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<article role="main">
<section id="math">
<span id="id1"></span><h1>Math<a class="headerlink" href="#math" title="Permalink to this heading">#</a></h1>
<p>Math Helper Functions</p>
<span class="target" id="module-nerfstudio.utils.math"></span><dl class="py class">
<dt class="sig sig-object py" id="nerfstudio.utils.math.Gaussians">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">Gaussians</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mean</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">dim'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">cov</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">dim</span> <span class="pre">dim'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#Gaussians"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.Gaussians" title="Permalink to this definition">#</a></dt>
<dd><p>Bases: <code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></p>
<p>Stores Gaussians</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>mean</strong> – Mean of multivariate Gaussian</p></li>
<li><p><strong>cov</strong> – Covariance of multivariate Gaussian.</p></li>
</ul>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.columnwise_squared_l2_distance">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">columnwise_squared_l2_distance</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*M</span> <span class="pre">N'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*M</span> <span class="pre">N'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'N</span> <span class="pre">N'</span></span><span class="p"><span class="pre">]</span></span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#columnwise_squared_l2_distance"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.columnwise_squared_l2_distance" title="Permalink to this definition">#</a></dt>
<dd><p>Compute the squared Euclidean distance between all pairs of columns.
Adapted from <a class="reference external" href="https://github.com/google-research/multinerf/blob/5b4d4f64608ec8077222c52fdf814d40acc10bc1/internal/geopoly.py">https://github.com/google-research/multinerf/blob/5b4d4f64608ec8077222c52fdf814d40acc10bc1/internal/geopoly.py</a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x</strong> – tensor of floats, with shape [M, N].</p></li>
<li><p><strong>y</strong> – tensor of floats, with shape [M, N].</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>tensor of floats, with shape [N, N].</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>sq_dist</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.compute_3d_gaussian">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">compute_3d_gaussian</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">means</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dir_variance</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">radius_variance</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians"><span class="pre">Gaussians</span></a></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#compute_3d_gaussian"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.compute_3d_gaussian" title="Permalink to this definition">#</a></dt>
<dd><p>Compute gaussian along ray.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>directions</strong> – Axis of Gaussian.</p></li>
<li><p><strong>means</strong> – Mean of Gaussian.</p></li>
<li><p><strong>dir_variance</strong> – Variance along direction axis.</p></li>
<li><p><strong>radius_variance</strong> – Variance tangent to direction axis.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Oriented 3D gaussian.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians">Gaussians</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.conical_frustum_to_gaussian">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">conical_frustum_to_gaussian</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">origins</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">starts</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ends</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">radius</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians"><span class="pre">Gaussians</span></a></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#conical_frustum_to_gaussian"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.conical_frustum_to_gaussian" title="Permalink to this definition">#</a></dt>
<dd><p>Approximates conical frustums with a Gaussian distributions.</p>
<p>Uses stable parameterization described in mip-NeRF publication.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>origins</strong> – Origins of cones.</p></li>
<li><p><strong>directions</strong> – Direction (axis) of frustums.</p></li>
<li><p><strong>starts</strong> – Start of conical frustums.</p></li>
<li><p><strong>ends</strong> – End of conical frustums.</p></li>
<li><p><strong>radius</strong> – Radii of cone a distance of 1 from the origin.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Approximation of conical frustums</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians">Gaussians</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.cylinder_to_gaussian">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">cylinder_to_gaussian</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">origins</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">3'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">starts</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ends</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">radius</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">1'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians"><span class="pre">Gaussians</span></a></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#cylinder_to_gaussian"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.cylinder_to_gaussian" title="Permalink to this definition">#</a></dt>
<dd><p>Approximates cylinders with a Gaussian distributions.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>origins</strong> – Origins of cylinders.</p></li>
<li><p><strong>directions</strong> – Direction (axis) of cylinders.</p></li>
<li><p><strong>starts</strong> – Start of cylinders.</p></li>
<li><p><strong>ends</strong> – End of cylinders.</p></li>
<li><p><strong>radius</strong> – Radii of cylinders.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Approximation of cylinders</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference internal" href="#nerfstudio.utils.math.Gaussians" title="nerfstudio.utils.math.Gaussians">Gaussians</a></p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.expected_sin">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">expected_sin</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x_means</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x_vars</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tensor</span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#expected_sin"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.expected_sin" title="Permalink to this definition">#</a></dt>
<dd><p>Computes the expected value of sin(y) where y ~ N(x_means, x_vars)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x_means</strong> – Mean values.</p></li>
<li><p><strong>x_vars</strong> – Variance of values.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>The expected value of sin.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>torch.Tensor</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.generate_polyhedron_basis">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">generate_polyhedron_basis</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">basis_shape</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Literal</span><span class="p"><span class="pre">[</span></span><span class="s"><span class="pre">'icosahedron'</span></span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'octahedron'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">angular_tesselation</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">remove_symmetries</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">bool</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">0.0001</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tensor</span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#generate_polyhedron_basis"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.generate_polyhedron_basis" title="Permalink to this definition">#</a></dt>
<dd><p>Generates a 3D basis by tesselating a geometric polyhedron.
Basis is used to construct Fourier features for positional encoding.
See Mip-Nerf360 paper: <a class="reference external" href="https://arxiv.org/abs/2111.12077">https://arxiv.org/abs/2111.12077</a>
Adapted from <a class="reference external" href="https://github.com/google-research/multinerf/blob/5b4d4f64608ec8077222c52fdf814d40acc10bc1/internal/geopoly.py">https://github.com/google-research/multinerf/blob/5b4d4f64608ec8077222c52fdf814d40acc10bc1/internal/geopoly.py</a></p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>base_shape</strong> – string, the name of the starting polyhedron, must be either
‘icosahedron’ or ‘octahedron’.</p></li>
<li><p><strong>angular_tesselation</strong> – int, the number of times to tesselate the polyhedron,
must be &gt;= 1 (a value of 1 is a no-op to the polyhedron).</p></li>
<li><p><strong>remove_symmetries</strong> – bool, if True then remove the symmetric basis columns,
which is usually a good idea because otherwise projections onto the basis
will have redundant negative copies of each other.</p></li>
<li><p><strong>eps</strong> – float, a small number used to determine symmetries.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>a matrix with shape [3, n].</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>basis</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.intersect_aabb">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">intersect_aabb</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">origins</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">aabb</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_bound</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">10000000000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">invalid_value</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">10000000000.0</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tuple</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Tensor</span><span class="p"><span class="pre">]</span></span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#intersect_aabb"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.intersect_aabb" title="Permalink to this definition">#</a></dt>
<dd><p>Implementation of ray intersection with AABB box</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>origins</strong> – [N,3] tensor of 3d positions</p></li>
<li><p><strong>directions</strong> – [N,3] tensor of normalized directions</p></li>
<li><p><strong>aabb</strong> – [6] array of aabb box in the form of [x_min, y_min, z_min, x_max, y_max, z_max]</p></li>
<li><p><strong>max_bound</strong> – Maximum value of t_max</p></li>
<li><p><strong>invalid_value</strong> – Value to return in case of no intersection</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>t_min, t_max - two tensors of shapes N representing distance of intersection from the origin.</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.intersect_obb">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">intersect_obb</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">origins</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">directions</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">obb</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><a class="reference internal" href="../data/index.html#nerfstudio.data.scene_box.OrientedBox" title="nerfstudio.data.scene_box.OrientedBox"><span class="pre">OrientedBox</span></a></span></em>, <em class="sig-param"><span class="n"><span class="pre">max_bound</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">10000000000.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">invalid_value</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">10000000000.0</span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#intersect_obb"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.intersect_obb" title="Permalink to this definition">#</a></dt>
<dd><p>Ray intersection with an oriented bounding box (OBB)</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>origins</strong> – [N,3] tensor of 3d positions</p></li>
<li><p><strong>directions</strong> – [N,3] tensor of normalized directions</p></li>
<li><p><strong>R</strong> – [3,3] rotation matrix</p></li>
<li><p><strong>T</strong> – [3] translation vector</p></li>
<li><p><strong>S</strong> – [3] extents of the bounding box</p></li>
<li><p><strong>max_bound</strong> – Maximum value of t_max</p></li>
<li><p><strong>invalid_value</strong> – Value to return in case of no intersection</p></li>
</ul>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.k_nearest_sklearn">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">k_nearest_sklearn</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Tensor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">metric</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">str</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">'euclidean'</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tuple</span><span class="p"><span class="pre">[</span></span><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">k'</span></span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="pre">Int</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">k'</span></span><span class="p"><span class="pre">]</span></span><span class="p"><span class="pre">]</span></span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#k_nearest_sklearn"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.k_nearest_sklearn" title="Permalink to this definition">#</a></dt>
<dd><p>Find k-nearest neighbors using sklearn’s NearestNeighbors.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>x</strong> – input tensor</p></li>
<li><p><strong>k</strong> – number of neighbors to find</p></li>
<li><p><strong>metric</strong> – metric to use for distance computation</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>distances to the k-nearest neighbors
indices: indices of the k-nearest neighbors</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p>distances</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.masked_reduction">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">masked_reduction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input_tensor</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'1</span> <span class="pre">32</span> <span class="pre">mult'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mask</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Bool</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'1</span> <span class="pre">32</span> <span class="pre">mult'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">reduction_type</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Literal</span><span class="p"><span class="pre">[</span></span><span class="s"><span class="pre">'image'</span></span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'batch'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Tensor</span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#masked_reduction"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.masked_reduction" title="Permalink to this definition">#</a></dt>
<dd><p>Whether to consolidate the input_tensor across the batch or across the image
:param input_tensor: input tensor
:param mask: mask tensor
:param reduction_type: either “batch” or “image”</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>reduced input_tensor</p>
</dd>
<dt class="field-even">Return type<span class="colon">:</span></dt>
<dd class="field-even"><p>input_tensor</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.normalized_depth_scale_and_shift">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">normalized_depth_scale_and_shift</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">prediction</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'1</span> <span class="pre">32</span> <span class="pre">mult'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">target</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'1</span> <span class="pre">32</span> <span class="pre">mult'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mask</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Bool</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'1</span> <span class="pre">32</span> <span class="pre">mult'</span></span><span class="p"><span class="pre">]</span></span></span></em><span class="sig-paren">)</span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#normalized_depth_scale_and_shift"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.normalized_depth_scale_and_shift" title="Permalink to this definition">#</a></dt>
<dd><p>More info here: <a class="reference external" href="https://arxiv.org/pdf/2206.00665.pdf">https://arxiv.org/pdf/2206.00665.pdf</a> supplementary section A2 Depth Consistency Loss
This function computes scale/shift required to normalizes predicted depth map,
to allow for using normalized depth maps as input from monocular depth estimation networks.
These networks are trained such that they predict normalized depth maps.</p>
<p>Solves for scale/shift using a least squares approach with a closed form solution:
Based on:
<a class="reference external" href="https://github.com/autonomousvision/monosdf/blob/d9619e948bf3d85c6adec1a643f679e2e8e84d4b/code/model/loss.py#L7">https://github.com/autonomousvision/monosdf/blob/d9619e948bf3d85c6adec1a643f679e2e8e84d4b/code/model/loss.py#L7</a>
:param prediction: predicted depth map
:param target: ground truth depth map
:param mask: mask of valid pixels</p>
<dl class="field-list simple">
<dt class="field-odd">Returns<span class="colon">:</span></dt>
<dd class="field-odd"><p>scale and shift for depth prediction</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.random_quat_tensor">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">random_quat_tensor</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">N</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">int</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch</span> <span class="pre">4'</span></span><span class="p"><span class="pre">]</span></span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#random_quat_tensor"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.random_quat_tensor" title="Permalink to this definition">#</a></dt>
<dd><p>Defines a random quaternion tensor.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><p><strong>N</strong> – Number of quaternions to generate</p>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>a random quaternion tensor of shape (N, 4)</p>
</dd>
</dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="nerfstudio.utils.math.safe_normalize">
<span class="sig-prename descclassname"><span class="pre">nerfstudio.utils.math.</span></span><span class="sig-name descname"><span class="pre">safe_normalize</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">vectors</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch_dim</span> <span class="pre">N'</span></span><span class="p"><span class="pre">]</span></span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eps</span></span><span class="p"><span class="pre">:</span></span><span class="w"> </span><span class="n"><span class="pre">float</span></span><span class="w"> </span><span class="o"><span class="pre">=</span></span><span class="w"> </span><span class="default_value"><span class="pre">1e-10</span></span></em><span class="sig-paren">)</span> <span class="sig-return"><span class="sig-return-icon">&#x2192;</span> <span class="sig-return-typehint"><span class="pre">Float</span><span class="p"><span class="pre">[</span></span><span class="pre">Tensor</span><span class="p"><span class="pre">,</span></span><span class="w"> </span><span class="s"><span class="pre">'*batch_dim</span> <span class="pre">N'</span></span><span class="p"><span class="pre">]</span></span></span></span><a class="reference internal" href="../../../_modules/nerfstudio/utils/math.html#safe_normalize"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#nerfstudio.utils.math.safe_normalize" title="Permalink to this definition">#</a></dt>
<dd><p>Normalizes vectors.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>vectors</strong> – Vectors to normalize.</p></li>
<li><p><strong>eps</strong> – Epsilon value to avoid division by zero.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Normalized vectors.</p>
</dd>
</dl>
</dd></dl>
</section>
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<ul>
<li><a class="reference internal" href="#">Math</a><ul>
<li><a class="reference internal" href="#nerfstudio.utils.math.Gaussians"><code class="docutils literal notranslate"><span class="pre">Gaussians</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.columnwise_squared_l2_distance"><code class="docutils literal notranslate"><span class="pre">columnwise_squared_l2_distance()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.compute_3d_gaussian"><code class="docutils literal notranslate"><span class="pre">compute_3d_gaussian()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.conical_frustum_to_gaussian"><code class="docutils literal notranslate"><span class="pre">conical_frustum_to_gaussian()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.cylinder_to_gaussian"><code class="docutils literal notranslate"><span class="pre">cylinder_to_gaussian()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.expected_sin"><code class="docutils literal notranslate"><span class="pre">expected_sin()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.generate_polyhedron_basis"><code class="docutils literal notranslate"><span class="pre">generate_polyhedron_basis()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.intersect_aabb"><code class="docutils literal notranslate"><span class="pre">intersect_aabb()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.intersect_obb"><code class="docutils literal notranslate"><span class="pre">intersect_obb()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.k_nearest_sklearn"><code class="docutils literal notranslate"><span class="pre">k_nearest_sklearn()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.masked_reduction"><code class="docutils literal notranslate"><span class="pre">masked_reduction()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.normalized_depth_scale_and_shift"><code class="docutils literal notranslate"><span class="pre">normalized_depth_scale_and_shift()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.random_quat_tensor"><code class="docutils literal notranslate"><span class="pre">random_quat_tensor()</span></code></a></li>
<li><a class="reference internal" href="#nerfstudio.utils.math.safe_normalize"><code class="docutils literal notranslate"><span class="pre">safe_normalize()</span></code></a></li>
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