Hyperspectral Data Processing: Algorithm Design and Analysis

14.2 Abundance-Constrained LSMA (AC-LSMA)

When the abundance-unconstrained LSMA specified by (12.1) is considered, its unconstrained LSE solution to (14.2) is given by

(14.3)

A commonly used least squares method is to minimize the LSE problem specified by (12.1) over the abundance vector subject to the constraints imposed by ASC and/or ANC on α. If we impose either ASC or ANC or both on (14.1), three LSE problems derived from AC-LSMA can be formulated as follows (Heinz and Chang 2001; Chang, 2003b).

i. Abundance sum-to-one constrained LSMA (ASCLS-LSMA) problem:

(14.4)

ii. Abundance nonnegativity-constrained LSMA (ANCLS-LSMA) problem:

(14.5)

iii. Abundance fully constrained LSMA (AFCLS-LSMA) problem:

(14.6)

The solutions to (14.4)–(14.6) were well documented by Heinz and Chang (2001) and Chang (2003a) and will be referred to as SCLS, NCLS, and FCLS solutions, respectively.