Linear models, signal detection, and the Grassmann manifold - Schwickerath, Anthony Norbert, author; Kirby, Michael, advisor; Peterson, Chris, advisor; Scharf, Louis, committee member; Eykholt, Richard, committee member (Colorado State University. Libraries, 2014)
Standard approaches to linear signal detection, reconstruction, and model identification problems, such as matched subspace detectors (MF, MDD, MSD, and ACE) and anomaly detectors (RX) are derived in the ambient measurement space using statistical methods (GLRT, regression). While the motivating arguments are statistical in nature, geometric interpretations of the test statistics are sometimes developed after the fact. Given a standard linear model, many of these statistics are invariant under orthogonal transformations, have a constant false alarm rate (CFAR), and some are uniformly most powerful invariant (UMPI). These properties combined with the simplicity of the tests have led to their widespread use. In this dissertation, we present a framework for applying real-valued functions on the Grassmann manifold in the context of these same signal processing problems. Specifically, we consider linear subspace models which, given assumptions on the broadband noise, correspond to Schubert varieties on the Grassmann manifold. Beginning with increasing (decreasing) or Schur-convex (-concave) functions of principal angles between pairs of points, of which the geodesic and chordal distances (or probability distribution functions) are examples, we derive the associated point-to-Schubert variety functions and present signal detection and reconstruction algorithms based upon this framework. As a demonstration of the framework in action, we implement an end-to-end system utilizing our framework and algorithms. We present results of this system processing real hyperspectral images.
Keywords: Grassmann manifold; Schubert variety; minimum distance; geometric signal detection; geometric signal recovery; linear subspace model