formelsammlung.tex

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\begin{document}
\raggedright
\footnotesize
\begin{center}
	\Large{\underline{Phototechnik 2}} \\
\end{center}
\begin{multicols*}{3}

	\nonumber

	\setlength{\premulticols}{1pt}
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	\section{Beleuchtungsstaerke}
	\label{sec:beleuchtungsstaerke}

	\begin{framed}
		Die Beleuchtungsstarke oder Lichtstromdichte beschreibt den flaechenbezogenen Lichtstrom, der auf ein beleuchtetes Objekt trifft. Ihr steht gegenüber die Lichtstaerke, die den raumwinkelbezogenen Lichtstrom einer Lichtquelle beschreibt.
	\end{framed}

	\begin{align}
		&E=\frac{I}{d^2}*\Omega_0 \\
		&E=L * \pi *\Omega_0 && \text{fuer $d=0$} \\
		&E=\frac{I}{d^2}*\Omega_0*\cos{\alpha} \text{[lx]} \\
		&E= \frac{\Phi}{A_e} = L* \Omega_s\\
		&E_{grau} = E_{weiss}- P_{grau}\\
		&E = L*\pi*\Omega_0(\sin^2(\alpha_A)- \sin^2(\alpha_I)) && \text{Ring}\\
		&E = L*\pi*\Omega_0\sin^2\alpha && \text{Kreis} \\
		&E= L * \Omega_1 = \frac{A}{d^2} * L * \Omega_0 = L * \frac{\pi * D^2}{4 * d^2}* \Omega_0 && \text{Flaeche} 
	\end{align}

	\subsubsection{Am Strahler}
	\label{ssub:am_strahler}

	\begin{align}
		E_{max} &= L * \pi  * \Omega_0
	\end{align}

	\subsubsection{Scanner}
	\label{ssub:Scanner}

	\begin{align}
		E &= \frac{L_{Obj} * \pi}{4 * k_{eff}^2} * \Omega_0  \\
		E_{\alpha} &= \frac{L_{Obj} * \pi}{4 * k_{eff}^2} * \Omega_0  *\cos\alpha\\
	\end{align}

	\subsubsection{LW}
	\label{ssub:LW}
	
	\begin{align}
		E&= \frac{L_0 * \pi }{P_{LW}}	* \Omega_0
	\end{align}

	\columnbreak

	\section{Leuchtdichte}
	\label{sec:leuchtdichte}

	\begin{framed}
		Reprasendtiert ein Strahlungsmas fuer die Flaechenhelligkeit $L = \frac{I}{A * \cos \epsilon}$ und beruecksichtigt im Gegensatz zur Starahldichte noch die menschliche  Hellempfindlichkeit.
	\end{framed}
	\begin{align}
		L &= \frac{I}{A} \left[\frac{cd}{m^2}\right] \\
		L &= \frac{E*d^2}{A_1 * \Omega_0} \\
		L &= \frac{E*\rho_{diff}}{\pi* \Omega_0}
	\end{align}

	\begin{framed}
		Bei gerichteter Spiegelung kann Spiegelbild fuer die Kamera unsichbar sein, dann ist $L=0$
	\end{framed}

	\subsubsection{Durchmesser berechnen}
	\label{ssub:Durchmesser_berechnen}

	\begin{align}
		L &= \frac{I}{A_{min}} = \frac{I}{\pi * \frac{D_{min}^2}{4}} \\
		\Rightarrow D_{min} &= \sqrt{\frac{4I}{\pi * L}}
	\end{align}

	\section{Lichtstrom}
	\label{sec:lichtstrom}

	\begin{align}
		\Phi_{ges} &= E * A = \frac{L*\pi*\Omega_0}{\rho} * A \\
		I &= \frac{\Phi}{\Omega} = \frac{\Phi}{4\pi\Omega_0} && \text{Lichtstaerke}
	\end{align}

	\section{Im deutlichen Gesichtsfeld}
	\label{sec:im_deutlichen_gesichtsfeld}

	\begin{align}
		\beta' &= \frac{-h_{KB}}{h} \\
		\beta ' &= \frac{f'}{f'+a} \\
		a &= f'(\frac{1}{\beta'}-1) \\
		k &= \frac{-t_s * \beta '}{2*u'*(\frac{1}{\beta'}-1)} \\
		t_s &= - \frac{2u'k}{\beta'}(\frac{1}{\beta'}-1) \\
	\end{align}

	\section{Belichtung}
	\label{sec:belichtung}

	\begin{align}
		22 DIN &= 100 ASA\\
		24 DIN &= 200 ASA\\
		27 DIN &= 400 ASA\\
		+3 DIN &= 2*ASA\\
		t_{normal} &= \frac{t_{mess}}{2^{\Delta B} * l_{ref}}
		&& \text{mit $l_{ref} = 0,18$} \\
		A_{max} &= 2^{n_{bit}} -1 \\
		H_m &= E *t = 0.65 * \frac{L_{Obj}}{k^2 / t} * \Omega_0 \\
		    &= 0.65 * \frac{L_{Obj}}{2^B} * \Omega_0\\
		H_{sat} &= \frac{A_{max}}{A_{mittel}}\\
		S_{ISO\_Sat} &= \frac{78lx}{H_{Sat}}
	\end{align}

	\subsubsection{Objektmessung}
	\label{ssub:Objektmessung}

	\begin{align}
		t &= \frac{k^2 * C_R}{L * 10^{\frac{S_{DIN}}{10}}} \\
		C_R &= \frac{L * 10\frac{S_{DIN}}{10}}{2^B} \\
		2^B &= \frac{k^2}{t} = \frac{L_{Obj} * 10 ^{\frac{S_{DIN}}{10}}}{C_R} * 2^{\Delta B_{Anzeige}} \\
		L_{Obj} &= \frac{C_R * (\frac{k^2}{t})}{\frac{S_{DIN}}{10}} * 2^{\Delta B_{Anzeige}} \\
	\end{align}

	\subsubsection{Lichtmessung}
	\label{ssub:Lichtmessung}

	\begin{align}
		t &= \frac{k^2 * C_L}{E * 10^{\frac{S_{DIN}}{10}}} \\
		C_L &= \frac{E * 10\frac{S_{DIN}}{10}}{2^B}\\
		t &=t_0 *VF *2^{-\Delta B} \\
		2^B &=\frac{E * 10^{\frac{S_{DIN}}{10}}}{C_L} \\
		\Rightarrow E &= \frac{2^B * C_L}{10^{\frac{S_{DIN}}{10}}}
	\end{align}

	\subsubsection{Verlaengerungsfaktor}
	\label{ssub:Verlaengerungsfaktor}

	\begin{align}
		VF &= (1 - \beta ') ^2 \\
		\beta ' &= \frac{f'}{a+ f'}
	\end{align}


	\section{Optische Abbildung}
	\label{sec:optische_abbildung}

	\begin{align}
		f'_{ges} &= - \frac{1}{a_{fern}} + \frac{1}{a_{min}} \\
		f'_{ges} &= \frac{f_1' * f'_2}{f'_1 + f'_2 -d} \\
		\beta' &= \frac{\gamma'}{\gamma} && \text{$\gamma'$: Pixelabstand Sensor} \\
		a &= f'\left(\frac{1}{\beta'}-1\right) && \text{a Objektweite} \\
		a' &= a * \beta && \text{a': Bildweite} \\
		VF &= \left(1-\beta'\right)^2 && \text{Verlaengerungsfaktor}\\
		k_{eff} &= K_0\left( 1-\beta'\right) && \text{effektive Blende}
	\end{align}

	\subsubsection{Brennweite}
	\label{ssub:Brennweite}

	\begin{align}
		f' &= \frac{a'}{1-\beta} = \frac{a'}{1-\frac{\gamma'}{\gamma}}
	\end{align}

	\subsubsection{Austrittspuppile}
	\label{ssub:Austrittspuppile}

	\begin{align}
		k &= \frac{f'}{D_{EP}} \Rightarrow D_{AP} = D_{EP} = \frac{f'}{k} = f' * \text{OeV} && \text{Durchmesser} \\
	\end{align}

	\subsubsection{Aufloesungsvermoegen}
	\label{ssub:Aufloesungsvermoegen}

	\begin{align}
		AV &= \frac{1500}{k_{eff}} && \text{Bildebene}\\
		AV_{Sensor} &= \frac{1}{\text{Pixelabstand}} \\
		\beta' &= -\left(\frac{1500}{k_0 * AV_{Sensor}}-1 \right) && \text{$AV_{Sensor}$ genutzt} \\
		t_s &= - \frac{4*k}{\beta' * AV_{Sensor}}\left( \frac{1}{\beta'}-1\right) && \text{Schaerfentiefe}
	\end{align}

	\subsubsection{Hauptebenen}
	\label{ssub:Hauptebenen}

	\begin{align}
		H_2H'_{ges} &= f'_{ges} && \text{fuer $a = \infty$} \\
	\end{align}

	\subsubsection{Dioptrien}
	\label{ssub:Dioptrien}

	\begin{align}
		D &= \frac{1}{f'}\\
		\frac{1}{f_{korr}} & = \frac{1}{a'_{nah}} -  \frac{1}{a_{nah}} && \text{nach korrektur} \\
		\Rightarrow a'_{nah} &= \frac{a_{nah} * f'_{korr}}{a_{nah} + f'_{korr}}
	\end{align}

	\section{Konversionsfilter $\mu rd$ }
	\label{sec:konversionsfilter}

	\begin{align}
		&M = \frac{10^6}{T} \\
		&M_{Sensor} = M_{Licht} + M_{Filter} \\
		&\Delta M < 0 && \text{Blau-Filter} \\
		&\Delta M > 0 && \text{Orange-Gelb-Filter} \\
		&\tau(\lambda) \sim e^{\frac{-c_2}{\lambda}(\frac{1}{T_{Sensor}}-\frac{1}{T_{Licht}})} \\
		&\frac{\tau(\lambda_{g})}{\tau(\lambda_{b})} = e^{-c_2(\frac{1}{\lambda_g}- \frac{1}{\lambda_b})(\frac{1}{T_s}- \frac{1}{T_L})} && c_2 = 1.4388\times10^{-2}[mk] \\
		&\frac{\tau(\lambda_{1})}{\tau(\lambda_{2})} = e^{-c_2(\frac{1}{T_2}- \frac{1}{T_1})(\frac{1}{\lambda_1}- \frac{1}{\lambda_2})} \\
		&= e ^{c_2 * \frac{M_{Filter}}{10^6}*\left(\frac{1}{\lambda_1}-\frac{1}{\lambda_2}\right)} \\
		&\Rightarrow M_{Filter} = \frac{\log \frac{\tau(\lambda_1)}{\tau(\lambda_2)}*10^6}{c_2 * \left(\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right)} \\
		&\frac{V_g}{V_r} =  \frac{\tau_g}{\tau_r} && \text{Verstaerkungsfaktor} \\
		&Tf_{Lampe} = \frac{1}{\frac{1}{Tf_{Licht}}+ \frac{M_{dimm}}{10^6}}
	\end{align}
	\section{Lampengesetze}
	\label{sec:lampengesetze}

	\begin{align}
		\frac{X}{X_N}&=\left(\frac{U}{U_N}\right)^{\alpha x} && \text{mit $X \sim \Phi \sim P \sim \eta \sim T \sim z$} \\
		\alpha x &= \frac{\log\left(\frac{X}{X_N}\right)}{\log\left(\frac{U}{U_N}\right)}
	\end{align}

	\subsubsection{Strom halbieren}
	\label{ssub:strom_halbieren}

	\begin{align}
		\frac{E}{E_N} &\doteq \frac{1}{2} = \left(\frac{U}{U_N}\right)^{\alpha_{\Phi}} && \text{soll gelten}\\
		\Rightarrow U&=U_N*\left(\frac{1}{2}\right)^{\frac{1}{\alpha_\Phi}} \\
		\frac{\eta}{\eta_N}&= \frac{\frac{\Phi}{p}}{\frac{\Phi_N}{P_N}} = \frac{\Phi}{\Phi_N} * \left(\frac{P}{P_N}\right)^{-1} \\
		\Rightarrow &=\left(\frac{U}{U_N}\right)^{(\alpha_\Phi - \alpha_P)} && \text{Lichtausbeute}
	\end{align}

	\subsubsection{Lampenkoeffizienten}
	\label{ssub:Lampenkoeffizienten}


	\begin{align}
		\frac{E}{E_N} &= \frac{\Phi}{\Phi_N} = \left(\frac{U}{U_N}\right)^{\alpha_\Phi} \\
		\Rightarrow \alpha_\Phi  &= \frac{\log\left(\frac{E}{E_N}\right)}{\log\left(\frac{U}{U_N}\right)} \\
		\frac{U*I}{U_N*I_N} &= \frac{P}{P_N} = \left(\frac{U}{U_M}\right)^{\alpha_P} \\
		\Rightarrow \alpha_P &= \frac{\log\left(\frac{U*E}{U_N*I_N}\right)}{\log\left(\frac{U}{U_N}\right)}
	\end{align}

	\section{LED}
	\label{sec:led}

	\[
		I_v \sim I_D \sim I_e
	\]


	\section{Snellius}
	\label{sec:snellius}

	\begin{align}
		&n_1 * \sin\alpha = n_2 * \sin\beta && \text{n: Brechungsindex} \\
  &	n_\alpha> n_1 && \text{zum Lot}\\
  &	n_\alpha> n_1 && \text{vom Lot}
	\end{align}

	\section{Reflexion}
	\label{sec:reflexion}

	\begin{framed}
		Reflektierte Anteile sind vorzugsweise senkrecht zur Einfallsebene polarisiert.
	\end{framed}

	\subsubsection{Totalreflexion}
	\label{ssub:totalreflexion}


	\begin{align}
		\alpha_{grenz} = \arcsin\left(\frac{n_2}{n_1}\right)
	\end{align}
	\subsubsection{Brewster Winkel}
	\label{ssub:brewster_winkel}

	\begin{align}
		\tan\left(\alpha_B\right)&= \frac{n_1}{n_2} \\
		\alpha_B &= \arctan\left(\frac{n_2}{n_1}\right)
	\end{align}


	\subsubsection{Reflexionsgrade}
	\label{ssub:reflexionsgrade}

	\begin{align}
		\beta &= sin^{-1}\left(\frac{\sin\alpha}{n}\right)\\
		\rho_{s2} &= \frac{\sin^2\left(\alpha-\beta\right)}{\sin^2\left(\alpha+\beta\right)} && \text{senkrecht}\\
		\rho_{p2} &= \frac{\tan^2\left(\alpha-\beta\right)}{\tan^2\left(\alpha+\beta\right)} && \text{parallel}\\
		\rho_{unpol} &= \frac{\rho_{s}+ \rho_p }{2}
	\end{align}
	\begin{framed}
		Bei zweiter Greznflaeche (z.B. Austritt Glas) Reflexionsgrade vertauschen.
	\end{framed}

	\subsubsection{Verhaeltnis Reflexionsheligkeit}
	\label{ssub:verhaeltnis_reflexionsheligkeit}


	\begin{align}
		\frac{L_s}{L_p} = \frac{l_s}{l_p} = \frac{\frac{\sin^2(\alpha-\beta)}{\sin^2(\alpha+\beta}}{\frac{\tan^2(\alpha-\beta)}{\tan^2(\alpha+\beta}} = \frac{\cos^2(\alpha-\beta)}{\cos^2(\alpha+\beta)}
	\end{align}

	\begin{align}
		L_s &= L_{LQ,S} * l_s = \frac{1}{2}L_{LQ} * l_s && \text{$L_{LQ}$ unpolarisiert} \\
		L_p &= \frac{1}{2}L_{LQ} * l_p
	\end{align}



	\subsubsection{Helligkeit Reflexion}
	\label{ssub:helligkeit_reflexion}

	\begin{align}
		L &= L * \rho_{ger} && \text{gerichtet, Spiegelbild} \\
		L &= \frac{E*\rho_{diff}}{\pi * \Omega_0} && \text{diffus, Sekundaerstrahler}
	\end{align}

	\section{OECF}
	\label{sec:oecf}

	\begin{align}
		\frac{Y_1}{Y_2} &= \left(\frac{L_1}{L_2}\right)^{\frac{1}{\gamma}} \\
		\gamma &= \frac{\log(\frac{L_1}{L_2})}{\log(\frac{Y_1}{Y_2})} \\
		\frac{L}{L_{grau}} &= \frac{L}{L_{sat}} * \frac{L_{sat}}{L_{weiss}} * \frac{L_{weiss}}{L_{grau}} = (\frac{Y}{255})^\gamma * \sqrt{2} * \frac{1}{0,18}\\
		S_{ASA} &= S_{100} * \frac{L}{L{grau}}
	\end{align}


	\section{CCD Sensor}
	\label{sec:ccd_sensor}
	\begin{align}
		S_{rel}(\lambda) &= \frac{\lambda}{\lambda_0} && \text{rel spek Empf., $\lambda_0$ gegeben}
	\end{align}

	\subsubsection{Transmissionsverhaeltnisse bestimmen}
	\label{ssub:Transmissionsverhaeltnisse_bestimmen}
	\begin{align}
		\frac{\tau_r}{\tau_g} &= \frac{S_{rel}(\lambda_g) * Le_\lambda(\lambda_g, T_f)}{S_{rel}(\lambda_r) * Le_\lambda(\lambda_r, T_f)} \\
																	&= \frac{\lambda_r^4}{\lambda_g^4}*e^{\frac{c_2}{T}\left(\frac{1}{\lambda_r}-\frac{1}{\lambda_g}\right)} && \text{$\frac{\tau_r}{\tau_b}$ analog}
	\end{align}

	\section{Lichtstaerke}
	\label{sec:lichtstaerke}

	\section{Kirhoff}
	\label{sec:kirhoff}


	\section{Temperatur}
	\label{sec:temperatur}

	\begin{align}
		\frac{Le_{\lambda1}}{Le_{\lambda2}} &= \left(\frac{\lambda_2}{\lambda_1}\right)^5 * \frac{e^{\frac{c_2}{\lambda_2*T}}}{e^{\frac{c_2}{\lambda_1*T}}} =
		\left(\frac{\lambda_2}{\lambda_1}\right)^5 * e^{\frac{c_2}{T}  \left(\frac{1}{\lambda_2}-\frac{1}{\lambda1}\right)} \\
		\Rightarrow T&= \frac{c_2*\left(\frac{1}{\lambda_2}-\frac{1}{\lambda_1}\right)}
		{\log\left(\frac{Le_{\lambda1}}{Le_{\lambda2}}\right)+5\log\left(\frac{\lambda_1}{\lambda_2}\right)} && \text{$c_2 = 1,4388*10^{-2}mk$}
	\end{align}


	\subsubsection{Bolzman}
	\label{ssub:Bolzman}

	\begin{align}
		L &\sim \Phi \sim T^n\\
		\Phi &= U * I  =  P = P_{zu} = \Phi_e - \Phi_{abs} = (T_{su}^n - T_u^n) \\
		\Phi_{em} &=  o *T^4 * A * \epsilon
	\end{align}


	\subsubsection{Wiensches Verscheibungsgesetz}
	\label{ssub:Wiensches_Verscheibungsgesetz}

	\begin{align}
		\lambda_{max1} * T_{f1} &=\lambda_{max2}*T_{f2}\\
		\Rightarrow T_{f2} &= \frac{\lambda_{max1}}{\lambda_{max2}}* T_{f1} \\
		\lambda_{max} * T &= 2.898*10^{-3} && \text{[mk]} \\
		\Rightarrow T &= \frac{2.898*10^{-3}}{\lambda_{max}}
	\end{align}

	\section{Strahldichteverhaeltnis}
	\label{sec:strahldichteverhaeltnis}


	\section{Blende}
	\label{sec:blende}


	\begin{framed}
		\textbf{foerderliche Blende}: kleinste Blendeneinstellung, bei der Beugungsunschaerfae gerade noch nicht sichtbar ist
	\end{framed}

	\begin{align}
		W_{min} &= 1' = \frac{2\pi}{360 * 60} = \frac{u_{repro/2}}{d_B}\\
		\Rightarrow u_{repro} &= d_B * \frac{4\pi}{360*60} && \text{Zerstreuungskreis}\\
		AV_{repro} &= \frac{2}{u_{repro}} && \text{Aufloesungsvermoegen} \\
		AV_{Sensor} &= \frac{1500}{k_{eff}}[mm^{-1}] \doteq AV_{repro} * \beta_{repro} \\
		\Rightarrow k_{eff} &= \frac{1500}{AV_{repro} * \beta_{repro}} && \text{$\beta$: Abb.-massstab} \\
		a_h &= \frac{a*h}{h-(a+f')} \\
		\Rightarrow h&= \frac{a+f '}{1-\frac{a}{a_h}} \\
		u' &= -\frac{f'^2}{h*k} = \frac{u_{repro}}{\beta_{repro}}  = 2'd_B*\beta_{repro}&& \text{Unschaerfedurchmesser} \\
		\beta_{repro} &= \frac{h_{sensor}}{h}
	\end{align}


\end{multicols*}
\end{document}