Improved Manifold Coordinate Representations of Large-Scale Hyperspectral Scenes

Abstract

In recent publications, we have presented a data-driven approach to representing the nonlinear structure of hyperspectral imagery using manifold coordinates. The approach relies on graph methods to derive geodesic distances on the high-dimensional hyperspectral data manifold. From these distances, a set of intrinsic manifold coordinates that parameterizes the data manifold is derived. Scaling the solution relied on divide-conquer-and-merge strategies for the manifold coordinates because of the computational and memory scaling of the geodesic coordinate calculations. In this paper, we improve the scaling performance of isometric mapping (ISOMAP) and achieve full-scene global manifold coordinates while removing artifacts generated by the original methods. The CPU time of the enhanced ISOMAP approach scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(N log^2(N))$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> is the number of samples, while the memory requirement is bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(Nlog(N))$</tex> . Full hyperspectral scenes of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(10^6)$</tex> samples or greater are obtained via a reconstruction algorithm, which allows insertion of large numbers of samples into a representative “backbone” manifold obtained for a smaller but representative set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(10^5)$</tex> samples. We provide a classification example using a coastal hyperspectral scene to illustrate the approach.