Analytical fully constrained least squares linear spectral mixture analysis

This chapter revisits a well-known fully constrained least squares (FCLS) method developed by Heinz and Chang for linear spectral unmixing. Due to the two physical constraints, abundance sum-to-one constraint (ASC) and abundance non-negativity constraint (ANC), FCLS does not have analytic solutions. This is because ANC is an inequality constraint and the Lagrange multiplier method cannot be applied for this purpose. So, in order to find nuAbstractmerical solutions, FCLS utilizes the Kuhn-Tucker conditions to develop a numerical algorithm to find approximate solutions. As an intriguing twist, a modified FCLS (MFCLS) developed by Ren and Chang converts the inequality-based ANC to an equality constraint, called absolute abundance sum-to-one constraint (AASC) such that the Lagrange multiplier method can be used to find numerical solutions. However, AASC is not differentiable. To resolve this dilemma, MFCLS particularly derives two iterative equations to be used for finding numerical solutions. Although both FCLS and MFCLS make use of different constraints, they still must rely on numerical algorithms to find approximate solutions. This chapter further derives an analytic approach, to be called analytical FCLS (AFCLS) which can find FCLS solutions analytically in closed forms via Cramer's rule. Accordingly, the AFCLS-unmixed results using analytical solutions are more accurate than FCLS-unmixed and MFCLS-unmixed results resulting from numerical solutions.