We present a flag manifold based method for detecting chemical plumes in long-wave infrared hyperspectral movies. The method encodes temporal and spatial information related to a hyperspectral pixel into a flag, or nested sequence of linear subspaces. The technique used to create the flags pushes information about the background clutter, ambient conditions, and potential chemical agents into the leading elements of the flags. Exploiting this temporal information allows for a detection algorithm that is sensitive to the presence of weak signals. This method is compared to existing techniques qualitatively on real data and quantitatively on synthetic data to show that the flag-based algorithm consistently performs better on data when the SINR dB is low, and beats the ACE and MF algorithms in probability of detection for low probabilities of false alarm even when the SINR dB is high.
This paper concerns the minimax center of a collection of linear subspaces. When the subspaces are k-dimensional subspaces of R n , this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold, Gr(k, n). For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in this context leads to a candidate minimax center on each of the constituent manifolds, but does not inherently provide intuition about which candidate is the best representation of the data. Additionally, the solutions of different rank are generally not nested so a deflationary approach will not suffice, and the problem must be solved independently on each manifold. We propose and solve an optimization problem parametrized by the rank of the minimax center. The solution is computed using a subgradient algorithm on the dual. By scaling the objective and penalizing the information lost by the rank-k minimax center, we jointly recover an optimal dimension, k * , and a central subspace, U * ∈ Gr(k * , n) at the center of the minimum enclosing ball, that best represents the data.
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