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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.
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| \end{equation}
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< < | |
> > | |
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- 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.
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< < |
- A background of Poissonian distributed random variables with mean 160 pixel values was also added.
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- 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.
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> > |
- The resulting field and its background:
- When run through our floodfill algorithm and it found 47 stars. Following stars were found:
- It removed well shaped stars but left some stars that resemble galaxies. Must fix this.
- Width ratios shown below and calculated using:
- The distances between found centers and generated centers is shown below:
- Largest distance is due to the following star:
- Disrepancy is due to neighbouring star.
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| Skeweness of data
- Last week we saw that the brightest star appeared to be skewed and was not fitted properly.
- Fitted:
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- Here is a zoomed in brightest star in the image just for reference:
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< < |
- Fitted skewed Gaussians to brightest and dimmest stars and compared them to unskewed Gaussians:
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> > | Maximum Likelihood fitting
- The predicted pixel value for a pixel is the following:
- where
is pixel coordinates, is the Gaussian, is the total number of pixel values.
- However, no integration was perfomed, the Gaussian was just calculated over the pixel.
- The maximum likelihood (Poissonian):
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| %BEGINLATEX% |
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< < | \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} ] |
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\end{equation}
%ENDLATEX% |
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> > |
- where
are the maximum number of pixels, is the data pixel value for individual pixel.
- The fitting done using it:
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-- ElenaCukanovaite - 20 Jan 2016 |
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