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authorAndreas MĂĽller <andreas.mueller@ost.ch>2021-06-23 10:02:46 +0200
committerGitHub <noreply@github.com>2021-06-23 10:02:46 +0200
commitc0526b5fd264bedb0e89fd703cbdee16f3294af3 (patch)
tree88611f0d5f5f101cca0739df03436d92fd4e9b39 /buch/papers/ifs
parentfix paper/ifs/references.bib (diff)
parentminor changes (diff)
downloadSeminarMatrizen-c0526b5fd264bedb0e89fd703cbdee16f3294af3.tar.gz
SeminarMatrizen-c0526b5fd264bedb0e89fd703cbdee16f3294af3.zip
Merge pull request #25 from LordMcFungus/master
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-rw-r--r--buch/papers/ifs/images/farncolor2.eps1137
-rw-r--r--buch/papers/ifs/images/farnnotweight.eps2179
-rw-r--r--buch/papers/ifs/images/farnrightwight.eps1027
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-rw-r--r--buch/papers/ifs/images/zurich.pngbin71780 -> 0 bytes
-rw-r--r--buch/papers/ifs/references.bib33
-rw-r--r--buch/papers/ifs/teil0.tex4
-rw-r--r--buch/papers/ifs/teil1.tex92
-rw-r--r--buch/papers/ifs/teil2.tex122
-rw-r--r--buch/papers/ifs/teil3.tex132
17 files changed, 6599 insertions, 130 deletions
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+d setpagedevice
+} if
+} bd
+/RE { % /NewFontName [NewEncodingArray] /FontName RE -
+ findfont dup length dict begin
+ {
+ 1 index /FID ne
+ {def} {pop pop} ifelse
+ } forall
+ /Encoding exch def
+ /FontName 1 index def
+ currentdict definefont pop
+ end
+} bind def
+%%EndResource
+%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0
+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
+/BeginEPSF { %def
+/b4_Inc_state save def % Save state for cleanup
+/dict_count countdictstack def % Count objects on dict stack
+/op_count count 1 sub def % Count objects on operand stack
+userdict begin % Push userdict on dict stack
+/showpage { } def % Redefine showpage, { } = null proc
+0 setgray 0 setlinecap % Prepare graphics state
+1 setlinewidth 0 setlinejoin
+10 setmiterlimit [ ] 0 setdash newpath
+/languagelevel where % If level not equal to 1 then
+{pop languagelevel % set strokeadjust and
+1 ne % overprint to their defaults.
+{false setstrokeadjust false setoverprint
+} if
+} if
+} bd
+/EndEPSF { %def
+count op_count sub {pop} repeat % Clean up stacks
+countdictstack dict_count sub {end} repeat
+b4_Inc_state restore
+} bd
+%%EndResource
+%%EndProlog
+%%Page: 1 1
+%%PageBoundingBox: 0 0 420 315
+%%BeginPageSetup
+[1 0 0 -1 0 315] CT
+%%EndPageSetup
+GS
+[0.75 0 0 0.75 0 0] CT
+1 GC
+N
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+f
+GR
+GS
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+[1 0 0 1 0 0] CT
+N
+0 -236 M
+875 -236 L
+875 420 L
+0 420 L
+0 -236 L
+cp
+clip
+GS
+0 0 translate
+560 420 scale
+%AXGBeginBitmap: java.awt.image.BufferedImage
+{{
+/RawData currentfile /ASCII85Decode filter def
+/Data RawData /FlateDecode filter def
+/DeviceRGB setcolorspace
+<<
+ /ImageType 1
+ /Decode [0 1 0 1 0 1]
+ /DataSource Data
+ /Height 420
+ /ImageMatrix [560 0 0 420 0 0]
+ /Width 560
+ /BitsPerComponent 8
+>> image
+} stopped {handleerror} if
+ RawData flushfile
+} exec
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+} stopped {handleerror} if
+ RawData flushfile
+} exec
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+} stopped {handleerror} if
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+} exec
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+} bind def
+%%EndResource
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+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
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+} if
+} bd
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+} bd
+%%EndResource
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+%%BeginPageSetup
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+%%EndPageSetup
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+0 1323 L
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+N
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+4000 -651 L
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+ RawData flushfile
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diff --git a/buch/papers/ifs/images/faroe.png b/buch/papers/ifs/images/faroe.png
deleted file mode 100644
index 183702a..0000000
--- a/buch/papers/ifs/images/faroe.png
+++ /dev/null
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deleted file mode 100644
index 7396ed3..0000000
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+++ /dev/null
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deleted file mode 100644
index b2aff49..0000000
--- a/buch/papers/ifs/images/faroe1.PNG
+++ /dev/null
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deleted file mode 100644
index 50f9564..0000000
--- a/buch/papers/ifs/images/faroe5.PNG
+++ /dev/null
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diff --git a/buch/papers/ifs/images/rapperswil0.PNG b/buch/papers/ifs/images/rapperswil0.PNG
deleted file mode 100644
index 3eba43a..0000000
--- a/buch/papers/ifs/images/rapperswil0.PNG
+++ /dev/null
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diff --git a/buch/papers/ifs/images/rapperswil1.PNG b/buch/papers/ifs/images/rapperswil1.PNG
deleted file mode 100644
index 6c085db..0000000
--- a/buch/papers/ifs/images/rapperswil1.PNG
+++ /dev/null
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diff --git a/buch/papers/ifs/images/rapperswil4.PNG b/buch/papers/ifs/images/rapperswil4.PNG
deleted file mode 100644
index 56d1331..0000000
--- a/buch/papers/ifs/images/rapperswil4.PNG
+++ /dev/null
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diff --git a/buch/papers/ifs/images/zurich.png b/buch/papers/ifs/images/zurich.png
deleted file mode 100644
index bb70f7d..0000000
--- a/buch/papers/ifs/images/zurich.png
+++ /dev/null
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diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib
index fbf75f4..817c5a4 100644
--- a/buch/papers/ifs/references.bib
+++ b/buch/papers/ifs/references.bib
@@ -4,15 +4,6 @@
% (c) 2020 Autor, Hochschule Rapperswil
%
-@online{ifs:bibtex,
- title = {BibTeX},
- url = {https://de.wikipedia.org/wiki/BibTeX},
- date = {2020-02-06},
- year = {2020},
- month = {2},
- day = {6}
-}
-
@online{ifs:chaos,
title = {Chaosspiel},
url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel},
@@ -38,16 +29,6 @@
isbn = {0-471-92287-0},
}
-@article{ifs:mendezmueller,
- author = { Tabea Méndez and Andreas Müller },
- title = { Noncommutative harmonic analysis and image registration },
- journal = { Appl. Comput. Harmon. Anal.},
- year = 2019,
- volume = 47,
- pages = {607--627},
- url = {https://doi.org/10.1016/j.acha.2017.11.004}
-}
-
@Inbook{ifs:Rousseau2012,
author= {Rousseau, Christiane
and Saint-Aubin, Yvan
@@ -64,4 +45,18 @@
url={https://doi.org/10.1007/978-3-642-30092-9_11}
}
+@article{ifs:pifs,
+ title = {Applications of Partitioned Iterated Function Systems in Image and Video Compression},
+ journal = {Journal of Visual Communication and Image Representation},
+ volume = 7,
+ number = {2},
+ pages = {144-154},
+ year = 1996,
+ issn = {1047-3203},
+ doi = {https://doi.org/10.1006/jvci.1996.0014},
+ url = {https://www.sciencedirect.com/science/article/pii/S1047320396900140},
+ author = {Guojun Lu and Toon Lin Yew},
+ abstract = {Iterated function systems (IFS) have been used to compress image data. Because of difficulty in finding IFS in natural images, a technique based on partitioned IFS (PIFS) has been proposed for image compression. In this technique, an image to be compressed is divided into nonoverlapping blocks. For each block an affine transformation is found in the image. This set of affine transformations (called PIFS) corresponds to a unique image. In the simplest case, images are partitioned into fixed size blocks. In this paper, we investigate image and video compression techniques using variable block sizes based on the quadtree partition. One property of images generated using PIFS is scalability: they have fine detail in any scale. We exploit this property to reduce required compression time and improve compression performance. There are large amounts of temporal redundancy between fames of a video sequence. We describe a method to remove temporal redundancies effectively using a quadtree partitioning technique. We have implemented the above schemes to compress image and video sequences and will report our experimental results.}
+}
+
diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex
index 7cb218f..833748c 100644
--- a/buch/papers/ifs/teil0.tex
+++ b/buch/papers/ifs/teil0.tex
@@ -3,9 +3,9 @@
%
% (c) 2020 Prof Dr Andreas MĂĽller, Hochschule Rapperswil
%
-\section{Teil 0\label{ifs:section:teil0}}
+\section{Einleitung \label{ifs:section:teil0}}
\rhead{Was ist ein Iteriertes Funktionsschema}
-Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, komplexe Bilder beschreiben.
+Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden.
In der Regel sind diese Bilder Fraktale.
Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen.
diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex
index 54089ec..a75b529 100644
--- a/buch/papers/ifs/teil1.tex
+++ b/buch/papers/ifs/teil1.tex
@@ -6,29 +6,30 @@
\section{Fraktale
\label{ifs:section:teil1}}
\rhead{Problemstellung}
-Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an.
+Bevor wir die IFS ansehen, schauen wir uns Fraktale genauer an.
Ăśber die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig.
In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt.
-Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten:
+Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten:
\begin{enumerate}
\item $F$ hat eine unendlich feine Struktur
\item $F$ kann nicht mit der klassischen Geometrie beschrieben werden.
- \item Oftmals haf $F$ eine Form von Selbstähnlichkeit.
- \item Die 'fraktale Dimension' ist grösser als die Topologische Dimension
- \item Viele Fraktale lassen sich einfach beschrieben
+ \item Oftmals hat $F$ eine Form von Selbstähnlichkeit.
+ \item Die 'fraktale Dimension' ist grösser als die topologische Dimension
+ \item Viele Fraktale lassen sich auf eine simple Art definieren. Es genĂĽgen zum Beispiel nur wenige Funktionen, welche rekursiv ausgefĂĽhrt werden, um ein Fraktal zu definieren.
\end{enumerate}
\subsection{Koch Kurve
\label{ifs:subsection:lilkoch}}
-Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen.
-In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht sie aus lauter kleineren Kopien von sich selber.
+Diese Eigenschaften möchten wir nun am Beispiel der Koch Kurve näher anschauen.
+In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht aus lauter kleineren Kopien von sich selber.
Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt.
Gestartet wird mit einer einzelnen Strecke der Länge $a$.
-Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt.
+Diese wird in ersten Schritt durch vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt.
In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich.
Dieser Schritt wird nun fĂĽr jeden der resultierten Streckenabschnitten wiederholt.
-Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist.
+Die Kurve besteht also aus vier kleineren Kopien der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist.
+Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind.
\begin{figure}
@@ -54,41 +55,80 @@ Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch
\end{figure}
Die resultierende Kurve hat ein paar interessante Eigenschaften.
-Die Länge der Kurve lasst sich einfach berechnen.
+Die Länge der Kurve der jeweiligen Iteration lässt sich mit
\begin{align*}
- l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad ... , \quad
- l_n = a * \left( \frac{4}{3}\right)^n \quad
+ l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad \cdots , \quad
+ l_n = a \cdot \left( \frac{4}{3}\right)^n \quad
\Rightarrow \quad
\lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty
\end{align*}
-In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Somit divergiert die Länge gegen Unendlich.
+berechnen.
+In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert.
+
+
Die Fläche unter der Kurve lässt sich folgendermassen berechnen
\begin{align*}
- A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\
- A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\
- A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1
+ A_0 &= 0 \\
+ A_1 &= \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\
+ A_2 &= A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\
+ A_3 &= A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1.
\end{align*}
Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist.
-Daraus resultiert eine konvergierende Geometrische Reihe.
+Die Gesamtfläche ist daher gegeben durch die konvergierende geometrische Reihe,
\begin{align*}
A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\
- \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2
\end{align*}
-Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang.
+mit dem Grenzwert
+\begin{align*}
+ \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2.
+\end{align*}
+Wie wir sehen ist die Koch-Kurve ein Objekt mit endlicher Fläche, aber unendlichem Umfang.
+
+
Zu guter Letzt bestimmen wir die Dimension der Kurve.
-Es gibt viele verschiedene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern.
-Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten.
+Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern.
+Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten.
In diesem Beispiel werden wir die Ă„hnlichkeits-Dimension \cite{ifs:fractal-geometry}.
+Die Ähnlichkeits-Dimension $D$ ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$
+
\begin{align*}
- D = - \frac{log(N)}{log(\epsilon)}
+ D = - \frac{\log N}{\log \epsilon }.
\end{align*}
Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen.
-Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge.
+Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}.
Somit hat das Dreieck die Dimension $D = 2$.
-Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon = 1/3$.
+Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$.
+Ihre Ă„hnlichkeits-Dimension ist somit
\begin{align*}
- D = - \frac{log(N)}{log(\epsilon)} = - \frac{log(4)}{log(1/3)} \approx 1.2619
+ D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619.
\end{align*}
-Wie wir nun sehen besitzt die Kochkurve alle oben beschriebenen Eigenschaften von Fraktalen.
+Wie wir nun sehen besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen.
Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten.
+\begin{figure}
+ \centering
+ \begin{tikzpicture}
+
+ % draw the background
+ \draw [line width=1.5pt, fill=gray!2] (0,0) -- (60:4) -- (4,0) -- cycle;
+
+ \coordinate[label=left:$A$] (A) at (0,0);
+ \coordinate[label=right:$B$] (B) at (4,0);
+ \coordinate[label=above:$C$] (C) at (2,3.464);
+
+ \coordinate[label=below:$l$](c) at ($ (A)!.5!(B) $);
+ \coordinate[label=left:$l$] (b) at ($ (A)!.5!(C) $);
+ \coordinate[label=right:$l$](a) at ($ (B)!.5!(C) $);
+
+ \coordinate[label=below:$l/2$](d) at ($ (b)!.5!(a)$);
+
+ % the triangle
+ \draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle;
+ \draw [line width=0.5pt] (a) -- (b);
+ \draw [line width=0.5pt] (a) -- (c);
+ \draw [line width=0.5pt] (c) -- (b);
+
+ \end{tikzpicture}
+ \caption{Selbstähnlichkeit eines gleichseitigen Dreiecks}
+ \label{ifs:trinagle}
+\end{figure}
diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex
index 143317a..fd10634 100644
--- a/buch/papers/ifs/teil2.tex
+++ b/buch/papers/ifs/teil2.tex
@@ -14,13 +14,13 @@ Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck.
\caption{Sierpinski-Dreieck}
\label{ifs:sierpinski10}
\end{figure}
-Wenn man das Dreieck genau anschaut, erkennt man schnell, dass es aus drei kleineren Kopien seiner selbst besteht.
-Es ist also ein Selbstähnliches Konstrukt.
+Es besteht aus drei kleineren Kopien von sich selbst.
+Es ist also ein Selbstähnliches Gebilde.
Diese Eigenschaft wollen wir uns zunutze machen.
Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$.
-Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet
+Ausserdem bestimmen wir drei Funktionen
\begin{align*}
f_1(x,y)
=
@@ -63,13 +63,15 @@ Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer
\begin{pmatrix}
\frac{1}{4} \\
\frac{1}{2}
- \end{pmatrix}\\
+ \end{pmatrix},
\end{align*}
+welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet.
$f_1$ bildet das Dreieck auf das TeilstĂĽck unten links ab, $f_2$ auf das TeilstĂĽck unten rechts und $f_3$ auf das obere TeilstĂĽck.
-Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wieder ein Sierpinski-Dreieck.
+Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an
\begin{align*}
- X = \bigcup\limits_{i = 1}^{3} f_i(X)
+ X = \bigcup\limits_{i = 1}^{3} f_i(X),
\end{align*}
+entsteht also wieder ein Sierpinski-Dreieck.
Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck.
\begin{figure}
\centering
@@ -90,37 +92,44 @@ Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktion
\label{ifs:sierpconst}
\end{figure}
Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird.
-Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Iterationen gegen null.
+Der Abstand zum Original wird immer kleiner, und konvergiert gegen null.
\subsection{Iterierte Funktionensysteme
-\label{ifs:subsection:bonorum}}
-In diesem Unterkapitel wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern.
+\label{ifs:subsection:IteratedFunktionensysteme}}
+In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern.
-$S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt
+$S_1,\dots,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt
\begin{align}
|S_i(x) - S_i(y)| \leq c_i|x - y|
\end{align}
-fĂĽr jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge $F$ fĂĽr die gilt
+fĂĽr jedes i mit einem $c_i < 1$.
+Der Banachsche Fixpunktsatz besagt, dass fĂĽr solche Kontraktionen ein Eindeutiges $A$ existiert, fĂĽr das $S(A) = A$ gilt.
+Den Beweis kann man in \cite{ifs:Rousseau2012} nachlesen.
+Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeutige kompakte Menge $F$ fĂĽr die gilt
\begin{equation}
- F = \bigcup\limits_{i = 1}^{m} S_i(F)
+ F = \bigcup\limits_{i = 1}^{m} S_i(F).
\end{equation}
-Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge.
+Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leere Menge
\begin{equation}
- S(E) = \bigcup\limits_{i = 1}^m S_i(E)
+ S(E) = \bigcup\limits_{i = 1}^m S_i(E).
+ \label{ifs:transformation}
\end{equation}
-Wird diese Transformation Iterativ ausgefĂĽhrt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und fĂĽr jedes $i$ $S_i(E) \subset E$, gilt
+Wird diese Transformation Iterativ ausgefĂĽhrt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt
\begin{equation}
F = \bigcap\limits_{k = 1}^{\infty} S^k(E).
+ \label{ifs:ifsForm}
\end{equation}
In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgefĂĽhrt, gegen eine eindeutige Menge konvergiert.
-Dies fĂĽr jede Startmenge, solange diese ihre Transformierten wieder beinhaltet.
-Auf den Beweis wird verzichtet.
+Diese Menge ist auch als Attraktor eines IFS bekannt.
+Der Beweis fĂĽr die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben.
+
\subsection{Beispiel: Barnsley-Farn}
-Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein weiteres Fraktal, welches mit einem IFS generiert werden kann.
+Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann.
Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben.
-\begin{align*}
- {S_1(x,y)}
+Die vier affinen Transformationen
+\begin{align}
+ & {S_1(x,y)}
=
\begin{pmatrix}
0 & 0 \\
@@ -129,9 +138,9 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross
\begin{pmatrix}
x\\
y\\
- \end{pmatrix}, \quad
+ \end{pmatrix}, \quad &
{S_2(x,y)}
- =
+ &=
\begin{pmatrix}
0.85 & 0.04 \\
-0.04 & 0.85 \\
@@ -145,7 +154,7 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross
0 \\
1.6
\end{pmatrix}\\
- {S_3(x,y)}
+ & {S_3(x,y)}
=
\begin{pmatrix}
0.2 & -0.26 \\
@@ -159,9 +168,9 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross
\begin{pmatrix}
0 \\
1.6
- \end{pmatrix}, \quad
+ \end{pmatrix}, \quad &
{S_4(x,y)}
- =
+ &=
\begin{pmatrix}
-0.15 & 0.28 \\
0.26 & 0.24 \\
@@ -175,26 +184,51 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross
0 \\
0.44
\end{pmatrix}\\
-\end{align*}
-In der Abbildung \ref{ifs:farncolor} sehen wir die vier Transformationen farblich dargestellt.
-
+ \label{ifs:farnFormel}
+\end{align}
+, welche für die konstruktion des Farns benötigt werden sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt.
+Das gesamte Farnblatt ist in der schwarzen Box.
+Auf diese werden die Transformationen angewendet
$S_1$ erstellt den Stiel des Farnblattes (rot).
Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab.
$S_2$ (grĂĽn) erstellt den Hauptteil des Farnes.
Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$.
$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab.
-$S_4$ Spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab.
+$S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab.
+\subsection{Erzeugung eines Bildes zu einem IFS}
+Es gibt zwei verschiedene Methoden um das Bild zu einem IFS zu erzeugen.
+Die erste Methode ist wahrscheinlich die intuitivste.
+Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab.
+Das neue Bild, dass entsteht, ist die nächste Iterierte.
+Dieses wird wieder mit den Transformationen abgebildet.
+Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind.
+
+Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht.
+In Abbildung \ref{ifs:sierpinski10} ist die zehnte Iterierte zu sehen.
+Weitere Iterationen hätten in dieser Darstellungsgrösse kaum mehr einen Unterschied gemacht.
+
-Wir fĂĽhren im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um IFS auszufĂĽhren.
+Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}.
Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden.
-Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv.
-Eine Alternative ist das Chaosspiel \cite{ifs:chaos}.
-Bei dieser Methode werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt.
+Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist die erste Methode ziemlich rechenintensiv.
+Beim Chaosspiel werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt.
Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein.
Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet.
-Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet.
-Die Gewichtung erfolgt ĂĽber den Anteil der Gesamtmasse.
-Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgefĂĽhrt.
+Da, wie wir beim Barnsley-Farn gut sehen, nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet.
+Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es um die resultierende Teilabbildung darzustellen.
+Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgefĂĽhrt.
+Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grĂĽne Hauptteil des Blattes, $S_2$.
+
+In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet.
+Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn nicht so gut abgebildet wird.
+
+Am besten sieht man den Effekt einer schlechten Gewichtung in Abbildung \ref{ifs:farnrightWeight}.
+Hier wurde $S_4$, welches für das rechte untere Teilblatt zuständig ist, mit nur $1\%$ statt $7\%$ gewichtet.
+Man sieht, wie sich der Mangel an Punkten auf die anderen Abbildungen das Farnblattes auswirkt.
+In jeder Kopie des ganzen Farns fehlen die Punkte fĂĽr dieses rechte untere Teilblatt.
+
+
+
\begin{figure}
\centering
\makebox[\textwidth][c]{
@@ -204,7 +238,19 @@ Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$
\end{figure}
\begin{figure}
\centering
- \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor}
- \caption{Vier Transformationen des Barnsley-Farn}
+ \includegraphics[width=\textwidth]{papers/ifs/images/farncolor2}
+ \caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben}
\label{ifs:farncolor}
\end{figure}
+
+\begin{figure}
+ \centering
+ \subfigure[]{
+ \label{ifs:farnNoWeight}
+ \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}}
+ \subfigure[]{
+ \label{ifs:farnrightWeight}
+ \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}}
+ \caption{(a) Chaosspiel ohne Gewichtung (b) $S_4$ zu wenig gewichtet}
+ \label{ifs:farnweight}
+\end{figure}
diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex
index 24f0751..78fb935 100644
--- a/buch/papers/ifs/teil3.tex
+++ b/buch/papers/ifs/teil3.tex
@@ -9,33 +9,41 @@
Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren.
Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat.
Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat.
-In diesem Unterkapitel wollen wir eine Methode dafĂĽr anschauen.\cite{ifs:Rousseau2012}
+In diesem Unterkapitel wollen wir eine Methode dafĂĽr anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist.
-
-Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden.
-Dies muss jedoch nicht so sein.
-Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden.
-Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen.
-Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen.
Es ist wohl nicht falsch zu sagen, dass Ă„hnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind.
-Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben?
+Ein IFS, wie wir es in \ref{ifs:subsection:IteratedFunktionensysteme} definiert haben, wird uns also nicht weiter helfen.
+Die Lösung dazu sind Partitionierte IFS (PIFS) \cite{ifs:pifs}.
+In \ref{ifs:transformation} wurde definiert, dass die Kontraktionen $S_i$ bei IFS auf die gesamte Menge $E$ angewendet werden.
+Bei einem PIFS wird der Attraktor in disjunkte Teilmengen aufgeteilt.
+Für jede dieser Teilmengen $R_i$ braucht es dann eine grössere Teilmenge, welche mit einer affinen Transformation eine zu $R_i$ ähnliche Menge bildet.
+Wir mĂĽssen nicht mehr Ă„hnlichkeiten zum ganzen Bild finden, sondern zwischen Teilen des Bildes.
+Doch wie finden wir das PIFS, welches das Bild als Attraktor hat?
\subsection{das Kompressionsverfahren
\label{ifs:subsection:malorum}}
-In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert.
+Wir beschränken das Verfahren für Graustufenbilder. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert.
+Ein Graustufenbild kann man als Pixelraster mit einer x und y Achse verstehen.
+Jedem dieser Pixel wird ein Grauwert zugeordnet.
+Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist
+\begin{align*}
+ z = f(x,y).
+\end{align*}
+Wir suchen ein PIFS welches das zu komprimierende Bild als Attraktor hat.
In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$
Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$.
Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist.
+Zwei Beispiele wie solche Domain-, und Range-Block Paare aussehen können, sehen wir in Abbildung \ref{ifs:FIC}
\subsubsection{Finden des ähnlichsten $D_j$}
-Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden.
+Zuerst brauchen wir die Transformation
\begin{align*}
- T(x,y,z) =
+ T_i(x,y,z) =
\begin{pmatrix}
- a & b & 0 \\
- c & d & 0 \\
- 0 & 0 & s
+ a_i & b_i & 0 \\
+ c_i & d_i & 0 \\
+ 0 & 0 & s_i
\end{pmatrix}
\begin{pmatrix}
x \\
@@ -44,52 +52,86 @@ Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element vo
\end{pmatrix}
+
\begin{pmatrix}
- \alpha \\
- \beta \\
- g
+ \alpha_i \\
+ \beta_i \\
+ g_i
\end{pmatrix}
\end{align*}
-Diese Transformation bildet den Pixel $P$ auf Koordinate $(x,y)$ und Graustufe $z$ auf den Pixel $P'$ ab.
-
-Da wir mit Pixeln arbeiten, sind die Transformationen in der Ebene Beschränkt.
-Diese wird durch die Parameter $a,b,c$ und $d$ bestimmt.
-Mögliche Transformationen sind auf folgende Liste Beschränkt:
+um ein Element aus $D$ auf ein Element von $R$ Abzubilden.
+Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung
+\begin{align}
+ t_i(x,y) =
+ \begin{pmatrix}
+ a_i & b_i \\
+ c_i & d_i
+ \end{pmatrix}
+ \begin{pmatrix}
+ x \\
+ y
+ \end{pmatrix}
+ +
+ \begin{pmatrix}
+ \alpha_i \\
+ \beta_i
+ \end{pmatrix}.
+\label{ifs:affTrans}
+\end{align}
+Da wir mit Pixeln arbeiten, ist die Auswahl der möglichen Abbildungen begrenzt.
+Wir sind auf folgende acht Abbildungen beschränkt:
\begin{itemize}
\item Identische Transformation, keine Ă„nderung
\item Drehung um 90, 180 oder 270 Grad.
\item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen.
\end{itemize}
-$\alpha$ und $\beta$ verschieben den Pixel an die richtige Stelle.
Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren.
Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen.
-Skaliert und transformiert erhalten wir $\tilde{D_j}$
-Die Parameter $s$ und $g$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit.
-$s$ und $g$ werden mit der linearen Regression ermittelt.
+
+Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion
\begin{align*}
- z' = sz + g \\
- f(\tilde{D_j}) \text{, Funktion um das Bild eins Blockes zu erhalten} \\
- s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\
- g = E(f(R_i)) - s E(f(\tilde{D_j}))
+ z' = s_i z + g_i.
\end{align*}
+FĂĽr die Bestimmung dieser Parameter fĂĽhren wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein.
+$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$.
+
+Wir suchen $s_i$ und $g_i$ so das
+\begin{align*}
+ f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}.
+\end{align*}
+Die Parameter lassen sich mit
+\begin{align*}
+ s = \frac{\operatorname{cov}(f_{R_i}), f(\tilde{f_{R_i}}))}{\operatorname{var}(\tilde{f_{R_i}})} \\
+ g = E(f_{R_i}) - s E(f(\tilde{f_{R_i}}))
+\end{align*}
+berechnen.
Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt.
-Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$.
+Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Abstand
\begin{align*}
- e = d(f(R_i), f(T(D_j)))
+ e = d(f_{R_i}, \bar{f_{R_i}}).
\end{align*}
Dieser Abstand sollte so klein wie möglich sein.
-Die beste Kombination von $D_j$ und $T_i$ ist also diese, welche den kleinsten Abstand zum Block $R_i$ hat, und somit am ähnlichsten ist.
-Am Ende des Verfahrens haben wir also für jeden $R_i$ einen passenden $D_i$ mit der zugehörigen Abbildung $T_i$ gefunden.
+Wir bestimmen die Parameter $s$ und $g$ für jede der acht möglichen affinen Abbildungen und das mit jedem Domain-Block.
+Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist die beste.
+
+Diese Schritte fĂĽhren wir fĂĽr jeden Range-Block $R_i$ aus.
+Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden.
+
+\begin{figure}
+ \centering
+ \includegraphics[width=\textwidth]{papers/ifs/images/FIC}
+ \caption{Domain-, und Range-Block Paare in GrĂĽn und Rot}
+ \label{ifs:FIC}
+\end{figure}
\subsubsection{Rekonstruktion des Bildes}
-Mit den Gefundenen Abbildungen lässt sich das Bild generieren.
+Mit den gefundenen Abbildungen lässt sich das Bild generieren.
Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge.
In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse.
Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$.
Dies wird verkĂĽrzt als Operator $W$ geschrieben.
So erhalten wir ein neues Bild $f_1 = W(f_0)$.
-Dieses Vorgehen fĂĽhren wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied feststellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht.
+Dieses Vorgehen fĂĽhren wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen Unterschied feststellen. Die Iteration hat nun ihren Attraktor, das Bild, erreicht.
\subsubsection{Farbbilder}
Dieses Verfahren mit Graustufenbilder lässt sich ganz einfach auf Farbbilder erweitern.
@@ -98,20 +140,20 @@ Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein
Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen.
\subsubsection{Performance des Verfahren}
-Dieser Grundalgorithmus der Fraktalen Bildkompression ist offensichtlich recht langsam und skaliert auch schlecht mit grösseren Bilder.
-Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das jpeg verfahren.
+Dieser Grundalgorithmus der fraktalen Bildkompression ist recht langsam und skaliert auch schlecht für grössere Bilder.
+Dies resultiert aus eigenen Experimenten.
+Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nicht so schnell wie zum Beispiel das JPEG-Verfahren.
+Es wurden bessere Algorithmen der fraktalen Bildkompression entwickelt, doch auch diese können, vor allem in der Laufzeit, noch nicht mit herkömmlichen Komprimierungsverfahren mithalten.
\subsection{Beispiel}
-Kommen wir nun zu einem Beispiel.
-Wir Verwenden dafĂĽr den oben beschriebenen Algorithmus.
+Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert.
+Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren.
Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$.
Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht.
Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}.
-Der Algorithmus liefert uns für jeden Range-Block die benötigten Parameter.
-Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren.
-
-Als Startbild wird ein mittelgraues 360x360px Bild gewählt, Abbildung \ref{ifs:bild0}.
-Nun lassen wir das IFS laufen.
+Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}.
+Es kann jedoch ein beliebiges Startbild
+Nun lassen wir das PIFS laufen.
Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar.
Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden.
\begin{figure}