From 44fe683fb854dbe6ecb105624a31828c1b14db5a Mon Sep 17 00:00:00 2001 From: wieerwill Date: Fri, 6 May 2022 14:16:31 +0200 Subject: [PATCH] Digitale Bilder --- Assets/Bildverarbeitung-2d-dreiecksgitter.png | Bin 0 -> 16413 bytes Assets/Bildverarbeitung-2d-hexagongitter.png | Bin 0 -> 11787 bytes Assets/Bildverarbeitung-2d-quadratgitter.png | Bin 0 -> 1760 bytes Assets/Bildverarbeitung-4er-nachbarschaft.png | Bin 0 -> 1678 bytes Assets/Bildverarbeitung-8er-nachbarschaft.png | Bin 0 -> 1656 bytes ...eitung-bildrepräsentation-falschfarben.png | Bin 0 -> 157781 bytes ...rarbeitung-bildrepräsentation-grauwert.png | Bin 0 -> 113800 bytes ...dverarbeitung-bildrepräsentation-pixel.png | Bin 0 -> 35759 bytes ...-digitalisierung-grauwertquantisierung.png | Bin 0 -> 22055 bytes ...rbeitung-digitalisierung-ortsvariablen.png | Bin 0 -> 50473 bytes ...g-digitalisierung-quantisierungsstufen.png | Bin 0 -> 231431 bytes .../Bildverarbeitung-euklidische-distanz.png | Bin 0 -> 22081 bytes 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+Abtastung: $g_A(x,y)=s(x,y)*g(x,y)=\sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} g(m*\Delta x, n*\Delta y)* \delta(x-m*\Delta x, y-n*\Delta y)= A_{\Delta x, \Delta y}\{g(x,y)\}$ + +Repräsentation des ortsdiskreten Bildes unabhängig von $\Delta x, \Delta y$: $\rightarrow g_A(m,n)$ (,,2D-Zahlenfolge'') +![](Assets/Bildverarbeitung-digitalisierung-ortsvariablen.png) + +##### Digitalisierung: Quantisierung der Grauwerte +$q(g)=[\frac{g-g_{min}}{g_{max}-g_{min}} * (q_{max}-q_{min}) +q_{min}]_{mathbb{N}}$ (Runden auf nächste natürliche Zahl) + +häufig wird $q_{min}=0$ gewählt: $q(g)=[\frac{g-g_{min}}{g_{max}-g_{min}}*q_{max}]_{mathbb{N}}$ + +$q_{max}=2^N -1$, $N$ Auflösung des AD-Wandlers + +![](Assets/Bildverarbeitung-digitalisierung-grauwertquantisierung.png) + +![Quelle: Pratt, ,,Digital Image Processing''](Assets/Bildverarbeitung-digitalisierung-quantisierungsstufen.png) + +##### Digitale Bildrepräsentation +Pixel ![](Assets/Bildverarbeitung-bildrepräsentation-pixel.png) + +Grauwertbild ![](Assets/Bildverarbeitung-bildrepräsentation-grauwert.png) + +Falschfarbendarstellung ![](Assets/Bildverarbeitung-bildrepr%C3%A4sentation-falschfarben.png) + +#### Nachbarschaft, Pfad, Zusammenhang und Distanzmaße +Nachbarschaften +- 4er-Nachbarschaft + - Nachbarpixel: gemeinsame Kante + - ![](Assets/Bildverarbeitung-4er-nachbarschaft.png) +- 8er-Nachbarschaft + - Nachbarpixel: gemeinsame Kante oder Ecke + - ![](Assets/Bildverarbeitung-8er-nachbarschaft.png) +- Regelmäßige 2D-Gitter + - Dreieckgitter + - ![](Assets/Bildverarbeitung-2d-dreiecksgitter.png) + - 3-Nachbarschaft + - 12-Nachbarschaft + - Quadratisches Gitter + - ![](Assets/Bildverarbeitung-2d-quadratgitter.png) + - 4-Nachbarschaft + - 8-Nachbarschaft + - Hexagonales Gitter + - ![](Assets/Bildverarbeitung-2d-hexagongitter.png) + - 6-Nachbarschaft + +**Pfad:** +- Zwei Pixel $P_A(m_A,n_A)$ und $P_B(m_B,n_B)$ sind durch einen Pfad verbunden, falls es eine Folge von benachbarten Pixeln $(P_A, P_1, ..., P_B)$ gibt, für die eine Homogenitätsbedingung (z.B. alle Pixel haben gleichen Grauwert, d.h. $g(P_A)=g(P_1)=...=g(P_B))$ gilt. +- Offener Pfad: $P_A \not= P_B$ +- Geschlossener Pfad: $P_A = P_B$ +- Pfade sind an Nachbarschaftsdefinitionen gebunden! + +**Zusammenhang:** Eine Menge von Pixeln ist zusammenhängend, wenn zwischen zwei beliebigen Pixeln ein Pfad existiert. + +Zusammenhang: Definition gemäß Nachbarschaftsbeziehung +- Zusammenhang gemäß 4-Nachbarschaft + - Die beiden grauen Regionen sind unter 4-Nachbarschaft nicht zusammenhängend (kein verbindender Pfad vorhanden). + - Der Hintergrund ist unter Annahme der komplementären8-Nachbarschaft zusammenhängend. + - ![](Assets/Bildverarbeitung-nachbarschaft-definition-4.png) +- Zusammenhang gemäß 8-Nachbarschaft + - Die beiden grauen Regionen sind unter 8-Nachbarschaft zusammenhängend (verbindender Pfad vorhanden). + - Der Hintergrund ist unter Annahme der komplementären4-Nachbarschaft nicht zusammenhängend. + - ![](Assets/Bildverarbeitung-nachbarschaft-definition-8.png) +- Die Nachbarschaftsdefinitionen in Vorder-und Hintergrund sollten komplementär sein! + +**Rand:** +- Der Rand einer zusammenhängenden Pixelmenge $M$ ist eine Folge von Pixeln in $M$, die mindestens einen Nachbarn haben, der nicht zu $M$ gehört. +- Die Randpixel gehören somit zu $M$ dazu. +- Der Rand ist ein zusammenhängender Pfad und deshalb auch an eine Nachbarschaftsdefinition gebunden. + +Rand: Definition gemäß Nachbarschaftsbeziehung +- Rand in 4 - Nachbarschaft zum Hintergrund + - Jedes Randpixel hat mind. einen Nachbarn in 4-Nachbarschaft, der nicht zu M gehört. + - Rand = zusammenhängender Pfad gemäß 8-Nachbarschaft + - ![](Assets/Bildverarbeitung-rand-4-nachbarschaft.png) +- Rand in 8 - Nachbarschaft zum Hintergrund + - Jedes Randpixel hat mind. einen Nachbarn in 8-Nachbarschaft, der nicht zu M gehört. + - Rand = zusammenhängender Pfad gemäß 4-Nachbarschaft + - ![](Assets/Bildverarbeitung-rand-8-nachbarschaft.png) + +##### Distanzmaße zwischen zwei Pixeln +Euklidische Distanz +- Länge der direkten Verbindung +- $D_E= ||P_1-P_2||_2=\sqrt{(m_1-,_2)^2 + (n_1-n_2)^2}$ +- Euklidische Norm $N=2$, $p=2$ +- ![](Assets/Bildverarbeitung-euklidische-distanz.png) + +Manhattan-Distanz (City-Block) +- Länge des kürzesten Pfades unter 4er-Nachbarschaft +- $D_4=||P_1-P_2||_1=|m_1 - m_2|+|n_1-n_2|$ +- Summennorm $N=2$, $p=1$ +- ![](Assets/Bildverarbeitung-manhatten-distanz.png) + +Schachbrett-Distanz +- Länge des kürzesten Pfades unter 8er-Nachbarschaft +- $D_8 = ||P_1-P_2||_{\infty} = max\{|m_1-m_2|, |n_1 -n_2|\}$ +- Maximalnorm $N=2$, $p=\infty$ +- ![](Assets/Bildverarbeitung-schachbrett-distanz.png) + +Normangabe der Distanzmaße +- p-Norm: $||x||_p = (\sum_{i=1}^N |x_i|^p )^{\frac{1}{p}}$ mit $x_1=m_1-m_2$ und $x_2=n_1-n_2$ +- Euklidische Norm: $N=2, p=2$, $D_E=(\sum_{i=1}^2 |x_i|^2)^{frac{1}{2}}=\sqrt{x_1^2 + x_2^2}$ +- Summennorm: $N=2, p=1$, $D_4=(\sum_{i=1}^2 |x_i|)= x_1+x_2$ +- Maximalnorm: $N=2, p=\infty$, $D_8=lim_{p\rightarrow\infty}(x^p_{max} + (ax_{max})^p)^{\frac{1}{p}} = lim_{p\rightarrow\infty} (x_{max}^p (1+a^p))^{\frac{1}{p}} = x_{max}$ mit $a<1$ weil $lim_{p\rightarrow\infty}(a^p)=0$ +- $D_8\leq D_E \leq D_4$ + - ![](Assets/Bildverarbeitung-schachbrett-distanz.png) + - ![](Assets/Bildverarbeitung-euklidische-distanz.png) + - ![](Assets/Bildverarbeitung-manhatten-distanz.png) +- Schachbrett Distanz $\leq$ Euklidische Distanz $\leq$ Manhatten Distanz + # Literaturempfehlungen - Wilhelm Burger and MarkJ. Burge, ,,Digitale Bildverarbeitung – eine algorithmische Einführung mit Java'', Springer, 3. Auflage, 2015 - Klaus D. Tönnies, ,,Grundlagen der Bildverarbeitung'', Pearson Studium, 1. Auflage, 2005

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