Partially Geometric-Constrained Sequential Endmember Finding: Convex Cone Volume Analysis

Abstract

N-FINDR discussed in Chap. 6 uses Simplex Volume (SV) as a criterion to find endmembers which specify the vertices of a simplex with maximal SV. A simplex can be considered as a convex set within which all data sample vectors are fully constrained by its vertices via linear convexity. From a Linear Spectral Mixture Analysis (LSMA) viewpoint, the data sample vectors within a simplex can be linearly mixed by its vertices with full abundance constraints, Abundance Sum-to-one Constraint (ASC) and Abundance Non-negativity Constraint (ANC). This chapter presents an approach, called called Convex Cone Volume Analysis (CCVA) developed by Chang et al. (2016) that uses one fewer abundance constraint by only imposing ANC without ASC for finding endmembers. It is a partially abundance-constrained (more specifically, ANC) technique which implements Convex Cone Volume (CCV) as a criterion instead of SV used by N-FINDR. As shown in this chapter, finding the maximal volume of a convex cone in the original data space is equivalent to finding the maximal volume of a simplex formed by the projection of the convex cone on a specific hyperplane, referred to as Convex Cone Projection (CCP) whose dimensionality is reduced by one from the original data dimensionality. This makes sense because a simplex requires an additional ASC imposed on its convexity structure and projecting a convex cone on a hyperplane is equivalent to imposing ASC on CCP, which is actually a simplex on a hyperplane. As a result, CCVA can take full advantage of whatever is developed for N-FINDR in Chap. 6 to derive its counterpart for CCVA.