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start [2019/09/23 21:08]
tstibor 		start [2020/12/09 15:27] (current)
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-Just a test: $1 + 2 + \ldots + N = \sum_{n=1}^{N} n = \frac{n (n + 1)}{2}$ +  * Photometry 
- +  * [[spectroscopy|Spectroscopy]] 
-====== Sensor Parameters and Linearity Check ====== +  * [[equipment|Equipment]]
- +
-===== ATIK 414EX mono ===== +
- +
-^Gain [e/ADU]^Readout noise [e]^Readout noise [ADU]^Dark current [e/sec]^Fullwell capacity^Dynamic range [steps]^Sensor temperature (median) [celcius]^ +
-|0.261 | 5.687 | 21.781 | 0.044 | 17110.913 | 3008.800 | 0| +
- +
-===== Canon EOS 700D ===== +
-^ISO^Gain [e/ADU]^Readout noise [e]^Readout noise [ADU]^Dark current [e/sec]^Fullwell capacity^Dynamic range [steps]^Sensor temperature (median) [celcius]^ +
-| 100  | 2.2   | 15.783 | 7.173  | 0.113 | 36049.763 | 2284.028 | 29.5 | +
-| 200  | 1.126 | 8.517  | 7.562  | 0.057 | 18451.242 | 2166.487 | 29   | +
-| 400  | 0.567 | 5.204  | 9.180  | 0.017 | 9287.538  | 1784.674 | 28   | +
-| 800  | 0.291 | 3.703  | 12.732 | 0.006 | 4765.202  | 1286.805 | 28   | +
-| 1600 | 0.143 | 3.013  | 21.052 | 0.004 | 2344.487  | 778.228  | 27.5 | +
- +
-====== Equipment ====== +
- +
-===== Telescope ===== +
-^Model^Type^Diameter [mm]^Focal length [mm]^Photographic speed^ +
-| TS-Optics UNC Carbon tube | Newton               | 150 | 600 | f/4   | +
-| TS-Optics Quadruple       | Achromatic Refractor | 65  | 420 | f/6.5 | +
-| TS-Optics Guide Scope     | Achromatic Refractor | 60  | 240 | f/4   | +
- +
-===== Mount ===== +
-^Model^Type^Maximum instrument capacity [kg]^ +
-|Celestron AVX | Equatorial mount | 14 | +
- +
-===== Camera ===== +
-^Model^Pixel size [μm]^Sensor size [mm]^Resolution [pixel]^Sensor^Modification +
-| Canon EOS 700D ((BCF-Filter modified by Baader Planetarium)) | 4.3  | 22.3 x 14.9 | 5184 x 3456| +
-| ATIK 414EX mono | 6.45 | 8.98 x 6.71 | 1391 x 1039| SONY ICX825ALA | +
-| ZWO ASI130mm    | 5.2  | 6.66 x 5.32 | 1280 x 1024| MT9M001C12STM | +
- +
-====== CCD/CMOS Resolution ====== +
-^Camera^Telescope^Resolution [arc seconds / pixel]^FOV [degree]^FOV [arc minutes] +
-| Canon EOS 700D  | TS-Optics UNC Carbon tube | 1.48 | 2.13 x 1.42 | 127.82 x 85.4 | +
-| Canon EOS 700D  | TS-Optics Quadruple       | 2.11 | 3.04 x 2.03 | 182.59 x 122  | +
-| ATIK 414EX mono | TS-Optics UNC Carbon tube | 2.22 | 0.86 x 0.64 | 51.47 x 38.46 | +
-| ATIK 414EX mono | TS-Optics Quadruple       | 3.17 | 1.23 x 0.91 | 73.53 x 54.94 | +
- +
-====== Fitting Gaussian Function with Gradient Descent ====== +
- +
-===== Gaussian Function ===== +
-The Gaussian function is defined as follows +
-\begin{equation} +
-  \label{eq:univariate.gaussian.func} +
-  G(x ; b, I, \mu, \sigma) = b + \frac{I}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{(x - \mu)^2}{2 \sigma^2} \right) +
-\end{equation} +
-where parameter $b \in \mathbb{R}$ denotes the background, that is, the offset from the abscissa. Parameter $I \in \mathbb{R}$ the intensity, the area under the curve to the background $b$, +
-$\mu \in \mathbb{R}$ the mean and $\sigma \in \mathbb{R}$ standard deviation --- sometimes also called the //width//. +
-Note, for $I = 1$ and $b = 0$, term (\ref{eq:univariate.gaussian.func}) is the probability density function of the normal distribution which we denote as $p(x; \mu, \sigma) \widehat{=} G(x; 0, 1, \mu, \sigma)$. +
-For the sake of simplicity we denote $\theta := (b, I, \mu, \sigma)$. +
- +
-===== Fitting Data ===== +
-Giving a sample $\left\{x^{(n)}, y^{(n)}\right\}_{n=1}^N$ of size $N$ where $x^{(n)}, y^{(n)}\in \mathbb{R}$ our goal is to +
-minimize the least square error +
-\begin{eqnarray} +
-  \label{eq:error.function} +
-  E(x) & = & \frac{1}{2 N}\sum_{n=1}^N \left[G(x^{(n)} ; \Theta) - y^{(n)}\right]^2 +
-\end{eqnarray} +
-and thus inferring parameter $\Theta$. +
- +
-===== Derivatives ===== +
-The first order partial derivatives of~(\ref{eq:univariate.gaussian.func}) are +
-\begin{eqnarray} +
-  \label{eq:deriv.1d.I} +
-  \frac{\partial G(x ; \overline{\theta})}{\partial I} & = & \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right) = p(x; \mu, \sigma)\\ +
-  \label{eq:deriv.1d.b} +
-  \frac{\partial G(x ; \overline{\theta})}{\partial b} & = & 1 \\ +
-  \label{eq:deriv.1d.mu} +
-  \frac{\partial G(x ; \overline{\theta})}{\partial \mu} & = & I \, \frac{(x - \mu)}{\sigma^3  \sqrt{2 \pi}} \exp\left(-\frac{(x - \mu)^2}{2  \sigma^2} \right) = I \frac{(x - \mu)}{\sigma^2} \, p(x; \mu, \sigma)\\ +
-  \label{eq:deriv.1d.sigma} +
-  \frac{\partial G(x ; \overline{\theta})}{\partial \sigma} & = & I \, \frac{((x - \mu)^2 - \sigma^2)}{\sigma^4 \sqrt{2  \pi}  } \exp\left(-\frac{(x - \mu)^2}{2  \sigma^2} \right) = I \frac{((x - \mu)^2 - \sigma^2)}{\sigma^3} \, p(x; \mu, \sigma) +
-\end{eqnarray} +
- +
  
  
start.1569272936.txt.gz · Last modified: 2020/12/09 15:30 (external edit)

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