Topological Data Analysis - Point Cloud Data

Point Cloud Data

Data is often represented as points in a Euclidean n-dimensional space E_n. The global shape of the data may provide information about the phenomena that the data represent.

One type of data set for which global features are certainly present is the so-called point cloud data coming from physical objects in 3D. E.g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object. Point cloud data refers to any collection of points in E_n or a (perhaps noisy) sample of points on a lower-dimensional subset.

For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of computer graphics and statistics. Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections to faithfully represent the features of the point cloud.

To convert a point cloud in a metric space into a global object, use the point cloud as the vertices of a graph whose edges are determined by proximity, then turn the graph into a simplicial complex and use algebraic topology to study it. An alternative approach is the minimum spanning tree-based method in the geometric data clustering. If a group of data points forms a cluster, then the geometry of this point cloud can be determined.