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<p>Hi, </p>
<p><br>
</p>
<p>yes, it is indeed an interior- / region of interest problem in the sense that we only are interested in the fully illuminated centre region. <br>
I have previously had success in getting good reconstructions from interior data using variational formulation with tailored penalty term. However, this is implemented in Matlab and the aim is to use your implementation to get a C++ implementation of this method. </p>
<p>But for now, I am just figuring how to do a simple reconstruction to get a grasp on the library. </p>
<p><br>
</p>
<p>Best regards, </p>
<p><br>
</p>
<p>Jacob</p>
<p><br>
<br>
</p>
<p><br>
</p>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>Fra:</b> simon.rit@gmail.com <simon.rit@gmail.com> på vegne af Simon Rit <simon.rit@creatis.insa-lyon.fr><br>
<b>Sendt:</b> 16. maj 2017 16:59:12<br>
<b>Til:</b> Jacob Frøsig<br>
<b>Cc:</b> rtk-users@openrtk.org<br>
<b>Emne:</b> Re: [Rtk-users] Reconstructing from fan-beam projections</font>
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<div>Hi,<br>
</div>
If this is true, then the sinogram indicates that you are dealing with an interior problem since you have high line integrals at the border (see enclosed profile). Is that indeed the case? If yes, that's a difficult problem...<br>
</div>
Simon<br>
</div>
<div class="gmail_extra"><br>
<div class="gmail_quote">On Tue, May 16, 2017 at 3:16 PM, Jacob Frøsig <span dir="ltr">
<<a href="mailto:jafs@force.dk" target="_blank">jafs@force.dk</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div link="blue" vlink="purple" lang="DA">
<div class="m_-2204890646387654464WordSection1">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif">Hi,
<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"><u></u> <u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="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 style="font-size:13.0pt;font-family:"Calibri",sans-serif" lang="EN-US">]</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="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.<u></u><u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt;text-align:justify"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">j</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> is the index for the
projections, i.e., <b>j</b> </span><span style="font-size:10.5pt;font-family:"Cambria Math",serif;color:black;background:#f8f9fa" lang="EN-US">∈ {</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">1,2,…,360} since we take
one projection for each whole angle. <u></u><u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">I hope this explains the sinogram and again; thank you for your time!
<u></u><u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Best regards,
<u></u><u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Jacob<u></u><u></u></span></p>
<p class="MsoNormal" style="margin-top:12.0pt"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"><u></u> <u></u></span></p>
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">
<a href="mailto:simon.rit@gmail.com" target="_blank">simon.rit@gmail.com</a> [mailto:<a href="mailto:simon.rit@gmail.com" target="_blank">simon.rit@gmail.com</a>]
<b>On Behalf Of </b>Simon Rit<br>
<b>Sent:</b> 16. maj 2017 14:54<br>
<b>To:</b> Jacob Frøsig <<a href="mailto:jafs@force.dk" target="_blank">jafs@force.dk</a>><br>
<b>Cc:</b> <a href="mailto:rtk-users@openrtk.org" target="_blank">rtk-users@openrtk.org</a></span></p>
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<div class="h5"><br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections<u></u><u></u></div>
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<p></p>
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<div class="h5">
<p class="MsoNormal"><span lang="EN-US"><u></u> <u></u></span></p>
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<p class="MsoNormal">Hi Jacob,<u></u><u></u></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?<u></u><u></u></p>
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<p class="MsoNormal">Thanks,<u></u><u></u></p>
</div>
<p class="MsoNormal">Simon<u></u><u></u></p>
</div>
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<p class="MsoNormal"><u></u> <u></u></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:<u></u><u></u></p>
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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Hi again,</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">and thank you for the example!
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">I have some trouble getting a proper reconstruction. If you have the time, any help would be appreciated.
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">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><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Source to centre: 590 mm</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Source to detector: 1000 mm</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Detector length: 411mm</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Best regards,
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Jacob</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> Rtk-users [mailto:<a href="mailto:rtk-users-bounces@public.kitware.com" target="_blank">rtk-users-bounces@<wbr>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><u></u><u></u></p>
<div>
<div>
<p class="MsoNormal"><br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections<u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US">Hi,</span><u></u><u></u></p>
</div>
<p class="MsoNormal"><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.<wbr>php/FanBeam</span></a><u></u><u></u></p>
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<p class="MsoNormal"><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><u></u><u></u></p>
</div>
<p class="MsoNormal"><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><u></u><u></u></p>
</div>
<p class="MsoNormal"><span lang="EN-US">Best regards,</span><u></u><u></u></p>
</div>
<p class="MsoNormal"><span lang="EN-US">Simon</span><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<p class="MsoNormal"><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><u></u><u></u></p>
<blockquote style="border:none;border-left:solid #cccccc 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Hi again,
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Thank you for the quick answer! An example would be great, thanks. I have one concern about copying the fan-beam data:
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"><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><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Again, thanks!
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Best regards,
</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">Jacob</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">
</span><a href="mailto:simon.rit@gmail.com" target="_blank"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">simon.rit@gmail.com</span></a><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> [mailto:</span><a href="mailto:simon.rit@gmail.com" target="_blank"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">simon.rit@gmail.com</span></a><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">]
<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 style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">ghostcz@hotmail.com</span></a><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">><br>
<b>Cc:</b> Jacob Frøsig <</span><a href="mailto:jafs@force.dk" target="_blank"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">jafs@force.dk</span></a><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">>;
</span><a href="mailto:rtk-users@public.kitware.com" target="_blank"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">rtk-users@public.kitware.com</span></a><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"><br>
<b>Subject:</b> Re: [Rtk-users] Reconstructing from fan-beam projections</span><u></u><u></u></p>
<div>
<div>
<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US">Hi,</span><u></u><u></u></p>
</div>
<p class="MsoNormal"><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><u></u><u></u></p>
</div>
<p class="MsoNormal"><span lang="EN-US">Simon</span><u></u><u></u></p>
</div>
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<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<p class="MsoNormal"><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><u></u><u></u></p>
<blockquote style="border:none;border-left:solid #cccccc 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt">
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<p class="MsoNormal"><span lang="EN-US">Hi,</span><u></u><u></u></p>
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<div id="m_-2204890646387654464m_-3903298939416454343m_-5086680236978975922m_2009876589595851285m_4474899766812565281AppleMailSignature">
<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<div id="m_-2204890646387654464m_-3903298939416454343m_-5086680236978975922m_2009876589595851285m_4474899766812565281AppleMailSignature">
<p class="MsoNormal"><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><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US">Let me know if it helps. </span><u></u><u></u></p>
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<div id="m_-2204890646387654464m_-3903298939416454343m_-5086680236978975922m_2009876589595851285m_4474899766812565281AppleMailSignature">
<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US">Best regards,</span><u></u><u></u></p>
</div>
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<p class="MsoNormal"><span lang="EN-US">Louie<br>
<br>
Sent from my iOS </span><u></u><u></u></p>
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<p class="MsoNormal" style="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><u></u><u></u></p>
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<p class="MsoNormal"><span lang="EN-US">Dear Rtk-users</span><u></u><u></u></p>
<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><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><u></u><u></u></p>
<p class="MsoNormal"><span lang="EN-US">Or, if anyone has experience using e.g. </span>
<span style="font-size:9.5pt;font-family:Consolas;color:#2b91af;background:white" lang="EN-US">FDKConeBeamReconstructionFilte<wbr>r</span><span lang="EN-US"> on projections from line-detectors (fan-beam instead of cone-beam).</span><u></u><u></u></p>
<p class="MsoNormal"><span lang="EN-US"> </span><u></u><u></u></p>
<p class="MsoNormal"><span lang="EN-US">I hope you guys can help.</span><u></u><u></u></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Tahoma",sans-serif" lang="EN-US">Yours sincerely</span><u></u><u></u></p>
<p class="MsoNormal"><b><span style="font-size:10.0pt;font-family:"Tahoma",sans-serif" lang="EN-US">Jacob Frøsig
</span></b><span lang="EN-US"><br>
</span><span style="font-size:7.5pt;font-family:"Tahoma",sans-serif" lang="EN-US"><br>
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