Point Clouds as 3D Point Sets

Point clouds keep raw 3D coordinates with no grid and no surface, which saves memory but leaves neighbor search to you.

Why Does This Exist?

Sensors output points, not shapes. The representation closest to the sensor keeps $(x, y, z)$ plus intensity with no quantization. Point cloud processing teaches the network side; this page teaches the representation trade: exact positions and tiny memory against no surface, no inside-outside test, and density that collapses with distance.

Think of It Like This

Stars plotted from a telescope log

A telescope log lists each star's coordinates and brightness with no lines between them. The list is tiny and exact, but finding constellations means searching neighbors yourself. Drawing grid lines over the sky would quantize every star.

Points are the log and voxels are the grid overlay. The analogy stops at motion: stars sit still while LiDAR points sweep.

How It Actually Works

A cloud is matrix $P \in \mathbb{R}^{N \times 3}$ with optional features. Memory is $M = N \times 3 \times 4$ bytes for float32. Density falls roughly as $1/r^2$ with range $r$ because a fixed angular beam spreads. Neighbor search needs a KD-tree or hashing since adjacency is not stored.

Worked example

Take $N = 100{,}000$ points. Memory is $100{,}000 \times 3 \times 4 = 1{,}200{,}000$ bytes, about $1.2$ MB. A car at $10$ m returns about $800$ points; the same car at $40$ m returns $800 \times (10/40)^2 = 50$ points. That $16$ -fold drop is why far objects need different thresholds than near ones.

Watch Out For

Treating density as uniform across range

A fixed neighbor radius of $0.2$ m finds $40$ neighbors near but $2$ far, so features shift with range. Scale the radius with $r$ or sample fixed- $k$ neighborhoods instead.

The Quick Version

Point clouds store exact positions with minimal memory and no quantization.
Density drops with range squared, so far objects look sparse.
No surface or occupancy answer comes free; search builds neighborhoods.
Do not use raw points when you need watertight surfaces; mesh or field instead.
$100{,}000$ points fit in about $1.2$ MB.