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<p class="MsoNormal"><span style="font-size:11.0pt">Hello!<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">I have some questions related to this post from 2015:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><a href="https://public.kitware.com/pipermail/rtk-users/2015-February/009750.html">https://public.kitware.com/pipermail/rtk-users/2015-February/009750.html</a><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">First, the referenced website “<a href="http://www.openrtk.org/Doxygen/classrtk_1_1NormalizedJosephBackProjectionImageFilter.html">http://www.openrtk.org/Doxygen/classrtk_1_1NormalizedJosephBackProjectionImageFilter.html</a>”
seems to not exist anymore. <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Second, I have some issues concerning backprojection and adjointness:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">For the following I am using these parameters:
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">sid = 500<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">sdd = 1000<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">n_projections = 4<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">angular_increment = 90.0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">I am using the python version of ITK and tried forward projection (CUDAForwardProjectionImageFilter) followed by backprojection (JosephBackProjectionImageFilter/CudaBackProjectionImageFilter/CudaRayCastBackProjectionImageFilter)
of an image with a single pixel value set to one:<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">img = np.zeros(image_size)<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">img[image_size[0] // 2, image_size[1] // 2, image_size[2] // 2] = 1.0<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<ol style="margin-top:0cm" start="1" type="1">
<li class="MsoListParagraph" style="margin-left:0cm;mso-list:l0 level1 lfo1"><span style="font-size:11.0pt">I do not get expected results for the 2D case:<o:p></o:p></span></li></ol>
<p class="MsoListParagraph"><span style="font-size:11.0pt">image_size = [3, 1, 3]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">pixel_size_mm = 1.0<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">slice_thickness_mm = 1.0<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">projection_size = [5, 1]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">projection_spacing = [1.0, 1.0]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">After forward projection, the backprojected image by JosephBackProjectionImageFilter is all
<b>zeros</b>. Is this expected behavior?<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<ol style="margin-top:0cm" start="2" type="1">
<li class="MsoListParagraph" style="margin-left:0cm;mso-list:l0 level1 lfo1"><span style="font-size:11.0pt">According to the definition of adjoint operator, the following should hold:
<o:p></o:p></span></li></ol>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><Ax, y> = <x, adjoint(A)y> (with <.,.> indicating the scalar product)<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">This works for 3D <b>CPU</b> case (JosephBackProjectionImageFilter):<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">image_size = [3, 3, 3]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">pixel_size_mm = 1.0<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">slice_thickness_mm = 1.0<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">projection_size = [5, 5]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">projection_spacing = [1.0, 1.0]<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"><Ax, y> = <fp, fp> = 9.0<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"><x, adjoint(A)y> = <img, bp> = 9.000001907348633<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">This discrepancy is within tolerance.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<ol style="margin-top:0cm" start="3" type="1">
<li class="MsoListParagraph" style="margin-left:0cm;mso-list:l0 level1 lfo1"><span style="font-size:11.0pt">However, in the 3D
<b>GPU</b> case:<o:p></o:p></span></li></ol>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><x, adjoint(A)y> = <img, bp> = 4.0 for CudaBackProjectionImageFilter<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt">and<o:p></o:p></span></p>
<p class="MsoListParagraph"><span style="font-size:11.0pt"><x, adjoint(A)y> = <img, bp> = 0.33841705322265625 for CudaRayCastBackProjectionImageFilter<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">What kind of filtering or normalization has been used on the projections/images? I could not verify adjointment with the aforementioned method of backprojecting a sinogram of ones
and applying division.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt">Is there a GPU backprojection filter for which adjointness holds?<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Thank you in advance for your reply and suggestions.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Best regards, Laura<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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