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<div class="section" id="merging-of-the-conventions">
<h1>Merging Of The Conventions<a class="headerlink" href="#merging-of-the-conventions" title="Permalink to this headline">ΒΆ</a></h1>
<p>The full implementation of all these elements ends up merging the SIP, DET2IM and FITS Distortion Paper
conventions to create a new version of the figure from the FITS Distortion Paper which illustrates the conversion
of detector coordinates to world coordinates. This implementation works in the following way:</p>
<blockquote>
<div><ol class="arabic simple">
<li>Apply detector to image correction (DET2IM) to input pixel values</li>
<li>Apply SIP coefficients to DET2IM-corrected pixel values</li>
<li>Apply lookup table correction to DET2IM-corrected pixel values</li>
<li>Add the results of the SIP and lookup table corrections</li>
<li>Apply the WCS transformation in the CD matrix to the summed results to get the intermediate world coordinates</li>
<li>Add the CRVAL keyword values to the transformed positions to get the final world coordinates</li>
</ol>
</div></blockquote>
<p>The computations to perform these steps can be described approximately using:</p>
<div class="math" id="equation-Equation 2">
<p><span class="eqno">(1)</span><span class="math">(x',y') &= DET2IM(x,y)</span></p>
</div><div class="math" id="equation-Equation 3">
<p><span class="eqno">(2)</span><span class="math">\binom{u'}{v'} &= \binom{x' - CRPIX1}{y' - CRPIX2}</span></p>
</div><div class="math" id="equation-Equation 4">
<span id="equation4"></span><p><span class="eqno">(3)</span><span class="math">\left( \begin{array}{ll}
\alpha \\
\delta \\
\end{array} \right) &=
\left( \begin{array}{ll}
CRVAL1 \\
CRVAL2\\
\end{array} \right) +
\left( \begin{array}{cc}
CD11 & CD12 \\
CD21 & CD22\\
\end{array} \right)
\left( \begin{array}{ll}
u' + f(u',v') + LT_x(x',y') \\
v' + g(u',v') + LT_y(x',y') \\
\end{array} \right)</span></p>
</div><p>where f(u’,v’) and g(u’,v’) represent the polynomial distortion correction specified as</p>
<div class="math" id="equation-Equation 5">
<p><span class="eqno">(4)</span><span class="math">f(u',v') = \sum_{p+q=2}^{AORDER} A_{pq} {u'}^{p} {v'}^{q}
\\
g(u',v') = \sum_{p+q=2}^{BORDER} B_{pq} {u'}^{p} {v'}^{q}</span></p>
</div><p>where</p>
<ul class="simple">
<li>x’, y’ are the initial coordinates x,y with the 68th column correction applied
through the DET2IM convention</li>
<li>u’,v’ are the DET2IM-corrected coordinates relative to CRPIX1,CRPIX2</li>
<li><span class="math">LT_{x}, LT_{y}</span> is the residual distortion in the lookup tables
written to the header using the FITS Distortion Paper lookup table convention</li>
<li>A, B are the SIP coefficients specified using the SIP convention</li>
</ul>
<p>These equations do not take into account the deprojection from the tangent plane to
sky coordinates. The complete Detector To Sky Coordinate Transformation is based on
the CTYPE keyword.</p>
<div class="figure" id="figure3">
<img alt="_images/pipeline.png" src="_images/pipeline.png" />
<p class="caption">Coordinate Transformation Pipeline</p>
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