Simplex Volume Calculation

Abstract

Using maximal simplex volume (SV) as an optimal criterion for finding endmembers is a common approach and has been widely adopted in the literature. However, very little work has been reported on how SV is calculated because it can be simply done by finding the determinant of a matrix formed by vertices of a simplex. Interestingly, it turns out that calculating SV is much more complicated and involved than one might think. This chapter investigates this issue from two different aspects, through eigenanalysis and a geometric approach. The eigenanalysis takes advantage of the Cayley–Menger determinant to calculate the SV, referred to as determinant-based SV (DSV) calculation. The major issue with this approach is that when the matrix is of ill-rank, calculating the determinant runs into a singularity problem. To deal with this issue, two methods are generally considered. One is to perform data dimensionality reduction to make the matrix of full rank. The drawback of this method is that the original volume has been shrunk and the found volume of a dimensionality-reduced simplex is not the true original SV. The other is to use singular value decomposition to find singular values for calculating the SV. An issue arising from this method is its instability in numerical calculations. An alternative to eigenanalysis is a geometric approach derived from the simplex structure whose volume can be calculated by the product of the base and height of a simplex. The resulting approach is called geometric SV (GSV) calculation. This chapter explores DSV and GSV calculations, with further discussions in Chaps. 11 and 12.