An Elastic-Window-Based Method for the Underdetermined Problem in Linear Spectral Unmixing to Enhance the Spatial Resolution of the Normalized Difference Vegetation Index Time Series

Inverting land cover reflectance or derived indices from low-spatial-resolution images to refine the spatial resolution of this data is cost-effective for land surface monitoring applications that face technical or budget limitations. Based on the linear spectral mixing model, many approaches have successfully unmixed coarse mixed pixels using high-spatial-resolution land cover maps in the past decades. However, in some cases, the solutions of linear systems composed of several mixed pixels may not be acquired due to the underdetermined problem. This study presents the causes of this problem and proposes an iterative approximation strategy to address it. An elastic-window-based algorithm was developed, where the initial size of the window was calculated based on the land cover of the mixed pixel. Mixed pixels of neighborhoods with similar land covers were then selected to form the unmixing linear system, which was examined through a simulation test to ensure it was not underdetermined. Otherwise, the window would expand to include more adjacent pixels. This process was repeated until a successful solution was obtained. A statistical analysis of sixty land cover maps from around the globe shows that the underdetermined problem exists at a low level but becomes more serious with an increase in mixed scale. The results demonstrate that the proposed algorithm effectively prevents the underdetermined problem for mixed pixels of different scales and can be integrated into the coarse NDVI downscaling procedure to refine spatial resolution. This study provides a reference for estimating underdetermined mixed pixels and benefits applications that require dealing with the inversion of land cover values directly.