# Difference: Photometry2016_Simulation (5 vs. 6)

#### Revision 614 Oct 2016 - AaronAndrews

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 META TOPICPARENT name="StewartBoogertPhotometry2016"

### Part One

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• Position of integrated peak bin offset from actual gaussian peak position - could estimate position using relative heights of adjacent bins.
Poisson Distribution of pixel values: Simulation_test4.py.txt
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• Each pixel value consists of a summation of n random variables (e.g. y=x1+...+xn). Each random variable corresponds to the contribution of a particular star (out of n stars in the image).
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• Each pixel value consists of a summation of n random variables (e.g. y=x1+...+xn). Each random variable corresponds to the contribution of a particular star (out of n stars in the image).

• Each random variable follows a Poisson distribution, with a mean value (E[x]) equal to the integrated value of a star's gaussian profile over the pixel area.
• We can assume that these random variables are not correlated (correlation coefficient P=0), therefore the total mean value will be the sum of individual mean values (E[y]=E[x1]+...+E[xn]).
• We can also assume that the summed pixel values (from star contributions only) also follow a Poisson distribution.
• Therefore, we can simulate te 'counting' of photons in each pixel by using the total intgrated pixel value as the mean to gain a random poisson distributed value.
• Example: projection (y=0) of a 16x1 pixel grid with a star generated at x=7.75 (A=100, sigma=2):
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• Test 1: 100x100 pixels, A=100, 10 stars
• Small pixel value range results in very obvious variations in signal data.
• Test 2: 100x100 pixels, A=65535 (pixel value range for camera), 10 stars
• Larger pixel value range results in less obvious variations - almost indistinguishable from earlier gaussian simulation.
-- AaronAndrews - 09 Oct 2016

 META FILEATTACHMENT attachment="PositionSimTest2.png" attr="" comment="random position test (100x100, 50 stars)" date="1476004481" name="PositionSimTest2.png" path="PositionSimTest2.png" size="19417" user="zyva010" version="1"
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 META FILEATTACHMENT attachment="Simulation_test3.py.txt" attr="" comment="Simulation test program 3 - implemented integration" date="1476222934" name="Simulation_test3.py.txt" path="Simulation_test3.py" size="4782" user="zyva010" version="1" attachment="latexc1c95c11b2b3f32a7f6946bc30b25bae.png" attr="h" comment="" date="1476280055" name="latexc1c95c11b2b3f32a7f6946bc30b25bae.png" user="zyva010" version="1" attachment="latex73933a957c41a4144be44ced293ef1e4.png" attr="h" comment="" date="1476280056" name="latex73933a957c41a4144be44ced293ef1e4.png" user="zyva010" version="1"
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 META FILEATTACHMENT attachment="Simulation_test4.py.txt" attr="" comment="Simulation test 4 - added poisson variations" date="1476354741" name="Simulation_test4.py.txt" path="Simulation_test4.py" size="5615" user="zyva010" version="1"
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 META FILEATTACHMENT attachment="Simulation_test4.py.txt" attr="" comment="Simulation test 4 - added poisson variations" date="1476445990" name="Simulation_test4.py.txt" path="Y:\PH4100\Simulation\Simulation_test4.py" size="5613" user="zyva010" version="2"

 META FILEATTACHMENT attachment="PoissonExample1.png" attr="" comment="Example of poisson distributed pixel values on a 16x1 pixel grid" date="1476354830" name="PoissonExample1.png" path="PoissonExample1.png" size="51505" user="zyva010" version="1"