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mathematics

Point cloud

Point cloud is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Point cloud rather than just read about it. In short: A point cloud is a discrete set of data points in space. The points may represent a 3D shape or object.

Point cloud — main illustration
Point cloud — illustration

Key takeaways

  • Point cloud belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Point cloud to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Point cloud from memory before moving on to harder problems.

Reference excerpt

A point cloud is a discrete set of data points in space. The points may represent a 3D shape or object. Each point position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than position such as RGB colors, normals, timestamps and others. Point clouds are generally produced by 3D scanners or by photogrammetry software, which measure many points on the external surfaces of objects around them. As the output of 3D scanning processes, point clouds are used for many purposes, including to create 3D computer-aided design (CAD) or geographic information systems (GIS) models for manufactured parts, for metrology and quality inspection, and for a multitude of visualizing, animating, rendering, and mass customization applications.

Alignment and registration

When scanning a scene in real world using Lidar, the captured point clouds contain snippets of the scene, which requires alignment to generate a full map of the scanned environment. Point clouds are often aligned with 3D models or with other point clouds, a process termed point set registration. The Iterative closest point (ICP) algorithm can be used to align two point clouds that have an overlap between them, and are separated by a rigid transform. Point clouds with elastic transforms can also be aligned by using a non-rigid variant of the ICP (NICP). With advancements in machine learning in recent years, point cloud registration may also be done using end-to-end neural networks. For industrial metrology or inspection using industrial computed tomography, the point cloud of a manufactured part can be aligned to an existing model and compared to check for differences. Geometric dimensions and tolerances can also be extracted directly from the point cloud.

Conversion to 3D surfaces

While point clouds can be directly rendered and inspected, point clouds are often converted to polygon mesh or triangle mesh models, non-uniform rational B-spline (NURBS) surface models, or CAD models through a process commonly referred to as surface reconstruction. There are many techniques for converting a point cloud to a 3D surface. Some approaches, like Delaunay triangulation, alpha shapes, and ball pivoting, build a network of triangles over the existing vertices of the point cloud, while other approaches convert the point cloud into a volumetric distance field and reconstruct the implicit surface so defined through a marching cubes algorithm. In geographic information systems, point clouds are one of the sources used to make digital elevation model of the terrain. They are also used to generate 3D models of urban environments. Drones are often used to collect a series of RGB images which can be later processed on a computer vision algorithm platform such as on AgiSoft Photoscan, Pix4D, DroneDeploy or Hammer Missions to create RGB point clouds from where distances and volumetric estimations can be made. Point clouds can also be used to represent volumetric data, as is sometimes done in medical imaging. Using point clouds, multi-sampling and data compression can be achieved.

MPEG point cloud compression MPEG began standardizing point cloud compression (PCC) with a Call for Proposal (CfP) in 2017. Three categories of point clouds were identified: category 1 for static point clouds, category 2 for dynamic point clouds, and category 3 for Lidar sequences (dynamically acquired point clouds). Two technologies were finally defined: G-PCC (Geometry-based PCC, ISO/IEC 23090 part 9) for category 1 and category 3; and V-PCC (Video-based PCC, ISO/IEC 23090 part 5) for category 2. The first test models were developed in October 2017, one for G-PCC (TMC13) and another one for V-PCC (TMC2). Since then, the two test models have evolved through technical contributions and collaboration, and the first version of the PCC standard specifications was expected to be finalized in 2020 as part of the ISO/IEC 23090 series on the coded representation of immersive media content.

See also Euclideon – 3D graphics engine which makes use of a point cloud search algorithm to render images MeshLab – open source tool to manage point clouds and convert them into 3D triangular meshes CloudCompare – open source tool to view, edit, and process high density 3D point clouds Point Cloud Library (PCL) – comprehensive BSD open source library for n-D point clouds and 3D geometry processing Point Set Processing in CGAL, the Computational Geometry Algorithms Library

References

Illustrations

Point cloud: A point cloud image of a torus
A point cloud image of a torus
Point cloud: Geo-referenced point cloud of Red Rocks, Colorado (by DroneMapper)
Geo-referenced point cloud of Red Rocks, Colorado (by DroneMapper)
Point cloud: An example of a 1.2 billion data point cloud render of Beit Ghazaleh, a heritage site in danger in Aleppo (Syria)[8]
An example of a 1.2 billion data point cloud render of Beit Ghazaleh, a heritage site in danger in Aleppo (Syria)[8]
Point cloud: Generating or reconstructing 3D shapes from single or multi-view depth maps or silhouettes and visualizing them in dense point clouds[9]
Generating or reconstructing 3D shapes from single or multi-view depth maps or silhouettes and visualizing them in dense point clouds[9]

Worked examples

Example 1 — a first encounter with Point cloud

Start with the simplest possible case. Write down what Point cloud claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Point cloud before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Point cloud ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Point cloud

In research
Point cloud appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Point cloud in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Point cloud is common in secondary-school and first-year university syllabi. It links to neighbouring topics 3D computer graphics, Geometry processing, so understanding it makes those chapters shorter.
In everyday life
Look for Point cloud outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Point cloud in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Point cloud means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Point cloud out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Point cloud in simple terms?

A point cloud is a discrete set of data points in space. The points may represent a 3D shape or object.

Why does Point cloud matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Point cloud?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Point cloud.

Tags

  • 3D computer graphics
  • Geometry processing

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