Determining the intrinsic dimension of a hyperspectral image is an important step in the spectral unmixing process and under- or overestimation of this number may lead to incorrect unmixing in unsupervised methods. In this paper, we discuss a new method for determining the intrinsic dimension using recent advances in random matrix theory. This method is entirely unsupervised, free from any user-determined parameters and allows spectrally correlated noise in the data. Robustness tests are run on synthetic data, to determine how the results were affected by noise levels, noise variability, noise approximation, and spectral characteristics of the endmembers. Success rates are determined for many different synthetic images, and the method is tested on two pairs of real images, namely a Cuprite scene taken from Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) and SpecTIR sensors, and a Lunar Lakes scene taken from AVIRIS and Hyperion, with good results.

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Cytotoxicity studies using a 3-(4,5 dimethylthiazol-2-yl)-2,5 diphenyl tetrazolium bromide (MT)-based in vitro toxicity assay revealed that McCoy cells exposed to low concentrations of mercuric (0.7 yM),cadmium (1 1M) and cupric chloride (3 tIM) exhibited significant increases in cellular activity. This increased activity, previously termed hormesis, coincided with the production ofhigh levels ofthe stress proteins, heat shock protein 70 (Hsp 70) and metallothionein, while the high constitutive expression of these proteins in cadmium-resistant mutant (CRM) cells corresponded to constitutive hormetic activity. Hormesis was found to obey uniform kinetics allowing for a mathematical description of this increased activity. These results suggest that hormetic activity is a specific cellular response, and most likely, a stress response to low but harmful levels of toxic agents and may therefore provide a rapid test for the presence of toxicants at concentrations associated with chronic toxicity.

Arising from courses taught by the authors, this largely self-contained treatment is ideal for mathematicians who are interested in applications or for students from applied fields who want to understand the mathematics behind their subject. Early chapters cover Fourier analysis, functional analysis, probability and linear algebra, all of which have been chosen to prepare the reader for the applications to come. The book includes rigorous proofs of core results in compressive sensing and wavelet convergence. Fundamental is the treatment of the linear system y=Φx in both finite and infinite dimensions. There are three possibilities: the system is determined, overdetermined or underdetermined, each with different aspects. The authors assume only basic familiarity with advanced calculus, linear algebra and matrix theory and modest familiarity with signal processing, so the book is accessible to students from the advanced undergraduate level. Many exercises are also included.

Numerical experiments with smooth surface extension and image inpainting using harmonic and biharmonic functions are carried out. The boundary data used for constructing biharmonic functions are the values of the Laplacian and normal derivatives of the functions on the boundary. Finite difference schemes for solving these harmonic functions are discussed in detail.

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Let (M, g 1 ) be a complete d-dimensional Riemannian manifold for d > 1. Let X n be a set of n sample points in M drawn randomly from a smooth Lebesgue density f supported in M . Let x, y be two points in M . We prove that the normalized length of the power-weighted shortest path between x, y through X n converges almost surely to a constant multiple of the Riemannian distance between x, y under the metric tensor g p = f 2(1−p)/d g 1 , where p > 1 is the power parameter.

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