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‎pages/documentation_matlab/AxesAlignment.html‎

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<head>
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<META http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<!--
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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-->
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<title>On Screen Coordinate System Alignment</title>
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<link rel="schema.DC" href="http://purl.org/dc/elements/1.1/">
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{% endhighlight %}
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{% highlight plaintext %}
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ans = plottingConvention
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outOfScreen: (0,0,1)
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north : (0,1,0)
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east : (1,0,0)
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{% endhighlight %}
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<p>The property <code class="language-plaintext highlighter-rouge">how2plot</code> is a handle class of type @plottingConvention and tells MTEX how to align the corresponding coordinate system on screen.</p>
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{% highlight matlab %}
@@ -58,7 +62,9 @@
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{% endhighlight %}
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{% highlight plaintext %}
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v1 = vector3d
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x y z
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1 1 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="AxesAlignment_02.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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ans = plottingConvention
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outOfScreen: (0,1,0)
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north : (0,0,-1)
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east : (1,0,0)
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{% endhighlight %}
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<center>
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{% include inline_image.html file="AxesAlignment_03.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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pC2 = plottingConvention
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outOfScreen: (0,0,1)
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north : (0,1,0)
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east : (1,0,0)
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{% endhighlight %}
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<center>
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{% include inline_image.html file="AxesAlignment_04.png" %}
@@ -118,7 +132,22 @@
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{% endhighlight %}
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{% highlight plaintext %}
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pf = PoleFigure (xyz)
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crystal symmetry : Quartz (321, X||a*, Y||b, Z||c*)
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h = (02-21), r = 72 x 19 points
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h = (10-10), r = 72 x 19 points
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h = (10-11)(01-11), r = 72 x 19 points
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h = (10-12), r = 72 x 19 points
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h = (11-20), r = 72 x 19 points
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h = (11-21), r = 72 x 19 points
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h = (11-22), r = 72 x 19 points
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ans = plottingConvention
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outOfScreen: (0,0,1)
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north : (0,1,0)
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east : (1,0,0)
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{% endhighlight %}
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<center>
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{% include inline_image.html file="AxesAlignment_06.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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ans = plottingConvention
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outOfScreen: (0,0,1)
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north : (0,1,0)
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east : (1,0,0)
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{% endhighlight %}
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</div>
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</body>

‎pages/documentation_matlab/OptimalKernel.html‎

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<head>
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<META http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<!--
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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-->
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<title>Optimal Kernel Selection</title>
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<link rel="schema.DC" href="http://purl.org/dc/elements/1.1/">
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% build a density function by combining a uniform texture with two
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% predefined texture components
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odf = 0.25*uniformODF(cs) + 0.25*unimodalODF(orientation.brass(cs)) + ...
36-
0.5*fibreODF(fibre.alpha(cs),'halfwidth',10*degree);
35+
odf = 0.25*uniformODF(cs) + ...
36+
0.25 * unimodalODF(orientation.byEuler([35,45,0]*degree,cs)) + ...
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0.5 * fibreODF(Miller(7,-1,10,cs),vector3d(7,-5,11),'halfwidth',10*degree);
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% plot the density function as six sigma sections
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plot(odf,'sections',6,'silent','sigma')
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mtexColorbar
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{% endhighlight %}
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{% highlight plaintext %}
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{% endhighlight %}
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<center>
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{% include inline_image.html file="OptimalKernel_01.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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ori = orientation (321 → xyz)
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size: 10000 x 1
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{% endhighlight %}
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<p>Next we estimate the optimal <a href="ODFShapes.html">kernel function</a> using the command <code class="language-plaintext highlighter-rouge"><a href="orientation.calcKernel.html">calcKernel</a></code> with the default settings.</p>
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{% highlight matlab %}
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psi = calcKernel(ori)
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{% endhighlight %}
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{% highlight plaintext %}
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psi = SO3DeLaValleePoussinKernel
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bandwidth: 43
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halfwidth: 5.7°
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{% endhighlight %}
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<p>This kernel can now be used to reconstruct the original ODF from the sampled points using the command <a href="DensityEsimation.html">density estimation</a>
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</p>
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{% endhighlight %}
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{% highlight plaintext %}
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odf_rec = SO3FunHarmonic (321 → xyz)
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bandwidth: 43
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="OptimalKernel_02.png" %}
@@ -111,20 +113,24 @@ <h2 id="5">Exploration of the relationship between estimation error and number o
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]);
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end
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{% endhighlight %}
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{% highlight plaintext %}
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{% endhighlight %}
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<p>Plot the error to the number of single orientations sampled from the original ODF.</p>
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{% highlight matlab %}
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% Plot the error to the number of single orientations sampled from the original ODF.
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close all;
122119
loglog(10.^(1:length(e)),e,'LineWidth',2)
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legend('Default','RuleOfThumb','magicRule')
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xlabel('Number of orientations (log scale)')
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ylabel('Estimation Error in degrees')
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title('Error between original ODF model and the reconstructed ODF','FontWeight','bold')
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{% endhighlight %}
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{% highlight plaintext %}
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RuleOfThumb: 75° KLCV: 24° magicRule: 31°
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RuleOfThumb: 30° KLCV: 12° magicRule: 22°
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RuleOfThumb: 18° KLCV: 8° magicRule: 16°
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RuleOfThumb: 10° KLCV: 6° magicRule: 11°
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RuleOfThumb: 9° KLCV: 5° magicRule: 8°
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RuleOfThumb: 7° KLCV: 4° magicRule: 6°
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{% endhighlight %}
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<center>
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{% include inline_image.html file="OptimalKernel_03.png" %}
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</center>

‎pages/documentation_matlab/RBFApproximationTheory.html‎

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<head>
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<META http-equiv="Content-Type" content="text/html; charset=UTF-8">
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<!--
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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This HTML was auto-generated from MATLAB code.
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To make changes, update the MATLAB code and republish this document.
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-->
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<title>RBF-Kernel Approximation from Discrete Data</title>
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<link rel="schema.DC" href="http://purl.org/dc/elements/1.1/">
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{% endhighlight %}
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{% highlight plaintext %}
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 5°
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center: 72018 orientations, resolution: 5°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_02.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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minValue =
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1.1635e-06
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meanValue =
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1.0000
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{% endhighlight %}
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<p>One has to keep in mind that we can not expect the error in the data nodes to be zero, because we compute a smooth function from the noisy input data.</p>
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{% highlight matlab %}
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norm(SO3F.eval(ori) - val) / norm(val)
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{% endhighlight %}
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{% highlight plaintext %}
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ans =
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0.1639
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{% endhighlight %}
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<p>In contrast, if we do not tell MTEX, that we try to approximate a density function, the solver has less information and the result is not denoised.</p>
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{% highlight matlab %}
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{% endhighlight %}
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{% highlight plaintext %}
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 5°
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center: 119088 orientations, resolution: 5°
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weight: 0.95
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_03.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 2°
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center: 1852941 orientations, resolution: 2°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_04.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 10°
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center: 70038 orientations, resolution: 5°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_05.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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S3G = SO3Grid (1 → xyz)
154+
grid: 119088 orientations, resolution: 5°
155+
Warning: Maximum number of iterations reached,
156+
result may not have converged to the optimum yet.
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 5°
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center: 72018 orientations, resolution: 5°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_06.png" %}
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147181
ylabel('relative error')
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{% endhighlight %}
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{% highlight plaintext %}
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_07.png" %}
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</center>
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{% endhighlight %}
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162193
{% highlight plaintext %}
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194+
SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: de la Vallee Poussin, halfwidth 7.5°
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center: 24649 orientations, resolution: 7.5°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_08.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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psi = SO3AbelPoissonKernel
213+
bandwidth: 48
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halfwidth: 5°
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SO3F = SO3FunRBF (1 → xyz)
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multimodal components
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kernel: Abel Poisson, halfwidth 5°
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center: 3989 orientations, resolution: 5°
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weight: 1
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_09.png" %}
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{% endhighlight %}
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{% highlight plaintext %}
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237+
Elapsed time is 65.813998 seconds.
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{% endhighlight %}
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<center>
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{% include inline_image.html file="RBFApproximationTheory_10.png" %}
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201246
{% endhighlight %}
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203248
{% highlight plaintext %}
204-
249+
ans =
250+
9.0528e-04
205251
{% endhighlight %}
206252
<p>Also, interpolation might not guarantee non-negativity of the function</p>
207253
{% highlight matlab %}
208254
minValue = min(SO3F)
209255
{% endhighlight %}
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211257
{% highlight plaintext %}
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258+
minValue =
259+
-7.6067
213260
{% endhighlight %}
214261
<h2 id="18">LSQR-Parameters</h2>
215262
<p>The <code class="language-plaintext highlighter-rouge">lsqr</code> solver and the <code class="language-plaintext highlighter-rouge">mlsq</code> solver, which are used to minimize the least squares problem from above has some predefined termination conditions. We can specify the method tolerance with the option <code class="language-plaintext highlighter-rouge">'tol'</code> (default 1e-3) and the maximum number of iterations by the option <code class="language-plaintext highlighter-rouge">'maxit'</code> (default 30/100).</p>
@@ -231,7 +278,13 @@ <h2 id="18">LSQR-Parameters</h2>
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{% endhighlight %}
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{% highlight plaintext %}
281+
Elapsed time is 1.065290 seconds.
282+
Number of iterations = 8
283+
Value of energy functional = 0.43171
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285+
Elapsed time is 1.607980 seconds.
286+
Number of iterations = 41
287+
Value of energy functional = 0.0015935
235288
{% endhighlight %}
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</div>
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</body>

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