Hyperspectral Data Processing: Algorithm Design and Analysis

7.2 Convex Geometry-Based Endmember Extraction

Due to the nature of a mixed sample vector in a linear mixing model, convex geometry has been used as a major criterion behind EEA design. As a simple illustrative example, assume that e₁ and e₂ are two endmembers. Any data sample vector x lying in a line segment connecting these two endmembers as end points can be expressed as a point linearly mixed by e₁ and e₂ and described in the form of , with based on convexity. For three endmembers e₁, e₂, and e₃, a data sample vector that is linearly mixed by these three endmembers must lie in a triangle with e₁, e₂, and e₃ as its three vertices. Similarly, a data sample vector within a triangular pyramid/tetrahedron, square-pyramid, and p-vertex simplex is also considered to be linearly mixed by its four vertices, five vertices, and p vertices, respectively, all of which are considered as endmembers. The OP and simplex volume are two principal criteria to materialize the concept of convex geometry.

7.2.1 Convex Geometry-Based Criterion: Orthogonal Projection

PPI is a popular technique widely used in endmember extraction due to its availability in the ENVI software provided by the Research Systems. Also due to its propriety and limited publication, its detailed implementation has never been made ...