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<h1>Voxel downsampling with geometry preservation</h1>
<p class="sub">How far can <code>bigpointcloud_001.ply</code> be reduced before its geometry starts to go?
Measured, not eyeballed.</p>

<div class="cards">
  <div class="card"><div class="k">Input</div><div class="v">11,357</div><div class="n">points, with normals + RGB</div></div>
  <div class="card"><div class="k">Native resolution</div><div class="v">0.140</div><div class="n">mean nearest-neighbour spacing</div></div>
  <div class="card"><div class="k">Safe voxel</div><div class="v">0.210</div><div class="n">8,832 pts — 78% kept</div></div>
  <div class="card"><div class="k">Aggressive voxel</div><div class="v">0.279</div><div class="n">5,592 pts — 49% kept</div></div>
</div>

<h2>Method</h2>
<p>Voxel size is chosen against an error budget rather than by guesswork. Every candidate is scored by
how far the surface actually moved — the distance from each original point to the nearest kept point —
plus how much the normals rotated and whether the bounding box shrank. The natural yardstick is the
cloud's own nearest-neighbour spacing (0.140): a downsample that perturbs the surface by less than
that has not degraded it beyond the noise the original sampling already carries.</p>
<ul>
  <li><strong>safe</strong> — largest voxel whose <em>95th-percentile</em> error stays under one NN spacing</li>
  <li><strong>aggressive</strong> — largest voxel whose <em>mean</em> error stays under one NN spacing</li>
</ul>

<h2>Voxel-size sweep</h2>
<div class="scroll">
<table>
<thead><tr>
  <th>voxel</th><th>× NN</th><th>points</th><th>kept</th><th>mean err</th><th>RMS</th>
  <th>p95</th><th>Hausdorff</th><th>normal dev</th><th>bbox shrink</th><th></th>
</tr></thead>
<tbody>
<tr><td>0.070</td><td>0.5×</td><td>11,320</td><td>99.7%</td><td>0.0002</td><td>0.0022</td><td>0.0000</td><td>0.0409</td><td>0.1°</td><td>0.00%</td><td class=note></td></tr>
<tr><td>0.105</td><td>0.7×</td><td>11,138</td><td>98.1%</td><td>0.0015</td><td>0.0080</td><td>0.0000</td><td>0.0788</td><td>0.4°</td><td>0.00%</td><td class=note></td></tr>
<tr><td>0.140</td><td>1.0×</td><td>10,705</td><td>94.3%</td><td>0.0058</td><td>0.0179</td><td>0.0540</td><td>0.0952</td><td>1.4°</td><td>0.18%</td><td class=note></td></tr>
<tr class="safe"><td>0.210</td><td>1.5×</td><td>8,832</td><td>77.8%</td><td>0.0315</td><td>0.0507</td><td>0.0990</td><td>0.1598</td><td>5.7°</td><td>0.28%</td><td class=note>safe</td></tr>
<tr class="aggr"><td>0.279</td><td>2.0×</td><td>5,592</td><td>49.2%</td><td>0.0829</td><td>0.0960</td><td>0.1536</td><td>0.2145</td><td>13.3°</td><td>0.35%</td><td class=note>aggressive</td></tr>
<tr><td>0.419</td><td>3.0×</td><td>2,620</td><td>23.1%</td><td>0.1467</td><td>0.1567</td><td>0.2357</td><td>0.3530</td><td>23.0°</td><td>1.15%</td><td class=note>geometry degrading</td></tr>
<tr><td>0.559</td><td>4.0×</td><td>1,511</td><td>13.3%</td><td>0.1999</td><td>0.2139</td><td>0.3175</td><td>0.4801</td><td>28.3°</td><td>1.50%</td><td class=note>geometry degrading</td></tr>
<tr><td>0.838</td><td>6.0×</td><td>655</td><td>5.8%</td><td>0.3102</td><td>0.3330</td><td>0.5034</td><td>0.7154</td><td>33.4°</td><td>4.69%</td><td class=note>geometry degrading</td></tr>
<tr><td>1.117</td><td>8.0×</td><td>385</td><td>3.4%</td><td>0.4017</td><td>0.4271</td><td>0.6343</td><td>0.9313</td><td>39.4°</td><td>11.05%</td><td class=note>geometry degrading</td></tr>
<tr><td>1.676</td><td>12.0×</td><td>163</td><td>1.4%</td><td>0.5899</td><td>0.6305</td><td>0.9446</td><td>1.2392</td><td>45.3°</td><td>15.01%</td><td class=note>geometry degrading</td></tr>
</tbody>
</table>
</div>
<p>Geometry holds to about 0.28 — half the points, surface displaced 0.083 units on average
(0.4% of the 20.7-unit bbox diagonal), normals within 13°. Past 0.42 it degrades quickly: normals
swing 23°+, and by voxel 1.1 the bounding box itself has lost 11%, meaning thin structure is being
erased outright rather than merely thinned.</p>

<figure>
  <img src="sweep.png" alt="Points kept, surface error and normal deviation versus voxel size">
  <figcaption>Point count falls off a cliff well before the error curves rise steeply — the region
  left of the green line buys a large reduction almost for free.</figcaption>
</figure>

<h2>Visual comparison</h2>
<figure>
  <img src="comparison.png" alt="Original, safe voxel, aggressive voxel and random baseline, in two views">
  <figcaption>Original · voxel 0.210 · voxel 0.279 · random subsample at the same budget. Colours are
  the cloud's own RGB; point size is identical across panels, so density differences are real.</figcaption>
</figure>
<p><a href="comparison.html">Open the interactive 3-D viewer →</a> (rotate/zoom, original vs downsampled side by side)</p>

<h2>Why voxel, not random</h2>
<div class="scroll">
<table>
<thead><tr><th>method (5,592 pts)</th><th>mean</th><th>RMS</th><th>p95</th><th>Hausdorff</th><th>normal dev</th></tr></thead>
<tbody>
  <tr class="aggr"><td>voxel 0.279</td><td>0.0829</td><td>0.0960</td><td>0.1536</td><td>0.2145</td><td>13.3°</td></tr>
  <tr><td>random subsample</td><td>0.0860</td><td>0.1261</td><td>0.2310</td><td>0.6478</td><td>12.9°</td></tr>
</tbody>
</table>
</div>
<p>Mean error is nearly the same; the tails are not. Random subsampling is 50% worse at p95 and
<strong>3× worse at the Hausdorff bound</strong> — it thins uniformly-dense regions and leaves holes
wherever the dice fall badly. Voxel downsampling caps the worst-case gap by construction, and averages
positions, colours and normals within each cell instead of picking a survivor.</p>

<figure>
  <img src="error_map.png" alt="Per-point error heat maps for the two voxel sizes and the random baseline">
  <figcaption>Original points coloured by distance to the kept surface, shared scale. Voxel error is
  dim and uniform; the random baseline is speckled with bright hotspots — the same average, a much
  worse worst case.</figcaption>
</figure>

<h2>Files</h2>
<ul class="files">
  <li><a href="bigpointcloud_001.ply">bigpointcloud_001.ply</a> — original, 11,357 pts <span class="size">(0.58 MB)</span></li>
  <li><a href="bigpointcloud_001_voxel0.210_safe.ply">bigpointcloud_001_voxel0.210_safe.ply</a> — 8,832 pts <span class="size">(0.45 MB)</span></li>
  <li><a href="bigpointcloud_001_voxel0.279_aggressive.ply">bigpointcloud_001_voxel0.279_aggressive.ply</a> — 5,592 pts <span class="size">(0.29 MB)</span></li>
  <li><a href="sweep.csv">sweep.csv</a> — full metric table</li>
  <li><a href="voxel_downsample.py">voxel_downsample.py</a> — the script that produced all of this</li>
</ul>
<p>Normals and colours survive the downsample (averaged per voxel), so the outputs drop straight into
a PointNet++ pipeline that expects them.</p>

<h2>Reproduce</h2>
<p><code>python voxel_downsample.py --input bigpointcloud_001.ply</code> — auto-selects both operating
points. Override with <code>--voxel 0.28</code> or <code>--target-points 3000</code>. Needs
<code>open3d</code>, <code>numpy</code>, <code>matplotlib</code>, and optionally <code>plotly</code>
for the interactive viewer.</p>

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<footer>Generated with Open3D 0.18 · figures rendered headless with Matplotlib.</footer>

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