Modeling and Mitigating Noise in Graph and Manifold Representations of Hyperspectral Imagery

Over the past decade, manifold and graph representations of hyperspectral imagery (HSI) have been explored widely in HSI applications. There are a large number of data-driven approaches to deriving manifold coordinate representations including Isometric Mapping (ISOMAP)(1), Local Linear Embedding (LLE)(2), Laplacian Eigenmaps (LE)(3), Diffusion Kernels (DK)(4), and many related methods. Improvements to specific algorithms have been developed to ease computational burden or otherwise improve algorithm performance. For example, the best way to estimate the size of the locally linear neighborhoods used in graph construction have been addressed(6) as well as the best method of linking the manifold representation with classifiers in applications. However, the problem of how to model and mitigate noise in manifold representations of hyperspectral imagery has not been well studied and remains a challenge for graph and manifold representations of hyperspectral imagery and their application. It is relatively easy to apply standard linear methods to remove noise from the data in advance of further processing, however, these approaches by and large treat the noise model in a global sense, using statistics derived from the entire data set and applying the results globally over the data set. Graph and manifold representations by their nature attempt to find an intrinsic representation of the local data structure, so it is natural to ask how can one best represent the noise model in a local sense. In this paper, we explore the approaches to modeling and mitigating noise at a local level, using manifold coordinates of local spectral subsets. The issue of landmark selection of the current landmark ISOMAP algorithm(5) is addressed and a workflow is proposed to make use of manifold coordinates of local spectral subsets to make optimal landmark selection and minimize the effect of local noise.