Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case
This provides theoretical justification for an empirical method in hyperspectral imaging, addressing a domain-specific problem for researchers in remote sensing and signal processing, but it is incremental as it builds on existing MVES approaches.
The paper tackles the theoretical understanding of why the minimum volume enclosing simplex (MVES) criterion works for blind hyperspectral unmixing without pure pixels, proving that MVES is robust to lack of pure pixels under conditions of not too heavy mixing and asymmetric spreading, with verification through numerical simulations.
In blind hyperspectral unmixing (HU), the pure-pixel assumption is well-known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no pure-pixel case, a good blind HU approach to consider is the minimum volume enclosing simplex (MVES). Empirical experience has suggested that MVES algorithms can perform well without pure pixels, although it was not totally clear why this is true from a theoretical viewpoint. This paper aims to address the latter issue. We develop an analysis framework wherein the perfect endmember identifiability of MVES is studied under the noiseless case. We prove that MVES is indeed robust against lack of pure pixels, as long as the pixels do not get too heavily mixed and too asymmetrically spread. The theoretical results are verified by numerical simulations.