Many flexible parameterizations exist to represent data on the sphere. In addition to the venerable spherical
harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper
we focus on the latter two: spherical wavelets developed for geophysical applications on the cubed sphere,
and the Slepian "tree", a new construction that combines a quadratic concentration measure with wavelet-like
multiresolution. We discuss the basic features of these mathematical tools, and illustrate their applicability in
parameterizing large-scale global geophysical (inverse) problems.
The hyperspectral subpixel detection and classification problem has been intensely studied in the downward-looking case, typically satellite imagery of agricultural and urban areas. In contrast, the hyperspectral imaging case when "looking up" at small or distant satellites creates new and unforeseen problems. Usually one pixel or one fraction of a pixel contains the imaging target, and spectra tend to be time-series data of a single object collected over some time period under possibly varying weather conditions; there is little spatial information available. Often, the number of collected traces is less than the number of wavelength bins, and a materials database with imperfect representative spectra must be used in the subpixel classification and unmixing process. A procedure is formulated for generating a "good" set of classes from experimentally collected spectra by assuming a Gaussian distribution in the angle-space of the spectra. Specifically, Kernel K-means, a suboptimal ML-estimator, is used to generate a set of classes. Covariance information from the resulting classes and weighted least squares methods are then applied to solve the linear unmixing problem. We show with cross-validation that Kernel K-means separation of laboratory material classes into "smaller" virtual classes before unmixing improves the performance of weighted least squares methods.