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</script></nav></div><div class="container mainContainer docsContainer"><div class="wrapper"><div class="post"><header class="postHeader"></header><article><div><span><h1><a class="anchor" aria-hidden="true" id="intersection-over-union-of-oriented-3d-boxes-a-new-algorithm"></a><a href="#intersection-over-union-of-oriented-3d-boxes-a-new-algorithm" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Intersection Over Union of Oriented 3D Boxes: A New Algorithm</h1>
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<p>Author: Georgia Gkioxari</p>
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<p>Implementation: Georgia Gkioxari and Nikhila Ravi</p>
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<h2><a class="anchor" aria-hidden="true" id="description"></a><a href="#description" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Description</h2>
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<p>Intersection over union (IoU) of boxes is widely used as an evaluation metric in object detection (<a href="http://host.robots.ox.ac.uk/pascal/VOC/">1</a>, <a href="https://cocodataset.org/">2</a>).
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In 2D, IoU is commonly applied to axis-aligned boxes, namely boxes with edges parallel to the image axis.
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In 3D, boxes are usually not axis aligned and can be oriented in any way in the world.
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We introduce a new algorithm which computes the <em>exact</em> IoU of two <em>oriented 3D boxes</em>.</p>
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<p>Our algorithm is based on the simple observation that the intersection of two oriented 3D boxes, <code>box1</code> and <code>box2</code>, is a convex polyhedron (convex n-gon in 2D) with <code>n > 2</code> comprised of connected <em>planar units</em>.
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In 3D, these planar units are 3D triangular faces.
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In 2D, they are 2D edges.
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Each planar unit belongs strictly to either <code>box1</code> or <code>box2</code>.
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Our algorithm finds these units by iterating through the sides of each box.</p>
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<ol>
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<li>For each 3D triangular face <code>e</code> in <code>box1</code> we check wether <code>e</code> is <em>inside</em> <code>box2</code>.</li>
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<li>If <code>e</code> is not <em>inside</em>, then we discard it.</li>
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<li>If <code>e</code> is <em>inside</em> or <em>partially inside</em>, then the part of <code>e</code> <em>inside</em> <code>box2</code> is added to the units that comprise the final intersection shape.</li>
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<li>We repeat for <code>box2</code>.</li>
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</ol>
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<p>Below, we show a visualization of our algorithm for the case of 2D oriented boxes.</p>
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<p align="center">
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<img src="assets/iou3d.gif" alt="drawing" width="400"/>
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</p>
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<p>Note that when a box's unit <code>e</code> is <em>partially inside</em> a <code>box</code> then <code>e</code> breaks into smaller units. In 2D, <code>e</code> is an edge and breaks into smaller edges. In 3D, <code>e</code> is a 3D triangular face and is clipped to more and smaller faces by the plane of the <code>box</code> it intersects with.
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This is the sole fundamental difference between the algorithms for 2D and 3D.</p>
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<h2><a class="anchor" aria-hidden="true" id="comparison-with-other-algorithms"></a><a href="#comparison-with-other-algorithms" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Comparison With Other Algorithms</h2>
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<p>Current algorithms for 3D box IoU rely on crude approximations or make box assumptions, for example they restrict the orientation of the 3D boxes.
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<a href="https://arxiv.org/abs/2012.09988">Objectron</a> provides a nice discussion on the limitations of prior works.
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<a href="https://arxiv.org/abs/2012.09988">Objectron</a> introduces a great algorithm for exact IoU computation of oriented 3D boxes.
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Objectron's algorithm computes the intersection points of two boxes using the <a href="https://en.wikipedia.org/wiki/Sutherland%E2%80%93Hodgman_algorithm">Sutherland-Hodgman algorithm</a>.
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The intersection shape is formed by the convex hull from the intersection points, using the <a href="http://www.qhull.org/">Qhull library</a>.</p>
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<p>Our algorithm has several advantages over Objectron's:</p>
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<ul>
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<li>Our algorithm also computes the points of intersection, similar to Objectron, but in addition stores the <em>planar units</em> the points belong to. This eliminates the need for convex hull computation which is <code>O(nlogn)</code> and relies on a third party library which often crashes with nondescript error messages.</li>
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<li>Objectron's implementation assumes that boxes are a rotation away from axis aligned. Our algorithm and implementation make no such assumption and work for any 3D boxes.</li>
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<li>Our implementation supports batching, unlike Objectron which assumes single element inputs for <code>box1</code> and <code>box2</code>.</li>
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<li>Our implementation is easily parallelizable and in fact we provide a custom C++/CUDA implementation which is <strong>450 times faster than Objectron</strong>.</li>
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</ul>
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<p>Below we compare the performance for Objectron (in C++) and our algorithm, in C++ and CUDA. We benchmark for a common use case in object detection where <code>boxes1</code> hold M predictions and <code>boxes2</code> hold N ground truth 3D boxes in an image and compute the <code>MxN</code> IoU matrix. We report the time in ms for <code>M=N=16</code>.</p>
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<p align="center">
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<img src="assets/iou3d_comp.png" alt="drawing" width="400"/>
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</p>
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<h2><a class="anchor" aria-hidden="true" id="usage-and-code"></a><a href="#usage-and-code" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Usage and Code</h2>
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<pre><code class="hljs css language-python"><span class="hljs-keyword">from</span> pytorch3d.ops <span class="hljs-keyword">import</span> box3d_overlap
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<span class="hljs-comment"># Assume inputs: boxes1 (M, 8, 3) and boxes2 (N, 8, 3)</span>
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intersection_vol, iou_3d = box3d_overlap(boxes1, boxes2)
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</code></pre>
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<p>For more details, read <a href="https://github.com/facebookresearch/pytorch3d/blob/main/pytorch3d/ops/iou_box3d.py">iou_box3d.py</a>.</p>
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<p>Note that our implementation is not differentiable as of now. We plan to add gradient support soon.</p>
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<p>We also include have extensive <a href="https://github.com/facebookresearch/pytorch3d/blob/main/tests/test_iou_box3d.py">tests</a> comparing our implementation with Objectron and MeshLab.</p>
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<h2><a class="anchor" aria-hidden="true" id="cite"></a><a href="#cite" aria-hidden="true" class="hash-link"><svg class="hash-link-icon" aria-hidden="true" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path fill-rule="evenodd" d="M4 9h1v1H4c-1.5 0-3-1.69-3-3.5S2.55 3 4 3h4c1.45 0 3 1.69 3 3.5 0 1.41-.91 2.72-2 3.25V8.59c.58-.45 1-1.27 1-2.09C10 5.22 8.98 4 8 4H4c-.98 0-2 1.22-2 2.5S3 9 4 9zm9-3h-1v1h1c1 0 2 1.22 2 2.5S13.98 12 13 12H9c-.98 0-2-1.22-2-2.5 0-.83.42-1.64 1-2.09V6.25c-1.09.53-2 1.84-2 3.25C6 11.31 7.55 13 9 13h4c1.45 0 3-1.69 3-3.5S14.5 6 13 6z"></path></svg></a>Cite</h2>
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<p>If you use our 3D IoU algorithm, please cite PyTorch3D</p>
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<pre><code class="hljs css language-bibtex">@article{ravi2020pytorch3d,
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author = {Nikhila Ravi <span class="hljs-keyword">and </span><span class="hljs-keyword">Jeremy </span>Reizenstein <span class="hljs-keyword">and </span>David Novotny <span class="hljs-keyword">and </span>Taylor Gordon
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<span class="hljs-keyword">and </span>Wan-Yen Lo <span class="hljs-keyword">and </span><span class="hljs-keyword">Justin </span><span class="hljs-keyword">Johnson </span><span class="hljs-keyword">and </span>Georgia Gkioxari},
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title = {Accelerating <span class="hljs-number">3</span>D Deep Learning with PyTorch3D},
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<span class="hljs-keyword">journal </span>= {arXiv:<span class="hljs-number">2007</span>.<span class="hljs-number">08501</span>},
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year = {<span class="hljs-number">2020</span>},
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}
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</code></pre>
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</span></div></article></div><div class="docs-prevnext"><a class="docs-prev button" href="/docs/cubify"><span class="arrow-prev">← </span><span>Cubify</span></a><a class="docs-next button" href="/docs/visualization"><span>Plotly Visualization</span><span class="arrow-next"> →</span></a></div></div></div><nav class="onPageNav"><ul class="toc-headings"><li><a href="#description">Description</a></li><li><a href="#comparison-with-other-algorithms">Comparison With Other Algorithms</a></li><li><a href="#usage-and-code">Usage and Code</a></li><li><a href="#cite">Cite</a></li></ul></nav></div><footer class="nav-footer" id="footer"><section class="sitemap"><div class="footerSection"><div class="social"><a class="github-button" href="https://github.com/facebookresearch/pytorch3d" data-count-href="https://github.com/facebookresearch/pytorch3d/stargazers" data-show-count="true" data-count-aria-label="# stargazers on GitHub" aria-label="Star PyTorch3D on GitHub">pytorch3d</a></div></div></section><a href="https://opensource.facebook.com/" target="_blank" rel="noreferrer noopener" class="fbOpenSource"><img src="/img/oss_logo.png" alt="Facebook Open Source" width="170" height="45"/></a><section class="copyright">Copyright © 2024 Meta Platforms, Inc<br/>Legal:<a href="https://opensource.facebook.com/legal/privacy/" target="_blank" rel="noreferrer noopener">Privacy</a><a href="https://opensource.facebook.com/legal/terms/" target="_blank" rel="noreferrer noopener">Terms</a></section></footer></div></body></html> |