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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;" class="elementToProof">Hi, we're having a slight issue with something in RTK 2.0.1 and are wondering about modifying the epsilon value in
</span><span style="font-family: Consolas, Courier, monospace;" class="elementToProof"><span style="background-color: rgb(255, 255, 255); font-family: Consolas, Courier, monospace; display: inline !important; font-size: 12pt;">rtk::ThreeDCircularProjectionGeometry::VerifyAngles()</span></span><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;" class="elementToProof">.
This is a bit cumbersome to describe so I'll go slow and kinda walk through it.</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;" class="elementToProof">We have some software that iterates source/detector positions for DRR generation via the CUDA forward projector.</span></div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;" class="elementToProof">For certain combinations of source/detector positions/orientations, rtk::ThreeDCircularProjectionGeometry::AddProjection() fails because rtk::ThreeDCircularProjectionGeometry::VerifyAngles()
fails and then subsequently rtk::ThreeDCircularProjectionGeometry::FixAngles() also fails. Ultimately, this all ends with a failure to generate our DRR. At first I thought maybe our detector vectors weren't orthonormal or something, but even after orthonormalizing
them we get the same result.</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;" class="elementToProof">Here's some sample inputs to </span><span style="font-family: Consolas, Courier, monospace;" class="elementToProof"><span style="background-color: rgb(255, 255, 255); font-family: Consolas, Courier, monospace; display: inline !important; font-size: 12pt;">AddProjection():</span></span></div>
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<span style="font-family: Consolas, Courier, monospace;" class="elementToProof"><span style="background-color:rgb(255, 255, 255);display:inline !important"><br>
</span></span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">sourcePosition {-1.6395187023763129, 1756.3965917941700, -144.39815739041538}</span><br>
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<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">detectorPosition {1.1837740829485037, -826.07679772360780, -144.71459973015834}</span><br>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">detectorRowVector {-0.99995122885867527, -0.0010919951074973716, -0.0098179685430691624}</span><br>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">detectorColumnVector {0.0098178296767637476, 0.00013326204342850774, -0.99995181030787128}</span><br>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">Detector normal vector is computed from row and column vectors, and those three vectors become the basis vectors of a 3x3 matrix. So far so good.</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">n = {0.0010932508471397015, -0.99999943265974056, -0.00012253452211328594}</span><br>
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<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="margin: 0px; font-size: 12pt; font-family: Consolas, Courier, monospace;">Matrix is decomposed into angles, which all seem sane enough:</span></div>
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<div style="margin:0px;font-size:12pt"><span style="margin: 0px; font-size: 12pt; font-family: Consolas, Courier, monospace;">inPlaneAngleRAD = 1.4493613948895230 (83.042290916569541537 deg)</span></div>
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<div style="margin:0px"><span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">outOfPlaneAngleRAD = -1.5697311129073750 (-89.938967740189397659 deg)</span><br>
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<div style="margin:0px;font-size:12pt"><span style="margin: 0px; background-color: rgb(255, 255, 255); display: inline !important; font-size: 12pt; font-family: Consolas, Courier, monospace;">gantryAngleRAD = -1.6824131994857836 (-96.395175727875965777 deg)</span><br>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">Then, inside VerifyAngles() we end up with the following matrices:</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">m = {</span><span style="color: rgb(0, 0, 0); font-family: Consolas, Courier, monospace; font-size: 12pt;"> {-0.99995124358590692, -0.0010573692766628138, -0.0098179642090855113},</span></div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<div><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> {0.0098178331830457710, 0.00012903646682954463, -0.99995179558875735},</span></div>
<div><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> {0.0010585851822123655, -0.99999943265974056, -0.00011864910030298149} }</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">rm = { {-0.99995122885867527, -0.0010919951074973716, -0.0098179685430691624},</span><br>
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<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> {0.0098178296767637476, 0.00013326204342850774, -0.99995181030787128},</span></div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> {0.0010932508471397015, -0.99999943265974056, -0.00012253452211328594} }</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">But then we fail when comparing the reference matrix with the reconstructed matrix:</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> for (int i = 0; i < 3; i++) // check whether matrices match</span></div>
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<div><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> for (int j = 0; j < 3; j++)</span></div>
<div><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> if (fabs(rm[i][j] - m[i][j]) > EPSILON)</span></div>
<div class="elementToProof"><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;"> return false;</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">because some of the matrix elements are just a bit uneq</span><span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">ual by more than EPSILON, which in this case is
defined as 1e-5.</span></div>
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<span style="font-family: Consolas, Courier, monospace; font-size: 12pt;">For instance, m[0][1] and rm[0][1] differ by </span><span style="font-family: Consolas, Courier, monospace; font-size: 12pt; background-color: rgb(255, 255, 255); display: inline !important;">0.0000346258
in magnitude. </span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">Bumping that epsilon up to 1e-4 gets us through the test, and the output seems good.</span></div>
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<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);" class="elementToProof">
<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">So finally, the question:</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">What are the ramifications of us bumping up the epsilon here? It seems like
</span><span style="font-family: Consolas, Courier, monospace; background-color: rgb(255, 255, 255); display: inline !important; font-size: 12pt;">VerifyAngles()</span><span style="font-size: 12pt; font-family: Consolas, Courier, monospace;"> is there because
the </span><span style="font-family: Consolas, Courier, monospace; background-color: rgb(255, 255, 255); display: inline !important; font-size: 12pt;">ThreeDCircularProjectionGeometry wants to store the angles for later use, and this is just a sanity check
to make sure the extracted angles capture the original vector parameters satisfactorily. If that's true I would expect that a tiny errorin the angles (at around 0.001 to 0.0001) would have a fairly miniscule impact on the actual DRR projection. Is this a valid
assumption, or are there potentially more perilous consequences of us upping the epsilon in VerifyAngles() a bit, either to 1e-4 or maybe even 1e-3?</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">Thanks in advance for any help/insights here.</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">Cheers,</span></div>
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<span style="font-size: 12pt; font-family: Consolas, Courier, monospace;">Brandon</span></div>
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