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<h1>Flat-field</h1>
<p class="abstract">
Overview of a general flat-fielding problem and description
of its solution by Munipack.
</p>
<p class="indent">
Munipack implements its own flat-field algorithm on base of
the standard photometry calibration rather than commonly used
methods (they uses of median of scaled flat-fields).
The presented approach enables to reach the maximal possible
accuracy which is limited by only statistical noise of the light.
The approach has been not found in any available literature.
</p>
<h1>Standing on the shoulders of flat-fields</h1>
<p>
Although a correct flat-field is the crucial tool for reaching
of suitable photometry accuracy, the care of acquiring
and processing of flat-fields is not commonly appropriate. This is
especially true for any flat-field post-processing.
</p>
<p class="indent">
A capacity of common semi-conductor detectors is limited
on values, say, 200k counts per pixels.
A good flat-field has its
mean level about half of full range, eg.
100k counts which gives its relative accuracy on value
√100k / 100k ≈ 0.003 per pixel, if <a href="https://en.wikipedia.org/wiki/Poisson_distribution">Poisson distribution</a> can by considered.
So for a star which occupy about
ten pixels (3×3), one will have a relative deviation over 0.01 magnitude
due to the flat-field. A small error in flat-field determination
can lead to measurable deviations of results.
</p>
<p class="indent">
To improve the precision, increase of a capacity of detectors
can help, but it have technical limitations. Also, it will not suppress
different light sensitivity of pixels including all the optical path.
In this case, the feasible way is averaging of frames as provides
Munipack <samp>flat</samp> utility.
</p>
<h1>The flat-fielding mystery revealing</h1>
<p>
Mean levels of flat-fields, captured using of an unstable light source
(during twilight), are unequal.
As a consequence, a direct average of that flat-fields
is impossible.
</p>
<p class="indent">
A common solution of the trouble is an initial normalisation of flat-fields
on an equal intensity level.
The approach key difficulty is the determination of a mean level of
every frame.
Single frame values has no
<a href="https://en.wikipedia.org/wiki/Normal_distribution">Normal</a>
distribution
which is leading to a staticticaly poor definition of the average level.
</p>
<!--
<p>
There are summarised key ideas of flat-field algorithm
implemented in Munipack.
</p>
-->
<figure>
<div class="twocolumn">
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<img class="figure" src="flatzero.png" alt="inital flat" title="inital flat">
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<div class="column">
<img class="figure" src="flatdebug_zero.svg" alt="Histogram" title="Histogram">
</div>
</div>
<figcaption>
A distribution of values of flat-field shows an asymmetric
non-Gaussian histogram
</figcaption>
</figure>
<p class="indent">
The difficulty comes due to the folded surface of flat-fields.
While it is possible to compute a mean level, the estimate
will not be optimal or accurate due to blending of statistical
distributions: the light noise and the surface of flat-field
itself.
</p>
<p class="indent">
The crucial point of Munipack approach is decomposition
of flat-field frames on single, independent pixels.
These pixels, having the same position and collected
over all frames, can be considered as sources of the light
like stars; a star calibration procedure can be subsequenty used.
Reference sources are initially unknown,
fortunately, they can be estimated by iterations.
</p>
<figure>
<img class="figure" src="wrinkledflat.svg" alt="wrinkled flat"
title="wrinkled flat" style="max-width:90%;">
<figcaption>
A blended distribution of values as result of a folded surface
of a flat-field
</figcaption>
</figure>
<p class="indent">
Munipack implements two-phase algorithm which determine a rough
flat-field during the first phase (equivalent to common
practice). In the second phase, it determines
the mean level of individual frames by scaling on to
the rough initial flat approximation.
The second phase of the approach works with Normally
distributed data giving accurate, reliable and statistically correct
results.
</p>
<figure>
<div class="twocolumn">
<div class="column">
<img class="figure" src="flatfine.png" alt="final flat" title="final flat">
</div>
<div class="column">
<img src="flatdebug_3.svg" alt="Histogram" title="Histogram">
</div>
</div>
<figcaption>
The final flat-field accepting folded (wrinkled) property. Resultant
histogram of residuals of an single frame is near of Normal distribution
(some per frame deviations are visible).
</figcaption>
</figure>
<p class="indent">
The developed algorithm solves a non-linear implicit equation
for all pixels of the final flat-field.
The approach is a variant of photon calibration,
the reference photon sources are iteratively established
during the computation.
</p>
<h1>Flat-fielding rules</h1>
<p>
There is a list of rules,
summarising of my long time experiences with flat-fielding,
which I recommends for flat-fielding:
</p>
<ul>
<li>The flat-field frame is very important, because
a quality of the flat-field determines photometry accuracy
of results.</li>
<li>Only twilight flat-frames are acceptable (any light gadgets
gives poor results – , if not agree, please send me some
counter-example).</li>
<li>The twilight flats can be acquired with this rules in mind:
<ul>
<li>catch its only on the clear sky, on the sky without any clouds,</li>
<li>any light pollution by an artificial light or Moon is
unacceptable,</li>
<li>use opposite directions on Sun,</li>
<li>choose fields in Earth's shadow,</li>
<li>acquire on field not too close to horizon,</li>
<li>keep <a href="https://en.wikipedia.org/wiki/Rayleigh_sky_model">polarisation</a>
due to Sun on minimum,
</li>
<li>select a field with sparse and faint stars,</li>
<li>switch-off sidereal motion of telescope mount's,</li>
<li>use shorter minimal exposure to prevent changes of illumination
during the twilight (but not such short to capture
shutter speed),</li>
<li>prefer levels of flat-fields around half of its full range (capacity).
</ul>
</li>
</ul>
<h2>See Also</h2>
<p>
<a href="man_flat.html">Flat-field manual</a>,
<a href="man_phcorr.html">Photometry corrections</a>.
<a href="https://en.wikipedia.org/wiki/Standing_on_the_shoulders_of_giants">Standing on the shoulders of giants</a>.
</p>
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