In the linear unmixing of hyperspectral images, the observation pixels form a simplex whose vertices correspond to the endmembers, hence finding the endmembers is equivalent to extracting these vertices. A common technique for determining vertices is to analyze the simplex volume, but it usually has a high computational complexity, resulting from the exhaustive searching of volume in the large hyperspectral data. This problem limits the practicability and real-time application. In this paper, we utilize triangular factorization (TF) to calculate the volume, deducing a method named simplex volume analysis based on TF (SVATF). It requires just one comparison through the data to succeed in finding the global optimal solution for all the endmembers, thus improving the searching efficiency. Dimensionality reduction transformation is not necessary, which is another advantage of this method. Moreover, since TF is a broad conception including different methods, SVATF is a framework including various implementations. Based on TF, we also propose a fast learning algorithm named abundance quantification based on TF to estimate the abundances, which further saves the computation by utilizing the intermediate values involved in SVATF. The abundance estimation method can rectify possible errors in the given endmembers by utilizing two important constraints (abundance nonnegative constraint and abundance sum-to-one constraint) of the linear mixture model, so it is useful for the imagery without pure pixels. Experimental results on synthetic and real hyperspectral data demonstrate that the proposed methods can obtain accurate results with much lower computational complexity, with respect to other state-of-the-art methods.
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