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zG?9{2R>IxLdLqDUK?w+&Z>O{)X%K*jw7BT03ox{8lIO9S_D87HC@J%aU9Zlq51D#>l@YG?r~Qx>xBwJxPB zQyBv9Ol93pRf4vy0%T=&PR{}Hqot0$p;OuCj7RwlS7LC8Ad7tT$0LhrFm-Ac8foR) zN?W71^p_;Qc*C|~q(qYv7+Q%541el7ikw_SZf7K#6 zj+DZ8pvzG^2Qvk!1|Ce1G4A{2C~M8F*1{m(ypH_ fT>jR7Np3{?dDEqXVTh4=SXg+FC@CeBB$56H+!p;G diff --git a/Computergrafik - Cheatsheet.tex b/Computergrafik - Cheatsheet.tex index cd10d2b..afa5e78 100644 --- a/Computergrafik - Cheatsheet.tex +++ b/Computergrafik - Cheatsheet.tex @@ -610,9 +610,9 @@ \item pro Rasterzeile \item pro Kante (möglichst viel vorberechnen, um pro Rasterzeile/Pixel Rechenzeit zu sparen) \end{itemize*} - \item Erweiterungen des inkrementellen Ansatzes für effiziente Berechnungen in der 3D-Grafik, z.B.: - \item Füllen des Z-Buffers (Tiefenwertberechnung) - \item lineare Interpolation beim Gouraud Shading (Farbwertberechnungen) + \item Erweiterungen des inkrementellen Ansatzes für effiziente Berechnungen in der 3D-Grafik, z.B.: + \item Füllen des Z-Buffers (Tiefenwertberechnung) + \item lineare Interpolation beim Gouraud Shading (Farbwertberechnungen) \end{itemize*} \paragraph{Füllmuster} @@ -633,7 +633,7 @@ \item verursachten Fehler $\delta$ jeweils nach Schema auf unbearbeitete Nachbarpixel in der Repräsentation verteilen \item bei kleinen Bildern mit hoher Auflösung ist Dithering kaum wahrnehmbar \end{itemize*} - + \paragraph{Dithering vs. Anti-Aliasing} \begin{itemize*} \item komplementär zueinander @@ -685,7 +685,7 @@ \item Quantisierbarkeit der Farben und Helligkeit \item Bezug zur Physik des Lichtes (Energie, Spektrum) \end{itemize*} - + \paragraph{RGB Farbraum} \begin{itemize*} \item Hypothese, dass Farbsehen auf drei Arten von Sinneszellen beruht (rot, grün, blau) (Young) @@ -719,9 +719,9 @@ \item drei linear-unabhängige Größen benötigt, zur Beschreibung und (technischen) Reproduktion der Farbempfindung \item zunächst werden folgende Werte gewertet \begin{itemize*} - \item die additive Mischung als Reproduktionsmethode - \item drei Primärfarben Rot, Grün, Blau - \item drei linear unabhängige Größen spannen stets einen 3D Raum auf + \item die additive Mischung als Reproduktionsmethode + \item drei Primärfarben Rot, Grün, Blau + \item drei linear unabhängige Größen spannen stets einen 3D Raum auf \end{itemize*} \item die RGB Werte werden den drei ortogonalen Achsen dieses Raumes zugeordnet \end{itemize*} @@ -756,342 +756,192 @@ Achtung: Dies gilt nur für die Bestrahlung mit weißem Licht. Wird beispielsweise ein gelbes Blatt mit blauem Licht bestrahlt, dann wirkt es schwarz, da das blaue Licht vom gelben Blatt absorbiert wird. \section{Licht \& Reflexion} - \subsection{ Strahlung} - Grundfrage: Was ist Licht? - \begin{itemize*} - \item Teil der elektromagnetischen Strahlung - \item ist für das menschliche Auge wahrnehmbar - \item Lichtspektrum liegen zwischen 380 nm und 780 nm - \item Farbe entspricht der Wellenlänge - \item längere Wellenlängen = weniger Photonenenergie - \item durch Überlagerungen vieler Frequenzen erscheint das Licht weiß - \end{itemize*} + \subsection{Strahlung} + \begin{description*} + \item[Licht] Teil der elektromagnetischen Strahlung + \item[Photon] Elementarteilchen der elektromagnetischen Wechselwirkung + \item[Radiometrie] Messung elektromagnetischer Strahlung + \item[Photometrie] Messverfahren im Wellenlängenbereich des sichtbaren Lichtes + \item[Strahlungsäquivalent] $K =\frac{\phi_v}{\phi_e}$] + \item[Lumen] 1 Lumen ist der Lichtstrom einer 1,464 mW starken 555-nm-Lichtquelle mit 100\% Lichtausbeute + \end{description*} - Radiometrie: - \begin{itemize*} - \item Wissenschaft von der Messung elektromagnetischer Strahlung - \item Größen sind physikalische Einheiten (ohne Berücksichtigung des menschl. Sehens) - \end{itemize*} + In der Radiometrie wird sich mit objektiven Messgrößen beschäftigt, in der Photometrie fließt spektrale Empfindlichkeit des menschlichen Auges mit ein. - Photometrie: - \begin{itemize*} - \item Messverfahren im Wellenlängenbereich des sichtbaren Lichtes (Messung mithilfe eines Photometers) - \item lassen sich aus den radiometrischen Größen, bei bekanntem Spektrum bestimmen - \item berücksichtigen die wellenlängenabhängige Empfindlichkeit des Auges - \end{itemize*} + \paragraph{Radiometrie (energetisch $_e$) } + \begin{description} + \item[Strahlungsenergie $Q$] durch Strahlung übertragene Energie $[J]$ + \item[Strahlungsleistung $\phi$] transportierte Strahlungsenergie in einer bestimmten Zeit $\phi = \frac{Q}{t} [W]$ + \item[Strahlstärke/Intensität $I$] Strahlungsleistung die in eine Raumrichtung mit Raumwinkel $\Omega$ emittiert wird $I=\frac{\phi}{\Omega}=\frac{W}{sr}$ + \item[Bestrahlungsstärke/Irradiance $E$] Strahlungsleistung durch die bestrahlte Fläche $A_i$ bzw. Strahlstärke die auf die Empfängerfläche trifft $E=\frac{W}{m^2}=\frac{\Phi}{A_i}$ + \item[Strahldichte/Radiance $L$] Strahlstärke von Sendefläche $A_r$ in eine bestimmte Richtung $L = \frac{I}{A'_r}=\frac{I}{\cos(\phi_r)*A_r} = \frac{\phi}{\cos(\phi_r)*A_r*\Omega}$; $\phi_r$ ist Winkel zwischen Normalen n und Abstrahlrichtung + \end{description} - Photon: - \begin{itemize*} - \item Elementarteilchen der elektromagnetischen Wechselwirkung - \item besitzen keine Masse - \item Energie und Impuls sind proportional zur Frequenz - \item kürzere Wellenlänge = höhere Frequenz = höhere Energie - \end{itemize*} + \paragraph{Photometrie (visuell $_v$ )} + \begin{description} + \item[Lichtmenge $Q$] Strahlungsleistung bewertet mit der spektralen Empfindlichkeitsfunktion des menschlichen Auges für das Hellempfinden $lm*s$ + \item[Lichtstrom (luminous flux) $\phi$] $[Lumen]$ + \item[Lichtstärke (luminous intensity) $I$] $[Candela]$ + \item[Beleuchtungsstärke $E$] $I_{in}\cos(\phi) [Lux]$ + \item[Leuchtdichte/Luminanz $L$] $[\frac{cd}{m^2}]$ + \end{description} - Strahlungsenergie (radiant energy): - \begin{itemize*} - \item durch Strahlung (elektromagnetische Wellen) übertragene Energie - \item entspricht dem Produkt von Photonenanzahl und der Energie der Photonen - \item Formelzeichen : Q - \item Einheit: J (Joule) - \item photometrisches Äquivalent: Lichtmenge (luminous energy) - \end{itemize*} - - Strahlungsleistung (auch Strahlungsfluss, engl. radiant flux, radiant power): - \begin{itemize*} - \item transportierte Strahlungsenergie in einer bestimmten Zeit - \item Formelzeichen : $\phi$ - \item Einheit: W (Watt) - \item Berechnung: $\phi = \frac{Q}{t}$ - \item photometrisches Äquivalent: Lichtstrom (luminous flux, luminous power) - \end{itemize*} - - Zusammenhang zwischen Radiometrie und Photometrie:\\ - In der Radiometrie wird sich mit objektiven Messgrößen beschäftigt, in der Photometrie gibt es jeweils eine entsprechende Messgrößen, bei denen die spektrale Empfindlichkeit des menschlichen Auges mit einfließt. - \begin{itemize*} - \item Beispiel: - \item radiometrisch: Strahlungsleistung $\phi_e$ gemessen in Watt W - \item photometrisch: Lichtstrom $\phi_v$ gemessen in Lumen lm - \item Verknüpfung von Radiometrie und Photometrie erfolgt über das photometrische Strahlungsäquivalent: $K =\frac{\phi_v}{\phi_e}$ - \item gibt die Empfindlichkeit des menschlichen Auges an - \item radiometrische Größe: Index $_e$ für energetisch - \item photometrische Größe: Index $_v$ für visuell - \end{itemize*} - Die radiometrischen Größen gewichtet mit dem photometrischen Strahlungsäquivalent K sind somit die photometrischen Größen. - - Ausbreitung eines Strahls: - \begin{itemize*} - \item geradlinig von einer Quelle zum Ziel, - \item Richtung ändert sich durch Brechung - \item an Oberflächen tritt Reflexion und Streuung auf - \item eine Strahlungsquelle sendet dabei Strahlen in alle Raumrichtungen unter einem gewissen Raumwinkel aus - \end{itemize*} + $$A=2\pi r^2; \Omega=\frac{A}{r^2}=2\pi ; I_e=\frac{\phi_e}{\Omega}= \frac{W}{sr}$$ \paragraph{Raumwinkel} Der Steradiant ist eine Maßeinheit für den Raumwinkel, der von der Mitte M einer Kugel mit Radius r aus gesehen eine Fläche von $r^2$ auf der Kugeloberfläche einnimmt. $\Omega=\frac{Flaeche}{Radius^2}=\frac{A}{r^2}sr$ - Eine komplette Kugeloberfläche $A_k$ beträgt allg. $A_k = 4\pi r^2$, entspricht also einem Raumwinkel $\Omega$ von $\frac{A_k}{r^2}= 4\pi r\approx 12,5sr$. Ein Steradiant =1sr entspricht einem Öffnungswinkel $\alpha$ von ca. $65,54^{\circ}$ - - \paragraph{Strahlstärke} - \begin{itemize*} - \item auch Intensität, engl. radiant intensity - \item Strahlungsleistung die in eine Raumrichtung mit Raumwinkel $\Omega$ emittiert wird - \item Formelzeichen : I - \item Berechnung: $I=\frac{\phi}{\Omega}$ - \item photometrisches Äquivalent: Lichtstärke (luminous intensity) - \end{itemize*} - - Beispiel: Berechnen Sie die Strahlstärke einer Lampe mit einem Öffnungswinkel von 180° und einer Strahlungsleistung von 20W. - $$\alpha=180^{\circ}\rightarrow A=2\pi r^2; \phi =20W; \Omega=\frac{A}{r^2}=2\pi ; I_e=\frac{\phi_e}{\Omega}=\frac{20}{2\pi}\approx 3,2 \frac{W}{sr}$$ + Eine komplette Kugeloberfläche beträgt allg. $A_k = 4\pi r^2$, entspricht also einem Raumwinkel $\Omega$ von $\frac{A_k}{r^2}= 4\pi r\approx 12,5sr$. Ein Steradiant $=1sr$ entspricht einem Öffnungswinkel $\alpha$ von ca. $65,54^{\circ}$ \paragraph{Räumliche Ausbreitung} Energieübertragung zwischen zwei Flächen: - Eine Fläche $A_r$ strahlt Licht auf eine Fläche $A_i$ ab.\\ - Frage: Wie viel Lichtleistung von einer infinitesimalen abstrahlenden Fläche $A_r$ wird auf einer Fläche $A_i$ empfangen? \begin{itemize*} - \item der Abstand zwischen den beiden infinitesimalen Flächen beträgt r - \item die Flächen stehen nicht notwendigerweise senkrecht zur Ausbreitungsrichtung des Lichts (gerade Verbindungslinie zwischen den Flächen) - \item Wir projizieren daher die abstrahlende und die empfangende Fläche jeweils in Ausbreitungsrichtung. Die projizierten Flächen nennen wir $A'_r$ und $A'_i$. - \item Wir betrachten Punktlichtquellen von der abstrahlenden Fläche $A_r$ , welche ihre Strahlungsleistung in den Raumwinkel $\Omega$ abgeben. + \item der Abstand zwischen den beiden Flächen beträgt r + \item die Flächen stehen nicht notwendigerweise senkrecht zur Ausbreitungsrichtung des Lichts + \item abstrahlende und empfangende Fläche jeweils in Ausbreitungsrichtung. Die projizierten Flächen sind $A'_r$ und $A'_i$. + \item betrachten Punktlichtquellen von der abstrahlenden Fläche $A_r$ , welche ihre Strahlungsleistung in den Raumwinkel $\Omega$ abgeben \item $\Omega$ ist somit die in Abstrahlrichtung reduzierte Fläche $A'_i$ , projiziert auf die Einheitskugel: $\Omega=\frac{A'_i}{r^2}$ \item Die übertragene Energie nimmt quadratisch zu r ab \end{itemize*} - \paragraph{Strahldichte} - \begin{itemize*} - \item engl. radiance - \item Strahlstärke welche von einer Sendefläche $A_r$ in eine bestimmte Richtung abgegeben wird - \item Formelzeichen : L - \item photometrisches Äquivalent: Leuchtdichte (auch Luminanz, engl. luminance) - \item Berechnung: $L = \frac{I}{A'_r}=\frac{I}{\cos(\phi_r)*A_r} = \frac{\phi}{\cos(\phi_r)*A_r*\Omega}$ - \item $\phi_r$ ist der Winkel zwischen der Normalen n und der Abstrahlrichtung (von der abstrahlenden Fläche $A_r$ zur empfangenden $A_i$) - \end{itemize*} - - Leuchtdichte (Luminanz) als Vorstufe der Helligkeit: - \begin{itemize*} - \item Strahlungsleistung bewertet mit der spektralen Empfindlichkeitsfunktion des menschlichen Auges für das Hellempfinden - \item Das menschliche Auge hat seine maximale Empfindlichkeit, bei einer Wellenlänge von 555 nm (gelbgrün) - \item 1 Lumen ist definiert als der Lichtstrom einer 1,464 mW starken 555-nm-Lichtquelle mit 100% Lichtausbeute. - \end{itemize*} - - - \paragraph{Bestrahlungsstärke} - \begin{itemize*} - \item auch Strahlungsflussdichte, engl. irradiance - \item Strahlungsleistung durch die bestrahlte Fläche $A_i$ bzw. Strahlstärke die auf die Empfängerfläche trifft - \item Formelzeichen : E - \item Berechnung: $E =\frac{\Phi}{A_i}$ - \item photometrisches Äquivalent: Beleuchtungsstärke (auch Lichtstromdichte, engl. illuminance) - \item erweitert: $E=\frac{\Phi}{A_i}=\frac{L*\cos(\phi_i)*\cos(\phi_r)*A_r}{r^2}$ - \end{itemize*} - - \paragraph{Zusammenfassung} - Radiometrische (physikalische) und Photometrische (unter Berücksichtigung des menschlichen Auges) Größen - - \begin{tabular}{ c | c | c } - Symbol & Radiometrie (energetisch $_e$) & Photometrie (visuell $_v$ ) \\ \hline - $Q$ & Strahlungsenergie $Joule$ & Lichtmenge $lm*s$ \\ - $\Phi$ & Strahlungsleistung Watt $W$ & Lichtstrom Lumen $lm$ \\ - $I$ & Strahlstärke $\frac{w}{sr}$ & Lichtstärke Candela $cd$ \\ - $E$ & Bestrahlungsstärke $\frac{W}{m^2}$ & Beleuchtungsstärke Lux $\frac{lm}{m^2}$ \\ - $L$ & Strahldichte $\frac{w}{sr*m^2}$ & Leuchtdichte $\frac{cd}{m^2}$ \\ - \end{tabular} - - - \subsection{ Reflexion} - Nach dem Auftreffen auf einer opaken Oberfläche wird die Strahlung spektral - unterschiedlich stark und geometrisch auf unterschiedliche Weise reflektiert. Es - können 2 Idealfälle der Reflexion unterschieden werden: + \subsection{Reflexion} + Nach Auftreffen auf einer opaken Oberfläche wird Strahlung spektral unterschiedlich stark und geometrisch auf unterschiedliche Weise reflektiert. Es können Fälle der Reflexion unterschieden werden: \begin{itemize*} \item ideal spiegelnde Reflexion (Einfallswinkel = Ausfallswinkel) \item ideal diffuse Reflexion + \item spekulär (diffus und gerichtete Reflexion) + \item gemischt: ideal diffus, gerichtet diffus und ideal spiegelnd \end{itemize*} - Aus den zwei Idealfällen der reflexion werden weitere (gemischte) Fälle abgeleitet: - \begin{itemize*} - \item spekuläre Reflexion (diffus und gerichtete Reflexion) - \item gemischte Reflexion: ideal diffus, gerichtet diffus und ideal spiegelnd - \end{itemize*} - - Bei der Betrachtung der Reflexion ist offensichtlich die Art der Bestrahlung und - insbesondere auch die Richtung der Einstrahlung zu beachten. - \paragraph{Diffuse Reflexion} - %![Diffuse Reflexion; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Diffuse_Reflexion.png) - - Lichtquelle im Unendlichen; Irradiance $E=\frac{A'_i}{A_i}I_{in}=I_{in}\cos(\phi)$ - - Eingestrahlte Strahlstärke I in durch $A'_i$ verteilt sich durch die Projektion auf die größere Fläche $A_i$ Die Bestrahlungsstärke E (Irradiance) ist dadurch proportional zum Vergrößerungsfaktor der Fläche abgeschwächt. - + Eingestrahlte Strahlstärke verteilt sich durch Projektion auf größere Fläche. Die Bestrahlungsstärke ist dadurch proportional zum Vergrößerungsfaktor der Fläche abgeschwächt. In Richtung Betrachter reflektierte Strahlstärke $I_{out}$ Aufgrund von Interferenz phasengleicher Lichtstrahlen $\rightarrow$ Projektion auf Normalenrichtung $\frac{I_{out}}{E_{refl}}=\cos(\phi)$ \begin{itemize*} \item Senkrecht zur Oberfläche: Maximale Kohärenz (Addition) \item Parallel zur Oberfläche: n Keine Kohärenz (Auslöschung) \end{itemize*} - %![Diffuse Reflexion Addition und Auslöschung; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Diffuse_Reflexion_2.png) - - Annahme kohärentes Licht: Parallel zur reflektierenden Oberfläche findet sich zu jeder Punktlichtquelle immer eine gleichphasige Punktlichtquelle im Abstand $\frac{\lambda}{2}$, Auslöschung parallel zur Fläche, - - %![Diffuse Reflexion Betrachter; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Diffuse_Reflexion_3.png) - $$\frac{A_r}{A'_r}=\frac{1}{\cos(\phi)} \rightarrow L=\frac{I_{out}}{\cos(\phi)}=I_{refl}$$ Ein Betrachter mit flachem Blickwinkel sieht Licht aus größerer Fläche $A_r$ durch Kombination dieser Effekte, kürzt sich der Einfluss des Betrachterwinkels $\cos(\phi)$ weg und es bleibt nur der Einfluss des Lichteinfallswinkels übrig: Strahldichte des reflektierten Lichtes: $L=I_{in}*k_d(\lambda)*\cos(\phi)$ \paragraph{Spekuläre Reflexion} - Spekuläre (gestreut spiegelnde) Reflexion: + (gestreut spiegelnd) \begin{itemize*} - \item Speckles (Fleckchen), bzw. (Micro-) Facetten sind einzeln jeweils "ideal" - \item spiegelnd: Einfallswinkel $\phi$ = neg. Ausfallswinkel = $-\phi$. - \item Die Ausrichtung der Microfacetten weichen von der Gesamtflächennormalen ab. $\rightarrow$ Statistische Abweichung von der Flächennormalen (z. B. Gauß-Verteilung) + \item Speckles bzw. Facetten sind einzeln jeweils "ideal" + \item spiegelnd: $\text{Einfallswinkel} \phi = \neg Ausfallswinkel = -\phi$ + \item Ausrichtung der Microfacetten weichen von Gesamtflächennormalen ab \item dadurch Streuung des Lichts (Keule) um den Winkel $\theta$ der idealen Spiegelung herum \item Je größer der Winkel $\theta$ zwischen idealer Spiegelrichtung und Richtung zum Betrachter, desto schwächer ist die Reflexion - \item Modellierung meist per $\cos^k(\theta)$ (Phong-Beleuchtungsmodell) - nicht physikalisch begründet. + \item Modellierung meist per $\cos^k(\theta)$ (Phong-Beleuchtungsmodell) \end{itemize*} - %![Spekuläre Reflexion; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Spekuläre_Reflexion.png) - Gestreute Spiegelung im Phong Modell mit $L=I*k_s*\cos^k(\theta)$ \begin{itemize*} - \item glänzende Oberfläche: großer Exponent k (16,...,128); kleine Streuung $\epsilon$ - \item matte Oberfläche: kleiner Exponent k (1,...,2); große Streuung $\epsilon$ + \item glänzende Fläche: großer Exponent k; kleine Streuung $\epsilon$ + \item matte Fläche: kleiner Exponent k; große Streuung $\epsilon$ \end{itemize*} - Energieerhaltung $\rightarrow$ Verhinderung der Abnahme bei großen Exponenten $\rightarrow$ Für die Energieerhaltung wird ein zusätzlicher Normierungsfaktor benötigt: + Für Energieerhaltung wird zusätzlicher Normierungsfaktor benötigt: \begin{itemize*} \item physikalisch nicht korrekt: $L=I*k_s*\cos^k(\theta)$ - \item gebräuchliche Normierung $\frac{k+2}{2\pi}$ somit: $L=I*k_s*\frac{k+2}{2\pi}*cos^k(\theta)$ + \item gebräuchliche Normierung $L=I*k_s*\frac{k+2}{2\pi}*cos^k(\theta)$ \end{itemize*} \paragraph{Remittierende Flächen} + ideal diffus remittierende weiße Flächen $(\beta(\lambda) = 1)$: \begin{itemize*} - \item Wegen der spektralen Unterschiede bei der Reflexion bleiben wir bei den spektralen physikalischen (radiometrischen) Größen! - \item Erst im Auge bzw. im Bildsensor erfolgt die Wandlung in die wellenlängenintegralen photometrischen (colorimetrischen) Größe! + \item von Quellen in Fläche $dA$ eingetragene Leistung führt zu Bestrahlungsstärke $E_{\lambda}$ + \item Bei vollständiger Reflexion $\beta(\lambda) = 1$ ist $E_{\lambda} = R_{\lambda}$ + \item zugehörige Strahlungsfluss $d\phi = R_{\lambda} * dA = E_{\lambda} * dA$ wird bei ideal diffusen streuenden Oberflächen gleichmäßig über den Halbraum verteilt, wobei die Strahldichte (Lambertsches Gesetz) konstant ist. \end{itemize*} - Zunächst ideal diffus remittierende weiße Flächen $(\beta(\lambda) = 1)$: + \subsection{BRDF: Bidirektionale Reflexionsverteilung} \begin{itemize*} - \item Die von den Quellen in die Fläche $dA$ eingetragene Leistung führt zu einer Bestrahlungsstärke $E_{\lambda}$ - \item Bei vollständiger Reflexion $\beta(\lambda) = 1$ ist $E_{\lambda} = R_{\lambda}$ (spektrale Radiosity, spezifische spektrale Ausstrahlung). - \item Der zugehörige spektrale Strahlungsfluss $d\phi = R_{\lambda} * dA = E_{\lambda} * dA$ wird bei ideal diffusen streuenden Oberflächen gleichmäßig über den Halbraum verteilt, wobei die Strahldichte (Lambertsches Gesetz) konstant ist. - \end{itemize*} - - - \subsection{ BRDF: Bidirektionale Reflexionsverteilungsfunktion} - \paragraph{Bidirektionale Reflexion} - \begin{itemize*} - \item englisch Bidirectional Reflectance Distribution Function, BRDF \item eine Funktion für das Reflexionsverhalten von Oberflächen eines Materials unter beliebigen Einfallswinkeln - \item Ziel: Oberfläche möglichst realistisch und physikalisch korrekt darstellen \item nach gewählter Genauigkeit sehr komplex - \item in der Computergrafik wird meist eine vereinfachte Variante gewählt um Rechenzeit zu sparen - \item erstmals 1965 definiert (Fred Nicodemus): $f_r(\omega_i, \omega_r)=\frac{dL_r(\omega_r)}{dE_i(\omega_i)}=\frac{dL_r(\omega_r)}{L_i(\omega_i)\cos(\theta_i)d\omega_i}$ - \item Eine BRDF beschreibt wie eine gegebene Oberfläche Licht reflektiert. - \item Das Verhältnis von reflektierter Strahldichte (radiance) $L_r$ in eine Richtung $\vec{\omega}_r$ zur einfallenden Bestrahlungsstärke (irradiance) $E_i$ aus einer Richtung $\vec{\omega}_i$ wird "bidirectional reflectance distribution function"(BRDF) genannt. + \item $f_r(\omega_i, \omega_r)=\frac{dL_r(\omega_r)}{dE_i(\omega_i)}=\frac{dL_r(\omega_r)}{L_i(\omega_i)\cos(\theta_i)d\omega_i}$ + \item BRDF beschreibt wie gegebene Oberfläche Licht reflektiert. \item $p(\lambda)=\frac{L_r}{E_i}=[\frac{1}{sr}]$ - \item Die BRDF (für jeden Punkt x) ist eine 5-dimensionale skalare Funktion: $p(\lambda, \phi_e, \theta_e, \phi_i, \theta_i)$ - \item Keine Energie-Einheiten, nur Verhältniszahl! - \item Kann durch Messung für verschiedene Materialien bestimmt werden (Messkamera/Normbeleuchtung) - \item Eigenschaften der BRDF: - \item Reziprozität: $\rho(\lambda)$ ändert sich nicht, wenn Einfalls- und Ausfallsrichtung vertauscht werden (wichtig für Ray-Tracing). - \item $\rho(\lambda)$ kann anisotrop sein, d.h. der Anteil des reflektierten Lichtes ändert sich, wenn bei gleicher Einfalls- undAusfallsrichtung die Fläche um die Normale gedreht wird (Textilien, gebürstete Metalle, Metalleffektlacke) + \item Die BRDF ist 5-dimensionale skalare Funktion: $p(\lambda, \phi_e, \theta_e, \phi_i, \theta_i)$ + \item Reziprozität: $\rho(\lambda)$ ändert sich nicht, wenn Einfalls- und Ausfallsrichtung vertauscht werden + \item $\rho(\lambda)$ kann anisotrop sein, d.h. der Anteil des reflektierten Lichtes ändert sich, wenn bei gleicher Einfalls- undAusfallsrichtung die Fläche um die Normale gedreht wird \item Superposition gilt, d.h. mehrere Quellen überlagern sich linear. \end{itemize*} - Es ist in der Computergrafik üblich, die bidirektionale Reflektivität als Gemisch von ambienten, diffusen und spekularen Komponenten $\rho_d, \rho_s$ aufzufassen und - einen ambienten Anteil $\rho_a$ zu addieren. Für eine Menge Q von Lichtquellen berechnen wir damit die gesamte reflektierte Strahlstärke: $L_r=p_a*E_a+\sum_{1\leq j \leq Q} E_j * (k_d*p_d + k_s*p_s)$ mit $k_d+k_s=1$ und Q= Anzahl der Lichtquellen + Für Menge Q von Lichtquellen die gesamte reflektierte Strahlstärke: $L_r=p_a*E_a+\sum_{1\leq j \leq Q} E_j * (k_d*p_d + k_s*p_s)$ mit $k_d+k_s=1$ \paragraph{Rendering-Equation} - Für ambiente und gerichtete Lichtquellen aus der Hemisphäre ergibt sich eine spezielle Form der BRDF, die Render-Gleichung (Jim Kajiya 1986): - \begin{itemize*} - \item eine BRDF mit Integral über alle Lichtquellen (bzw. Hemisphären) - \item $L_r=p_a + \int_{Omega} L*(k_d*p_d+k_s*p_s) \omega_i*n d\Omega$ - \end{itemize*} - - %![Rendering Equation; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Rendering_Equation.png) + Für ambiente und gerichtete Lichtquellen aus der Hemisphäre: + $L_r=p_a + \int_{Omega} L*(k_d*p_d+k_s*p_s) \omega_i*n d\Omega$ \paragraph{Strahlungsquellenarten} - \begin{itemize*} - \item Ambiente Strahlung: - \item es ist keine "eigentliche" Quelle zuordenbar - \item stark vereinfachtes Modell für die Streuung der Atmosphäre, für viele "durchmischte" Strahlungsquellen, für indirekte Reflexionen - \item Strahlung kommt von allen Seiten "Die Quelle ist überall und nirgends" - \item keine Abhängigkeit von Winkeln und Entfernungen - \item Beschreibung nur indirekt durch konstante Bestrahlungsstärke (Irradiance) von Flächen möglich - \item $E=\frac{\Phi}{A}=E_a$ - \item Parallele Strahlung: - \item Strahlung ist gerichtet und parallel (kollimiertes Licht, Strahlungsquelle im Unendlichen, Sonnenlicht) - \item für derartige Quellen lässt sich kein Ort (aber uneigentlicher Ort, Richtung) angeben - \item Wichtig sind die Richtung und die Strahlungsleistung, bezogen auf die senkrecht zur Ausbreitungsrichtung stehende Fläche (spezifische Ausstrahlung oder Radiosity $R_e$) $R=E_q=\frac{\Phi}{A_q}$ - \item für die Schattierungsrechnung lässt sich die Bestrahlungsstärke $E_e$ der Oberfläche (Flächenelement dA) berechnen: $E=\frac{\Phi}{A}=\frac{E_q*A_q}{A}=E_q*\cos(\phi) = E_q*V_I^T*n$ - \item Ideale Punktlichtquelle: - \item für die Punktquelle ist der Ort bekannt und die Strahlstärke in alle Richtungen konstant: $I=\frac{\Phi}{\Omega}=konstant$ - \item die Bestrahlungsstärke eines physikalischen vorliegenden, beliebig orientierten Flächenelementes A ergibt sich zu: - \item $E=\frac{\Phi}{A}=\frac{I*\Omega}{A}, \Omega=\frac{A}{r^2}*\cos(\phi)*\omega_r \rightarrow E=\frac{I}{r^2}*\cos(\phi)*\omega_r$ - \item zum Ausgleich der Adaptionsfähigkeit des menschlichen Auges wird in der Computergrafik oft der folgende Ansatz verwendet: - \item $E=\frac{I}{c_1+c_2*|r|+c_3*r^2}*\cos(\phi)*\omega_r$ - \item Remittierende Flächen (radiometrische Betrachtung): - \item Zur Berechnung der von der reflektierenden Fläche weitergegebenen Strahldichte L sind die weiter oben berechneten Bestrahlungsstärken E für die unterschiedlichen Quellen mit dem Faktor $\frac{\beta(\lambda)}{\pi\omega_r}$ zu bewerten - \end{itemize*} + \begin{description*} + \item[Ambiente Strahlung] + \begin{itemize*} + \item keine eigentliche Quelle zuordenbar + \item stark vereinfachtes Modell für die Streuung der Atmosphäre + \item Strahlung kommt von allen Seiten + \item keine Abhängigkeit von Winkeln und Entfernungen + \item Beschreibung nur indirekt durch konstante Bestrahlungsstärke + \item $E=\frac{\Phi}{A}=E_a$ + \end{itemize*} + \item[Parallele Strahlung] + \begin{itemize*} + \item Strahlung ist gerichtet und parallel + \item kein Ort für derartige Quellen + \item Wichtig sind Richtung und Strahlungsleistung, bezogen auf die senkrecht zur Ausbreitungsrichtung stehende Fläche $R=E_q=\frac{\Phi}{A_q}$ + \item für Schattierungsrechnung lässt sich Bestrahlungsstärke der Oberfläche berechnen: $E=\frac{\Phi}{A}=\frac{E_q*A_q}{A}=E_q*\cos(\phi) = E_q*V_I^T*n$ + \end{itemize*} + \item[Ideale Punktlichtquelle] + \begin{itemize*} + \item für Punktquelle ist Ort bekannt und Strahlstärke in alle Richtungen konstant $I=\frac{\Phi}{\Omega}=konstant$ + \item Bestrahlungsstärke eines physikalischen vorliegenden, beliebig orientierten Flächenelementes A ergibt sich zu $E=\frac{\Phi}{A}=\frac{I*\Omega}{A}, \Omega=\frac{A}{r^2}*\cos(\phi)*\omega_r \rightarrow E=\frac{I}{r^2}*\cos(\phi)*\omega_r$ + \item zum Ausgleich der Adaptionsfähigkeit des menschlichen Auges wird in der Computergrafik oft der folgende Ansatz verwendet $E=\frac{I}{c_1+c_2*|r|+c_3*r^2}*\cos(\phi)*\omega_r$ + \end{itemize*} + \item[Remittierende Flächen] Zur Berechnung von reflektierenden Fläche weitergegebenen Strahldichte L sind die weiter oben berechneten Bestrahlungsstärken E für unterschiedlichen Quellen mit dem Faktor $\frac{\beta(\lambda)}{\pi\omega_r}$ zu bewerten + \end{description*} - \begin{tabular}{c | c | c} - Quelle & Reflexion & Spektale Strahldichte $L(\lambda)$ \\\hline + \begin{tabular}{l | c | l} + Quelle & Ref. & Spektale Strahldichte $L(\lambda)$ \\\hline ambient & diffus & $L(\lambda)=\frac{E(\lambda)}{\pi\omega_r}*\beta(\lambda)$ \\ gerichtet & diffus & $L(\lambda)=\frac{E(\lambda)}{\pi\omega_r}*\cos(\phi)*\beta(\lambda)$ \\ punktförmig & diffus & $L(\lambda) = \frac{I(\lambda)}{\pi r^2 }*\cos(\phi)*\beta(\lambda)$ \\ gerichtet diffus & diffus & $L(\lambda)=\frac{I(\lambda)}{\pi r^2 }* \cos^m(\theta)*\cos(\phi)*\beta(\lambda)$ \\ \end{tabular} - \subsection{ Beleuchtungsmodelle} - Ein Beleuchtungsmodell ist eine Verfahren in der Computergrafik welches das Verhalten von Licht simuliert. Die Simulation unterscheidet dabei zwischen lokaler und globaler Beleuchtung: - \begin{itemize*} - \item Lokale Beleuchtungsmodelle: - \item simulieren das Verhalten von Licht auf den einzelnen Materialoberflächen - \item nur Beleuchtungseffekte welche direkt durch Lichtquellen auf einzelnen Objekt entstehen - \item indirekte Beleuchtung bleibt zunächst unberücksichtigt - \item Globale Beleuchtungsmodelle: - \item simulieren die Ausbreitung von Licht innerhalb der Szene - \item dabei wird die Wechselwirkung in der Szene beachtet (Schatttenwurf, Spiegelung, indirekte Beleuchtung) - \end{itemize*} + \subsection{Beleuchtungsmodelle} + \begin{description*} + \item[Lokale] simulieren Verhalten von Licht auf einzelnen Materialoberflächen; nur Beleuchtungseffekte die direkt durch Lichtquellen auf einzelnen Objekt entstehen + \item[Global] simulieren Ausbreitung von Licht innerhalb der Szene; dabei wird Wechselwirkung in der Szene beachtet (Schatttenwurf, Spiegelung, indirekte Beleuchtung) + \end{description*} \paragraph{Phong-Modell} \begin{itemize*} - \item lokales Beleuchtungsmodell (lässt sich durch BRDF beschreiben) + \item lokales Beleuchtungsmodell \item eignet sich zur Darstellung von glatten, plastikähnlichen Oberflächen - \item baut nicht auf physikalischen Grundlagen auf \item widerspricht dem Energieerhaltungssatz - \item Reflexion des Lichts = ambienter+ ideal diffuser + ideal spiegelnder Reflexion - \end{itemize*} - - %![Phong Modell; Quelle Computergrafik Vorlesung 2020](Assets/Computergrafik_Phong_Modell.png) - - \begin{itemize*} \item Allgemein: $L=I_{out}=I_{ambient}+I_{diffus}+I_{specular}$ - \item Ambiente: $I_{ambient}=I_a * k_a$ mit $I_a$ Intensität des Lichtes und $k_a$ Materialkonstante - \item Diffus: $I_{diffus}=I_{in}*k_d*\cos(\phi)$ mit $I_{in}$ Lichtstärke der Punktlichtquelle; $k_d$ empirischem Reflexionsfaktor; $\phi$ Winkel zwischen Oberflächennormale und Richtung des einfallenden Lichtstrahls - \item Spiegelnd: $I_{specular}=I_{in}*k_s*\frac{n+2}{2\pi}*\cos^n({\theta})$ mit - \item $I_{in}$ Lichtstärle des eingallendes Lichtstrahls der Punktlichtquelle - \item $k_s$ empirisch bestimmter Reflexionsfaktor - \item $\theta$ Winkel zwischen idealer Reflexionsrichtung des Lichtstrahls und Blickrichtung - \item $n$ konstante Exponent zur Beschreibung der Oberflächenbeschaffenheit + \item Ambiente: $I_{ambient}=I_a * k_a$ + \item Diffus: $I_{diffus}=I_{in}*k_d*\cos(\phi)$ + \item Spiegelnd: $I_{specular}=I_{in}*k_s*\frac{n+2}{2\pi}*\cos^n({\theta})$ + \begin{itemize*} + \item $I$ Lichtstärke/Intensität der Lichtquelle + \item $k_a$ Materialkonstante + \item $k_{d/s}$ empirischem Reflexionsfaktor + \item $\phi$ Winkel zwischen Oberflächennormale und Richtung des einfallenden Lichtstrahls + \item $\theta$ Winkel zwischen idealer Reflexionsrichtung des Lichtstrahls und Blickrichtung + \item $n$ konstante Exponent zur Beschreibung der Oberflächenbeschaffenheit + \end{itemize*} \item $\frac{n+2}{2\pi}$ Normalisierungsfaktor zur Helligkeitsregulierung - \item Vollständige Formel: $I_{out}=I_a*k_a+I_{in}*k_d*\cos(\phi)+I_{in}*k_s*\frac{n+2}{2\pi}*\cos^n(\theta)$ - \end{itemize*} - Unterschiedliche Definitionen sind möglich, z.B. mit mehrere Lichtquellen: - \begin{itemize*} - \item jeweiligen Komponenten für jede Lichtquelle separat berechnet - \item diese werden anschließend aufsummiert + \item Vollständig: $I_{out}=I_a*k_a+I_{in}*k_d*\cos(\phi)+I_{in}*k_s*\frac{n+2}{2\pi}*\cos^n(\theta)$ \end{itemize*} \paragraph{Cook-Torrance} \begin{itemize*} - \item Physik-basierte spekulare Reflexion: - \item Microfacetten: Grundidee ähnlich Phong-Modell - \item Statistische Abweichung der Microfacetten von der Flächennormalen (z. B. Beckmann-Verteilung) \item Streuung des Lichts (Keule) um den Winkel der idealen Spiegelung herum - \item Berücksichtigt auch die gegenseitigen Abschattung (insbesondere bei flachen Lichtstrahlen) - \item Vollständig physikbasiertes Modell, keine willkürlichen Reflexionskonstanten - \item Aufwendige Berechnung (verschiedene Näherungsformeln existieren) - \item Beckmann-Verteilung: $l_{spec}=\frac{exp(-\frac{tan^2(\alpha)}{m^2})}{\pi m^2 cos^4 (\alpha)}$, $\alpha=arccos(N*H)$ + \item Berücksichtigt auch die gegenseitigen Abschattung + \item Vollständig physikbasiertes Modell, keine willkürlichen Reflexionskonstanten, spekulare Reflexion + \item Aufwendige Berechnung + \item Beckmann-Verteilung: $l_{spec}=\frac{exp(-\frac{tan^2(\alpha)}{m^2})}{\pi m^2 cos^4 (\alpha)}$ mit $\alpha=arccos(N*H)$ \end{itemize*} \section{Schattierungsverfahren}