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Point cloud

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Consider this object: Point cloud1.jpg

It is in fact a triple torus.

But what if we zoom in:

Point cloud1.jpg

We realize that the "object" is just a bunch of points suspended in space! It is called a point cloud.

Another example:

Point cloud2.jpg

Where do point clouds come from?

They come from scanning:

Scanning and point cloud.jpg

They come from data:

Sine noisy.jpg A plot of 3-dimensional point cloud.jpg

Add edges and faces.jpg

Usually a simplicial complex is created to tell what's behind the point cloud.

How? A simplistic approach is to choose a threshold r and then

  • add an edge between any two points if they are within r from each other,
  • add a face spanning three points if the diameter of the triangle is less then r,


To capture all r's and to deal with noise, persistence is used.