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<!DOCTYPE HTML>
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<title>Munipack ‒ Flat-field</title>
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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
  precision which is limited by only statistical noise of light.
  The approach is unique and  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
  suitable photometry precision of observations, the care of acquiring
  and processing of flat-fields is not 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, 100k counts per pixels.
  A good flat-field has its
  mean level about 50k counts which gives its relative precision on value
  √50k / 50k ≈ 0.004 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 precision over 0.01 magnitude
  due to the flat-field. A small error in flat-field determination
  leads to measurable deviations of results.
</p>

<p class="indent">
  To improve the precision, some 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 normalisation of flat-fields
  on an unique intensity level preparatory to an averaging.
  The problem of the
  approach is determination of a mean level of every frame.
  Its values has no
  <a href="https://en.wikipedia.org/wiki/Normal_distribution">Normal</a>
  distribution
  which is leading to a poor definition of the average level.
</p>

<!--
<p>
  There are summarised key ideas of flat-field algorithm
  implemented in Munipack.
</p>
-->

<figure>
  <table>
    <tr>
      <td class="blank" style="width:50%">
	<img class="figure" src="flatzero.png" alt="inital flat" title="inital flat">
      </td>
      <td class="blank" style="width:50%">
	<img src="flatdebug_zero.svg" alt="Histogram" title="Histogram"
	     style="width:150%">
      </td>
    </tr>
  </table>
  <figcaption>
    A distribution of values of flat-field shows an asymmetric histogram
  </figcaption>
</figure>

<p class="indent">
  The main 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, with the same position but collected
  over all frames, can be considered as sources of light
  like stars and similar procedure as the star calibration can be used.
  Reference sources are initially unknown,
  but 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 is using two-phase algorithm which determine a rough
  flat-field during the first phase (equivalent to common
  practice). The second phase determines
  the mean level against to the rough flat followed by averaging.
  The approach makes the second phase to be work with Normally
  distributed data giving precise and reliable results.
</p>


<figure>
<table>
<tr>
  <td class="blank" style="width:50%">
<img class="figure" src="flatfine.png" alt="final flat" title="final flat">
</td>
<td class="blank" style="width:50%">
  <img src="flatdebug_3.svg" alt="Histogram" title="Histogram"
       style="width:150%">
  </td>
</tr>
</table>

<figcaption>
  The final flat-field accepting folded (wrinkled) property. Resultant
  histogram of residuals of an single frame is near Normal distribution
  (only on per frame basis deviations).
</figcaption>

</figure>

<p class="indent">
  The developed algorithm solves a non-linear implicit equation
  for both levels and all pixels of the resultan flat-field.
  The approach is a variant of photon calibration where
  the reference photon sources are iterative 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 precision
    of results.</li>
  <li>Only twilight flat-frames are acceptable (any light gadgets
    gives only poor results – please send  me counter-example).</li>
  <li>The twilight flats can be acquired with this rules in mind:
    <ul>
      <li>acquire only on clear sky without any clouds,</li>
      <li>additional 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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