Piecewise Convex Multiple-Model Endmember Detection and Spectral Unmixing

Context dependent spectral unmixing.

“…Another limitation of existing spectral unmixing algorithms is that they do not take into account the distribution of the data in the spectral space while unmixing. This is the case even for the piece-wise convex representation in [49].…”

“…However, it may be the case that multiple sets of endmembers, defining several overlapping convex regions, can better describe the hyperspectral image. This issue has been addressed in [49][50][51][52][53], where the linear mixing model has been extended to multiple sets of endmembers. Each endmember set is found using the convex geometry model resulting in a piece-wise convex representation of the hyperspectral data.…”

“…The first one is the Iterated Constrained Endmembers (ICE) algorithm [61] which fits a simplex to the data while penalizing its volume. The second one is the Piecewise Convex Multiple Model Endmember Detection (P-COMMEND) algorithm [49] which models a hyperspectral image using a piece-wise convex representation.…”

“…In [49], the authors showed that, using Lagrange multipliers optimization along with the Karush-Kuhn-Tucker (KKT) conditions, the objective function in (2.6) can be minimized by updating the endmembers, the proportions and the fuzzy memberships using…”

“…In the implementation of the P-COMMEND algorithm [49], the membership values are initialized using the Fuzzy C-Means algorithm [72] (which is, in turn, randomly initialized), and the endmember sets E i are initialized using the Minimum Volume Simplex Analysis (MVSA) algorithm [70]. The algorithm is stopped whenever the estimated parameters do not change significatively between successive iterations.…”