Difference: 20160120MSciPhotometryAnalysisOb1part6 (4 vs. 5)

Revision 522 Jan 2016 - ElenaCukanovaite

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META TOPICPARENT name="StewartBoogertPhotometry2015"

Mode vs. Median Background

  • The problem with using the mode is that if a number does not appear more than once then the mode just takes the first number in the list.
Line: 17 to 17
 \end{equation}

%ENDLATEX%

Changed:
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<
  • Generated 64 stars.
>
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  • Generated 50 stars.
 
  • For each randomly generated star position, a Gaussian was generated with the given centers and randomly choisen parameters from fitting images of Location 1 in focus filter (exposure time: 10 seconds).
  • The resulting Gaussian for one star was calculated over the entire image. This was done for all the stars.
Deleted:
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  • A background of Poissonian distributed random variables with mean 160 pixel values was also added.
 
  • Each pixel was then treated such that its pixel value up to this point was taken as the mean of a Poissonian distribution. A random Poissonian variables was then generated with that mean and this was set as the pixel value for each given pixels.
Added:
>
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  • The resulting field and its background:

log_gen_field.png

 bg.png
  • When run through our floodfill algorithm and it found 47 stars. Following stars were found:
removed_due_to_shape.png
  • It removed well shaped stars but left some stars that resemble galaxies. Must fix this.
  • Width ratios shown below and calculated using:
\begin{equation} \sigma_x = \sqrt{\frac{\sum_{i = 1}^N (x_i - x_{\rm center})^2 p_i}{\sum_{i=1}^N p_i}}\end{equation} \begin{equation} \sigma_y = \sqrt{\frac{\sum_{i = 1}^N (y_i - y_{\rm center})^2p_i}{\sum_{i=1}^N p_i}}\end{equation} <br />

histogram_width_ratios.png

  • The distances between found centers and generated centers is shown below:
distance_fixed.png
  • Largest distance is due to the following star:
distance_star.png
  • Disrepancy is due to neighbouring star.
 

Skeweness of data

  • Last week we saw that the brightest star appeared to be skewed and was not fitted properly.
  • Fitted:
Line: 41 to 58
 
1273_587.png  
  • Here is a zoomed in brightest star in the image just for reference:
zoomed_in_star.png
Changed:
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  • Fitted skewed Gaussians to brightest and dimmest stars and compared them to unskewed Gaussians:
>
>

Maximum Likelihood fitting

  • The predicted pixel value for a pixel is the following:
\begin{equation} \nu_{ij} = n_{\rm{tot}} \int \int_{ij} f(x,y,\vec{\theta}) dx dy \end{equation} <br />
  • where $i,j$ is pixel coordinates, $f(x,y,\vec{\theta})$ is the Gaussian, $n_{tot}$ is the total number of pixel values.
  • However, no integration was perfomed, the Gaussian was just calculated over the pixel.
  • The maximum likelihood (Poissonian):
 %BEGINLATEX%
Changed:
<
<		\begin{equation} f(x,y,\mu_x,\mu_y,\sigma_x,\sigma_y,\rho,a,d) = \frac{a}{2\pi\sigma_x\sigma_y\sqrt{(1-\rho^2)}} exp\Bigg\{-\frac{1}{2(1-\rho^2)}\Bigg[\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)^2 + \bigg(\frac{y_a y^2 + y_b y -\mu_y}{\sigma_y}\bigg)^2 -2\rho\bigg(\frac{x-\mu_x}{\sigma_x}\bigg)\bigg(\frac{y_a y^2 + y_b y -\mu_y}{\sigma_y}\bigg)\Bigg]\Bigg\}
>
>		\begin{equation} \sum^{N, M}_{i,j} [ n_{ij} \ln{\nu_{ij}} - \nu_{ij} ]
  \end{equation}


%ENDLATEX%

Added:
>
>
  • where $N, M$ are the maximum number of $i,j$ pixels, $n_{ij}$ is the data pixel value for individual pixel.
  • The fitting done using it:
unfixed_best.png
  -- ElenaCukanovaite - 20 Jan 2016
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