documentation/sampling.html at master · raysect/documentation · GitHub

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  <section id="monte-carlo-sampling">
<h1>Monte-Carlo Sampling<a class="headerlink" href="#monte-carlo-sampling" title="Link to this heading">¶</a></h1>
<p>The coordinate transformation is applied by multiplying the column vector for
the desired Point3D/Vector3D against the transformation matrix. For example,
if the original vector <span class="math notranslate nohighlight">\(\\vec{V_a}\)</span> is in space A and the transformation matrix
<span class="math notranslate nohighlight">\(\\mathbf{T_{AB}}\)</span> describes the position and orientation of Space A relative
to Space B, then the multiplication</p>
<section id="general-monte-carlo-integration">
<h2>General Monte-carlo integration<a class="headerlink" href="#general-monte-carlo-integration" title="Link to this heading">¶</a></h2>
<div class="math notranslate nohighlight">
\[\begin{split}Q_{N} = \\frac{1}{N} \\sum_{i=1}^{N} \frac{f(x_0) \\cos(\\theta)}{p(x_0)}\end{split}\]</div>
</section>
<section id="uniform-sampling">
<h2>Uniform Sampling<a class="headerlink" href="#uniform-sampling" title="Link to this heading">¶</a></h2>
<div class="math notranslate nohighlight">
\[\begin{split}p(x_0) = \\frac{1}{2 \\pi}\end{split}\]</div>
<div class="math notranslate nohighlight">
\[\begin{split}Q_{N} = \\frac{2 \\pi}{N} \\sum_{i=1}^{N} f(x_0) \\cos(\\theta)\end{split}\]</div>
</section>
<section id="cosine-weighted">
<h2>Cosine Weighted<a class="headerlink" href="#cosine-weighted" title="Link to this heading">¶</a></h2>
<div class="math notranslate nohighlight">
\[\begin{split}p(x_0) = \\frac{\\cos(\\theta)}{\\pi}\end{split}\]</div>
<div class="math notranslate nohighlight">
\[\begin{split}Q_{N} = \\frac{\\pi}{N \\cos(\\theta)} \\sum_{i=1}^{N} f(x_0) \\cos(\\theta) = \\frac{\\pi}{N} \\sum_{i=1}^{N} f(x_0)\end{split}\]</div>
</section>
<section id="irradiance-vs-radiance">
<h2>Irradiance vs Radiance<a class="headerlink" href="#irradiance-vs-radiance" title="Link to this heading">¶</a></h2>
<p>Irradiance at surface.</p>
<div class="math notranslate nohighlight">
\[\begin{split}I = \\frac{1}{N} \\sum_{i=1}^{N} \\frac{f(x_0) \\cos(\\theta)}{p(x)}\end{split}\]</div>
<p>Conversion to radiance:</p>
<div class="math notranslate nohighlight">
\[\begin{split}radiance = I / (2 \\pi)\end{split}\]</div>
</section>
<section id="implementation">
<h2>Implementation<a class="headerlink" href="#implementation" title="Link to this heading">¶</a></h2>
<p>Implementation in base observer:</p>
<div class="math notranslate nohighlight">
\[\begin{split}radiance = \\frac{1}{2 \\pi} \\frac{1}{N} \\sum_{i=1}^{N} w f(x)\end{split}\]</div>
<ul class="simple">
<li><p>f(x) is obtained from ray-tracing in base class</p></li>
<li><p>w is cos(theta)/p(x).</p></li>
<li><p>for an observer its this.</p></li>
<li><p>or a surface its this.</p></li>
</ul>
</section>
</section>


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