From a411c1012b48f9c7e4c85634ec3a86a2c156c1b4 Mon Sep 17 00:00:00 2001 From: Stefan Dresselhaus Date: Sat, 14 Oct 2017 13:45:22 +0200 Subject: [PATCH] ersten 3 Kapitel fertig. --- arbeit/ma.md | 52 ++++++++++++++----- arbeit/ma.pdf | Bin 404432 -> 405478 bytes arbeit/ma.tex | 98 +++++++++++++++++++++-------------- arbeit/settings/packages.tex | 2 +- 4 files changed, 99 insertions(+), 53 deletions(-) diff --git a/arbeit/ma.md b/arbeit/ma.md index 4b03d6a..1cdb821 100644 --- a/arbeit/ma.md +++ b/arbeit/ma.md @@ -180,14 +180,18 @@ continuous (given $d \ge 1$) as every $p_i$ gets blended in between $\tau_i$ and $\tau_{i+d}$ and out between $\tau_{i+1}$, and $\tau_{i+d+1}$ as can bee seen from the two coefficients in every step of the recursion. -\improvement[inline]{Weitere Eigenschaften erwähnen: -\newline Convex hull -\newline $\sum_i N_i = 1$? -\newline Bilder von Basisfunktionen zur Visualisierung. +This means that all changes are only a local linear combination between the +control--point $p_i$ to $p_{i+d+1}$ and consequently this yields to the +convex--hull--property of B-Splines --- meaning, that no matter how we choose +our coefficients, the resulting points all have to lie inside convex--hull of +the control--points. + +\improvement[inline]{ +Bilder von Basisfunktionen zur Visualisierung. } For a given number of points $v_1,\dots,v_n$ we can then calculate -the contributions $n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the +the contributions \linebreak[4]$n_{i,j}~:=~N_{j,d,\tau}$ of each control point $p_j$ to get the projection from the control--point--space into the object--space: $$ v_i = \sum_j n_{i,j} \cdot p_j = \vec{n}_i^{T} \vec{p} @@ -264,16 +268,21 @@ however, is very generic and we introduce it here in a broader sense. The general shape of an evolutionary algorithm (adapted from \cite{back1993overview}) is outlined in Algorithm \ref{alg:evo}. Here, $P(t)$ denotes the population of parameters in step $t$ of the algorithm. The -population contains $\mu$ individuals $a_i$ that fit the shape of the parameters -we are looking for. Typically these are initialized by a random guess or just -zero. Further on we need a so--called *fitness--function* $\Phi : I \mapsto M$\improvement{Was ist $I,M$?\newline Bezug Genotyp/Phenotyp} that can take -each parameter to a measurable space along a convergence--function $c : I \mapsto -\mathbb{B}$ that terminates the optimization. +population contains $\mu$ individuals $a_i$ from the possible individual--set +$I$ that fit the shape of the parameters we are looking for. Typically these are +initialized by a random guess or just zero. Further on we need a so--called +*fitness--function* $\Phi : I \mapsto M$ that can take each parameter to a measurable +space $M$ (usually $M = \mathbb{R}$) along a convergence--function $c : I \mapsto \mathbb{B}$ +that terminates the optimization. + +Biologically speaking the set $I$ corresponds to the set of possible *Genotypes* +while $M$ represents the possible observable *Phenotypes*. The main algorithm just repeats the following steps: - **Recombine** with a recombination--function $r : I^{\mu} \mapsto I^{\lambda}$ to - generate new individuals based on the parents characteristics. + generate $\lambda$ new individuals based on the characteristics of the $\mu$ + parents. This makes sure that the next guess is close to the old guess. - **Mutate** with a mutation--function $m : I^{\lambda} \mapsto I^{\lambda}$ to introduce new effects that cannot be produced by mere recombination of the @@ -332,9 +341,23 @@ least get suboptimal solutions fast, which then refine over time. ## Criteria for the evolvability of linear deformations \label{sec:intro:rvi} -\improvement[inline]{Nomenklatur. Was ist $\vec{U}$? Kurz Matrix--Darstellung -des Problems & Rückgriff auf FFD-Kapitel.} +As we have established in chapter \ref{sec:back:ffd}, we can describe a +deformation by the formula +$$ +V = UP +$$ +where $V$ is a $n \times d$ matrix of vertices, $U$ are the (during +parametrization) calculated deformation--coefficients and $P$ is a $m \times d$ matrix +of control--points that we interact with during deformation. +We can also think of the deformation in terms of differences from the original +coordinates +$$ +\Delta V = U \cdot \Delta P +$$ +which is isomorphic to the former due to the linear correlation in the +deformation. One can see in this way, that the way the deformation behaves lies +solely in the entries of $U$, which is why the three criteria focus on this. ### Variability @@ -550,7 +573,8 @@ entfernen kann?} For our tests we chose different uniformly sized grids and added gaussian noise onto each control-point^[For the special case of the outer layer we only applied -noise away from the object] to simulate different starting-conditions. +noise away from the object, so the object is still confined in the convex hull +of the control--points.] to simulate different starting-conditions. \unsure[inline]{verweis auf DM--FFD?} diff --git a/arbeit/ma.pdf b/arbeit/ma.pdf index 9b135c3be505301fa9ff224d89e0c6d991bb4217..0af94befe78f77bdaf132b02a56121de9fb6ec02 100644 GIT binary patch delta 109986 zcmZs?Q;;q|vn|-RwcECB+ctLFw)<<_wr%aUZQHi(p8wo46Lap|d8>$e$%@Qcxnix% zxmC=QT}(tWMNx4&CVDm)vZK4}H<+{n6mYTxR$M^TB@@J9cZpm?v+_UNiClH>{5>?% zKo!E2#$LZsw;pOSNss80=`7DjLINBvGHsvp#xoV 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