Boundary Compactness for Data of 3D or Higher Dimensions

Existing surface reconstruction methods only work for low-dimensional, mostly 2D or 3D, data. In real applications, high-dimensional data is difficult to interpret as it requires more dimensions to represent. As a dimension reduction method, manifold learning provides an explicit representation for the useful implicit information hidden in the original feature space. But the internal topological and differential structure has disappeared in the dimensionality reduction process. In order to investigate the topological and differential structure of a data set, we introduce boundary compactness, which is used to study the shape of the data set in the original high-dimensional feature space. Beginning with the Delaunay graph (for 3D data) or the complete graph (for data of higher dimensions), we select the edges based on the boundary compactness to make the relevance graph. The relevance graph is then used to reconstruct the surface of the data. Experiments show that the introduced technique based on the boundary compactness works well for data of 3D or higher dimensions.