking all singular values into account when
-capturing the notion of *regularity*.
+- Only focused on FFD, but will DM-FFD perform better?
+    - for RBF the indirect manipulation also performed worse than the direct one
+- Do grids with high regularity indeed perform better?
 
-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
-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.
+# Thank you
+
+Any questions?
diff --git a/presentation/template/template.html b/presentation/template/template.html
index a0999d7..2b7f936 100755
--- a/presentation/template/template.html
+++ b/presentation/template/template.html
@@ -65,6 +65,9 @@
             of: ['\\mkern{-2mu}\\left( #1 \\right\)', 1]
           }
       },
+      tex2jax: {
+          skipTags: ["script","noscript","style","textarea"],
+      },
       "HTML-CSS": {
           styles: { ".reveal section .MathJax_Display": { margin: "0.5em 0em" } },
           styles: { ".reveal table .MathJax_Display": { margin: "0em" } },
@@ -137,14 +140,17 @@ $body$
       center:       false,
       transition:   'none',
       viewDistance: 2, // otherwise videos start early
-      width:        1024,
-      height:       768,
+      width:        1280,
+      height:       1024,
       minScale:     0.2,
       maxScale:     5, // if this threshold is reached, the chalkboard drawing will be wrongly positioned. hence large threshold!
 
 
       // use local mathjax installation
-      math: { mathjax: '$template$/mathjax/MathJax.js', config: 'TeX-AMS_HTML-full' },
+      math: { mathjax: '$template$/mathjax/MathJax.js',
+              config: 'TeX-AMS_HTML-full',
+              extensions: ["content-mathml.js"]
+      },
 
 
       // setup chalkboard
@@ -191,10 +197,10 @@ $body$
 	  dependencies: [
 	  { src: '$template$/revealjs/plugin/math/math.js' },
 	  { src: '$template$/revealjs/plugin/notes/notes.js', async: true },
-	  { src: '$template$/revealjs/plugin/highlight/highlight.js', async: true, callback: function() {
+	  /*{ src: '$template$/revealjs/plugin/highlight/highlight.js', async: true, callback: function() {
         var code_blocks = document.querySelectorAll('code');
         for( var i = 0, len = code_blocks.length; i < len; i++ )  hljs.highlightBlock(code_blocks[i]);                                                      
-      }},
+      }},*/
       { src: '$template$/revealjs/plugin/menu/menu.js' },
       { src: '$template$/my-chalkboard/chalkboard.js' }, // do not load this async ('ready' event is missing, print wont work)
       { src: '$template$/my-zoom/zoom.js', async: true },