The input set has the underlying shape of a pyramid, to which several peaks and troughs of varying scale, as well as random noise, have been added. We model this set as a piecewise linear 2-manifold with boundary, and find a triangulation of the same as a subcomplex of a tetrahedralization of the 2 × 2 × 2 cube centered at the origin, within which the set is located. We use the method of constrained Delaunay tetrahedralization [41] implemented in the package TetGen [39] for this purpose. We then compute the multiscale simplicial flat norm decomposition of the input set at various scale ( λ ) values. At high values, e.g., when λ = 6, the optimal decomposition resembles the input set with the small kinks due to random noise smoothed out. At the other end, for λ = 0 . 01, the optimal decomposition resembles a flat “sheet”. For intermediate values of λ , the optimal decomposition captures features of the input set at varying scales. 

The entire 3-complex mesh modeling the cube in question consisted of 14,002 tetra- hedra and 28,844 triangles. For each λ , computation of the multiscale simplicial flat norm described above took only a few minutes on a regular PC using standard functions from MATLAB. This example demonstrates the feasibility of efficiently computing flat norm decompositions of large datasets in high dimensions, for the purposes of denoising or to recover scale information of the data. 

