[gnuastro-commits] master 5d2f8d45: Book: improved photon-counting noise description with analogy

[Mohammad Akhlaghi]

branch: master
commit 5d2f8d450c605362215007bd3f21553e73d20bf5
Author: Zahra Hosseini Shahisavandi <2hs.zahra@gmail.com>
Commit: Mohammad Akhlaghi <mohammad@akhlaghi.org>

    Book: improved photon-counting noise description with analogy
    
    Until now, there wasn't any description for adding an appropriate value of
    background when we used in '--background' options of 'astmkprof'. The
    connection of the background value to the photon-counting noise was also
    not very well elaborated in that section!
    
    With this commit, we bring up an analogy in the "Photon counting noise"
    subsection of the "Noise basics" section: comparing the background value to
    the depth of a sea (such that the height of waves/noise corresponds to the
    depth of the sea). We also explain the importance of background value in
    understanding this source of noise. For this aim, we further explain the
    relation between the background value and the free parameters of the
    noise's Gaussian distribution.
    
    This was done with the help of Sepideh Eskandarlou.
    
    The text was slightly edited by Mohammad Akhlaghi upon merging into the
    'master' branch of Gnuastro.
---
 doc/gnuastro.texi | 66 ++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 46 insertions(+), 20 deletions(-)

diff --git a/doc/gnuastro.texi b/doc/gnuastro.texi
index ff08a936..42a893a6 100644
--- a/doc/gnuastro.texi
+++ b/doc/gnuastro.texi
@@ -1913,6 +1913,7 @@ All this trouble was certainly worth it because now there 
is no dimming on the e
 
 The final step to simulate a real observation would be to add noise to the 
image.
 Sufi set the zero point magnitude to the same value that he set when making 
the mock profiles and looking again at his observation log, he had measured the 
background flux near the nebula had a magnitude of 7 that night.
+For more on how the background value deterimnes the noise, see @ref{Noise 
basics}.
 So using these values he ran MakeNoise:
 
 @example
@@ -21751,9 +21752,9 @@ Therefore in this section, we'll also discuss how you 
can set these two paramete
 @cindex Photon counting noise
 @cindex Poisson, Sim@'eon Denis
 With the very accurate electronics used in today's detectors, photon counting 
noise@footnote{In practice, we are actually counting the electrons that are 
produced by each photon, not the actual photons.} is the most significant 
source of uncertainty in most datasets.
-To understand this noise (error in counting), we need to take a closer look at 
how a distribution produced by counting can be modeled as a parametric function.
+To understand this noise (error in counting) and its effect on the images of 
astronomical targets, let's start by reviewing how a distribution produced by 
counting can be modeled as a parametric function.
 
-Counting is an inherently discrete operation, which can only produce positive 
(including zero) integer outputs.
+Counting is an inherently discrete operation, which can only produce positive 
integer outputs (including zero).
 For example we can't count @mymath{3.2} or @mymath{-2} of anything.
 We only count @mymath{0}, @mymath{1}, @mymath{2}, @mymath{3} and so on.
 The distribution of values, as a result of counting efforts is formally known 
as the @url{https://en.wikipedia.org/wiki/Poisson_distribution, Poisson 
distribution}.
@@ -21762,30 +21763,31 @@ Therefore some people suggest it should rightly be 
called the de Moivre distribu
 
 @cindex Probability density function
 Let's take @mymath{\lambda} to represent the expected mean count of something.
-Furthermore, let's take @mymath{k} to represent the result of one particular 
counting attempt.
+Furthermore, let's take @mymath{k} to represent the output of a counting 
attempt (hence @mymath{k} is a positive integer).
 The probability density function of getting @mymath{k} counts (in each 
attempt, given the expected/mean count of @mymath{\lambda}) can be written as:
 
 @cindex Poisson distribution
 @dispmath{f(k)={\lambda^k \over k!} e^{-\lambda},\quad k\in @{0, 1, 2, 3, 
\dots @}}
 
 @cindex Skewed Poisson distribution
-Because the Poisson distribution is only applicable to positive values (note 
the factorial operator, which only applies to non-negative integers), naturally 
it is very skewed when @mymath{\lambda} is near zero.
-One qualitative way to understand this behavior is that there simply aren't 
enough integers smaller than @mymath{\lambda}, than integers that are larger 
than it.
-Therefore to accommodate all possibilities/counts, it has to be strongly 
skewed when @mymath{\lambda} is small.
+Because the Poisson distribution is only applicable to positive integer values 
(note the factorial operator, which only applies to non-negative integers), 
naturally it is very skewed when @mymath{\lambda} is near zero.
+One qualitative way to understand this behavior is that for smaller values 
near zero, there simply aren't enough integers smaller than the mean, than 
integers that are larger.
+Therefore to accommodate all possibilities/counts, it has to be strongly 
skewed to the positive when the mean is small.
+For more on Skewness, see @ref{Skewness cased by signal and its measurement}.
 
 @cindex Compare Poisson and Gaussian
-As @mymath{\lambda} becomes larger, the distribution becomes more and more 
symmetric.
-A very useful property of the Poisson distribution is that the mean value is 
also its variance.
-When @mymath{\lambda} is very large, say @mymath{\lambda>1000}, then the 
@url{https://en.wikipedia.org/wiki/Normal_distribution, Normal (Gaussian) 
distribution}, is an excellent approximation of the Poisson distribution with 
mean @mymath{\mu=\lambda} and standard deviation @mymath{\sigma=\sqrt{\lambda}}.
-In other words, a Poisson distribution (with a sufficiently large 
@mymath{\lambda}) is simply a Gaussian that only has one free parameter 
(@mymath{\mu=\lambda} and @mymath{\sigma=\sqrt{\lambda}}), instead of the two 
parameters (independent @mymath{\mu} and @mymath{\sigma}) that it originally 
has.
+As @mymath{\lambda} becomes larger, the distribution becomes more and more 
symmetric, and the variance of that distribution is equal to its mean.
+In other words, the standard deviation is the square root of the mean.
+It can also be proved that when the mean is large, say @mymath{\lambda>1000}, 
the Poisson distribution approaches the 
@url{https://en.wikipedia.org/wiki/Normal_distribution, Normal (Gaussian) 
distribution} with mean @mymath{\mu=\lambda} and standard deviation 
@mymath{\sigma=\sqrt{\lambda}}.
+In other words, a Poisson distribution (with a sufficiently large 
@mymath{\lambda}) is simply a Gaussian that has one free parameter 
(@mymath{\mu=\lambda} and @mymath{\sigma=\sqrt{\lambda}}), instead of the two 
parameters that the Gaussian distribution originally has (independent 
@mymath{\mu} and @mymath{\sigma}).
 
 @cindex Sky value
 @cindex Background flux
 @cindex Undetected objects
-In real situations, the photons/flux from our targets are added to a certain 
background flux (observationally, the @emph{Sky} value).
+In real situations, the photons/flux from our targets are combined with 
photons from a certain background (observationally, the @emph{Sky} value).
 The Sky value is defined to be the average flux of a region in the dataset 
with no targets.
 Its physical origin can be the brightness of the atmosphere (for ground-based 
instruments), possible stray light within the imaging instrument, the average 
flux of undetected targets, etc.
-The Sky value is thus an ideal definition, because in real datasets, what lies 
deep in the noise (far lower than the detection limit) is never 
known@footnote{In a real image, a relatively large number of very faint objects 
can been fully buried in the noise and never detected.
+The Sky value is thus an ideal definition, because in real datasets, what lies 
deep in the noise (far lower than the detection limit) is never 
known@footnote{In a real image, a relatively large number of very faint objects 
can be fully buried in the noise and never detected.
 These undetected objects will bias the background measurement to slightly 
larger values.
 Our best approximation is thus to simply assume they are uniform, and consider 
their average effect.
 See Figure 1 (a.1 and a.2) and Section 2.2 in 
@url{https://arxiv.org/abs/1505.01664, Akhlaghi and Ichikawa [2015]}.}.
@@ -21794,24 +21796,48 @@ In a mock image/dataset, we have the luxury of 
setting the background (Sky) valu
 
 @cindex Simulating noise
 @cindex Noise simulation
-In each element of the dataset (pixel in an image), the flux is the sum of 
contributions from various sources (after convolution by the PSF, see 
@ref{PSF}).
-Let's name the convolved sum of possibly overlapping objects, @mymath{I_{nn}}.
-@mymath{nn} representing `no noise'.
+In summary, the value in each element of the dataset (pixel in an image) is 
the sum of contributions from various galaxies and stars (after convolution by 
the PSF, see @ref{PSF}).
+Let's name the convolved sum of possibly overlapping objects in each pixel as 
@mymath{I_{nn}}.
+@mymath{nn} represents `no noise'.
 For now, let's assume the background (@mymath{B}) is constant and sufficiently 
high for the Poisson distribution to be approximated by a Gaussian.
-Then the flux after adding noise is a random value taken from a Gaussian 
distribution with the following mean (@mymath{\mu}) and standard deviation 
(@mymath{\sigma}):
+Then the flux of that pixel, after adding noise, is @emph{a random value} 
taken from a Gaussian distribution with the following mean (@mymath{\mu}) and 
standard deviation (@mymath{\sigma}):
 
 @dispmath{\mu=B+I_{nn}, \quad \sigma=\sqrt{B+I_{nn}}}
 
+@cindex Bias level in detectors
+@cindex Dark level in detectors
+In astronomical instruments, @mymath{B} is enhanced by adding a ``bias'' level 
to each pixel before the shutter is even opened (for the exposure to start).
+As the exposure is ongoing and photo-electrons are accumulating from the 
astronomical objects, a ``dark'' current (due to thermal radiation of the 
instrument) also builds up in the pixels.
+The ``dark'' current will accumulate even when the shutter is closed, but the 
CCD electronics are working (hence the name ``dark'').
+This added dark level further enhaces the mean value in a real observation 
compared to the raw background value (from the atmosphere for example).
+
 Since this type of noise is inherent in the objects we study, it is usually 
measured on the same scale as the astronomical objects, namely the magnitude 
system, see @ref{Brightness flux magnitude}.
 It is then internally converted to the flux scale for further processing.
 
+The equations above clearly show the importance of the background value and 
its effect on the final signal to noise ratio in each pixel of a science image.
+It is therefore, one of the most important factors in understanding the noise 
(and properly simulating observations where necessary).
+An inappropriately bright background value can hide the signal of the mock 
profile hide behind the noise.
+In other words, a brighter background has larger standard deviation and vice 
versa.
+As a result, the only necessary parameter to define photon-counting noise over 
a mock image of simulated profiles is the background.
+For a complete example, see @ref{Sufi simulates a detection}.
+
+To better understand the correlation between the mean (or background) value 
and the noise standard deviation, let's use an analogy.
+Consider the profile of your galaxy to be analogous to the profile of a ship 
that is sailing in the sea.
+The height of the ship would therefore be analogous to the maximum flux 
difference between your galaxy's minimum and maximum values.
+Furthermore, let's take the depth of the sea to represent the background 
value: a deeper sea, corresponds to a brighter background.
+In this analogy, the ``noise'' would be the height of the waves that surround 
the ship: in deeper waters, the waves would also be taller (the squre root of 
the mean depth at the ship's position).
+
+If the ship is in deep waters, the height of waves are greater than when the 
ship is near to the beach (at lower depths).
+Therefore, when the ship is in the middle of the sea, there are high waves 
that are capable of hiding a significant part of the ship from our perspective.
+This corresponds to a brighter background value in astronomical images: the 
resulting noise from that brighter background can completely wash out the 
signal from a fainter galaxy, star or solar system object.
+
 @node Instrumental noise, Final noised pixel value, Photon counting noise, 
Noise basics
 @subsubsection Instrumental noise
 
 @cindex Readout noise
 @cindex Instrumental noise
 @cindex Noise, instrumental
-While taking images with a camera, a dark current is fed to the pixels, the 
variation of the value of this dark current over the pixels, also adds to the 
final image noise.
+While taking images with a camera, a bias current is fed to the pixels, the 
variation of the value of this bias current over the pixels, also adds to the 
final image noise.
 Another source of noise is the readout noise that is produced by the 
electronics in the detector.
 Specifically, the parts that attempt to digitize the voltage produced by the 
photo-electrons in the analog to digital converter.
 With the current generation of instruments, this source of noise is not as 
significant as the noise due to the background Sky discussed in @ref{Photon 
counting noise}.
@@ -21955,11 +21981,11 @@ This is done for future reproducibility.
 
 @item -b FLT
 @itemx --background=FLT
-The background value (per pixel) that will be added to each pixel value 
(internally) to estimate Poisson noise, see @ref{Photon counting noise}.
+The background value (per pixel) that will be added to each pixel value 
(internally) to simulate Poisson noise, see @ref{Photon counting noise}.
 By default the units of this value are assumed to be in magnitudes, hence a 
@option{--zeropoint} is also necessary.
-But if the background is in units of brightness, you need add 
@option{--bgisbrightness}, see @ref{Brightness flux magnitude}
+If the background is in units of counts, you need add 
@option{--bgisbrightness}, see @ref{Brightness flux magnitude}
 
-Internally, the value given to this option will be converted to brightness 
(@mymath{b}, when @option{--bgisbrightness} is called, the value will be used 
directly).
+Internally, the value given to this option will be converted to counts 
(@mymath{b}, when @option{--bgisbrightness} is called, the value will be used 
directly).
 Assuming the pixel value is @mymath{p}, the random value for that pixel will 
be taken from a Gaussian distribution with mean of @mymath{p+b} and standard 
deviation of @mymath{\sqrt{p+b}}.
 With this option, the noise will therefore be dependent on the pixel values: 
according to the Poission noise model, as the pixel value becomes larger, its 
noise will also become larger.
 This is thus a realistic way to model noise, see @ref{Photon counting noise}.