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/dev/null differ diff --git a/buch/papers/ifs/images/zurich.png b/buch/papers/ifs/images/zurich.png deleted file mode 100644 index bb70f7d..0000000 Binary files a/buch/papers/ifs/images/zurich.png and /dev/null differ 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 b8a678d..833748c 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -3,7 +3,7 @@ % % (c) 2020 Prof Dr Andreas Müller, 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 (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden. In der Regel sind diese Bilder Fraktale. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 385abcf..70b0b1b 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -6,18 +6,18 @@ \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. Über 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 hat $F$ eine Form von Selbstähnlichkeit. \item Die 'fraktale Dimension' ist grösser als die topologische Dimension - \item Viele Fraktale lassen sich einfach beschrieben TODO + \item Viele Fraktale lassen sich auf eine simple Art definieren. Es genügen zum Beispiel nur wenige Funktionen, welche rekursiv ausgeführt werden, um ein Fraktal zu definieren. \end{enumerate} \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} @@ -74,7 +74,7 @@ Die Fläche unter der Kurve lässt sich folgendermassen berechnen 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. -Die Gesamtfläche ist daher gegeben durch die 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 \\ \end{align*} @@ -89,7 +89,7 @@ Zu guter Letzt bestimmen wir die Dimension der Kurve. 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 Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. -Die Ähnlichkeits-Dimension ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ +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 }. diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index be3d354..0c957d6 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -95,7 +95,7 @@ Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jede Der Abstand zum Original wird immer kleiner, und konvergiert gegen null. \subsection{Iterierte Funktionensysteme -\label{ifs:subsection:bonorum}} +\label{ifs:subsection:IteratedFunktionensysteme}} In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. @@ -110,9 +110,10 @@ Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeu \begin{equation} F = \bigcup\limits_{i = 1}^{m} S_i(F). \end{equation} -Weiter definieren wir die Transformation S auf kompakte Mengen $E$ 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 ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt \begin{equation} @@ -122,7 +123,8 @@ Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. Diese Menge ist auch als Attraktor des IFS bekannt. Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. -Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann, +Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann. + \subsection{Beispiel: Barnsley-Farn} 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. @@ -194,14 +196,17 @@ $S_2$ (grün) 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. -\subsection{Erzeugung eines Bildes mit einem IFS} -Es gibt zwei verschiedene Methoden um ein Bild mit einem IFS zu erzeugen. +\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. Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}. @@ -216,8 +221,12 @@ Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne 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 kaum nicht so gut abgebildet ist. +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 für dieses rechte untere Teilblatt. @@ -234,10 +243,15 @@ Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn} \caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben} \label{ifs:farncolor} \end{figure} + \begin{figure} \centering - \makebox[\textwidth][c]{ - \includegraphics[width=1.4\textwidth]{papers/ifs/images/farnnotweight}} - \caption{Chaosspiel ohne Gewichtung} - \label{ifs:farnNoWeight} + \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 b3dff85..ebae0fb 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -11,14 +11,14 @@ Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Frac Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist. - -Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen, welche das IFS bilden, auf die gesamte Menge. -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 Ähnlichkeiten 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 müssen nicht mehr Ähnlichkeiten 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}} @@ -29,9 +29,12 @@ 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 @@ -114,6 +117,12 @@ Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist d Diese Schritte führen wir für 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 Grün und Rot} + \label{ifs:FIC} +\end{figure} \subsubsection{Rekonstruktion des Bildes} Mit den gefundenen Abbildungen lässt sich das Bild generieren. @@ -144,7 +153,7 @@ Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Bl Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}. Es kann jedoch ein beliebiges Startbild -Nun lassen wir das IFS laufen. +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} -- cgit v1.2.1