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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">Hi,
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">The attached sinogram consist of line integral values, i.e., sinogram(i,j) = -ln[
<b>I(i,j)/I_0(i)</b> </span><span lang="EN-US" style="font-size:13.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">]</span><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US"> where
<b>I </b>and <i></i><b>I_0 </b>are the measured intensities with and without an object, respectively.
<br>
Since we are measuring with a line detector, the “zero” (<b>I_0</b>) is in this case just an array.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt;text-align:justify"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">j</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">
is the index for the projections, i.e., <b>j</b> </span><span lang="EN-US" style="font-size:10.5pt;font-family:"Cambria Math",serif;color:black;background:#F8F9FA">∈ {</span><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">1,2,…,360}
since we take one projection for each whole angle. <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">I hope this explains the sinogram and again; thank you for your time!
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">Best regards,
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US">Jacob<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> simon.rit@gmail.com [mailto:simon.rit@gmail.com]
<b>On Behalf Of </b>Simon Rit<br>
<b>Sent:</b> 16. maj 2017 14:54<br>
<b>To:</b> Jacob Frøsig <jafs@force.dk><br>
<b>Cc:</b> rtk-users@openrtk.org<br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal">Hi Jacob,<o:p></o:p></p>
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<p class="MsoNormal">Thanks for the image of the sinogram. I don't understand this sinogram. Do you know how to convert each pixel value to a line integral? I.e., if it's x-ray imaging, do you have an image without object?<o:p></o:p></p>
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<p class="MsoNormal">Thanks,<o:p></o:p></p>
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<p class="MsoNormal">Simon<o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">On Tue, May 16, 2017 at 2:24 PM, Jacob Frøsig <<a href="mailto:jafs@force.dk" target="_blank">jafs@force.dk</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Hi again,</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">and thank you for the example!
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">I have some trouble getting a proper reconstruction. If you have the time, any help would be appreciated.
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Attached is my fan-beam sinogram as a .tif file. Here, projections are taken for 360 degrees and the
detector has 507 pixels. <br>
I believe the issue lies in setting the geometry and maybe the spacing between the pixels. The measurement geometry is as follows</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Source to centre: 590 mm</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Source to detector: 1000 mm</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Detector length: 411mm</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Best regards,
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Jacob</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> Rtk-users
[mailto:<a href="mailto:rtk-users-bounces@public.kitware.com" target="_blank">rtk-users-bounces@public.kitware.com</a>]
<b>On Behalf Of </b>Simon Rit<br>
<b>Sent:</b> 15. maj 2017 17:32<br>
<b>To:</b> <a href="mailto:rtk-users@openrtk.org" target="_blank">rtk-users@openrtk.org</a></span><o:p></o:p></p>
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<p class="MsoNormal"><br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections<o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Hi,</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">I have added an example here:<br>
</span><a href="http://wiki.openrtk.org/index.php/FanBeam" target="_blank"><span lang="EN-US">http://wiki.openrtk.org/index.php/FanBeam</span></a><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">The values in the first and the third rows are not used, as illustrated in this example. We add them to do a 2D interpolation but we do this 2D interpolation
exactly on the second row so that's not a problem.</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Note that it's a bit more tricky for iterative recon but we also have a solution for this if you need it.</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Best regards,</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Simon</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">On Mon, May 15, 2017 at 4:31 PM, Jacob Frøsig <</span><a href="mailto:jafs@force.dk" target="_blank"><span lang="EN-US">jafs@force.dk</span></a><span lang="EN-US">>
wrote:</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Hi again,
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Thank you for the quick answer! An example would be great, thanks. I have one concern about copying
the fan-beam data: </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"><br>
Then we would have 2D projections for which each column is the same fan-beam projection. By this, values of the same row are equal even though the cone-beam geometry indicates the outer columns correspond to intensities of rays with longer travel time through
the object and hence should have attenuated more. Is this neglectable? </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Again, thanks!
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Best regards,
</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">Jacob</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">
</span><a href="mailto:simon.rit@gmail.com" target="_blank"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">simon.rit@gmail.com</span></a><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"> [mailto:</span><a href="mailto:simon.rit@gmail.com" target="_blank"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">simon.rit@gmail.com</span></a><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">]
<b>On Behalf Of </b>Simon Rit<br>
<b>Sent:</b> 15. maj 2017 16:05<br>
<b>To:</b> louie L <</span><a href="mailto:ghostcz@hotmail.com" target="_blank"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">ghostcz@hotmail.com</span></a><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">><br>
<b>Cc:</b> Jacob Frøsig <</span><a href="mailto:jafs@force.dk" target="_blank"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">jafs@force.dk</span></a><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">>;
</span><a href="mailto:rtk-users@public.kitware.com" target="_blank"><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif">rtk-users@public.kitware.com</span></a><span lang="EN-US" style="font-size:11.0pt;font-family:"Calibri",sans-serif"><br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Hi,</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">I agree. What I do is do a 3 slice sinogram from the fan-beam projections by copying the same data in each slice but reconstruct one 2D slice. If you need an
example, I can quickly demonstrate this in a short Python script.</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Simon</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">On Mon, May 15, 2017 at 4:01 PM, louie L <</span><a href="mailto:ghostcz@hotmail.com" target="_blank"><span lang="EN-US">ghostcz@hotmail.com</span></a><span lang="EN-US">>
wrote:</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Hi,</span><o:p></o:p></p>
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<div id="m_-3903298939416454343m_-5086680236978975922m_2009876589595851285m_4474899766812565281AppleMailSignature">
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">I think you can carefully pad zeros to your projections. Reconstruct as if it is a 3d object. Take the middle slice as your fanbeam result. </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Let me know if it helps. </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
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<div id="m_-3903298939416454343m_-5086680236978975922m_2009876589595851285m_4474899766812565281AppleMailSignature">
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Best regards,</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Louie<br>
<br>
Sent from my iOS </span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;margin-bottom:12.0pt"><span lang="EN-US"><br>
Am 15.05.2017 um 15:55 schrieb Jacob Frøsig <</span><a href="mailto:jafs@force.dk" target="_blank"><span lang="EN-US">jafs@force.dk</span></a><span lang="EN-US">>:</span><o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Dear Rtk-users</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">I was wandering if Rtk includes an implementation of a reconstruction method to reconstruct a 2D representation of an object given fan-beam projections.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">Or, if anyone has experience using e.g.
</span><span lang="EN-US" style="font-size:9.5pt;font-family:Consolas;color:#2B91AF;background:white">FDKConeBeamReconstructionFilter</span><span lang="EN-US"> on projections from line-detectors (fan-beam instead of cone-beam).</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US">I hope you guys can help.</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span lang="EN-US" style="font-size:10.0pt;font-family:"Tahoma",sans-serif">Yours sincerely</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><b><span lang="EN-US" style="font-size:10.0pt;font-family:"Tahoma",sans-serif">Jacob Frøsig
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