A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, July 14, 2015. / Hyperspectral images nd application in many areas of modern society, we use them for land surveying, core sample analysis, in the conservation and forestry industries and many more. A major problem in hyperspectral images is how to deal with noise. Many methods that analyse hyperspectral images either need clean images or accurate estimations of the noise statistics in the images. The goal of this dissertation is to present and compare methods for statistic estimation and noise removal. We use an arti cial hyperspectral image to study some existing methods and develop some new ones based on existing methods, speci cally the BM3D algorithm. We test methods that estimate the level of the noise present in an image, methods that estimate the structure of the noise and methods that remove noise. We analyse all the methods under an additive noise model and consider spectrally correlated and uncorrelated noise. Within our investigations we investigate di erent types of correlation. We will show the strengths that the various methods have and establish a way to approach treating a hyperspectral image with no information beyond the image itself. Using our observations and insights from the experiments on the arti cial data we analyse some radiance data from the AVIRIS instrument. We show that the additive signal independent part of the noise is small but not negligible. We also show some evidence for the structure of the noise in the AVIRIS instrument.

Multiple gene phylogenetic analysis is of interest since single gene analysis often results in poorly resolved trees. Here the use of spectral techniques for analyzing multi-gene data sets is explored. The protein sequences are treated as categorical time series and a measure of similarity between a pair of sequences, the spectral covariance, is used to build trees. Unlike other methods, the spectral covariance method focuses on the relationship between the sites of genetic sequences. We consider two methods with which to combine the dissimilarity or distance matrices of multiple genes. The first method involves properly scaling the dissimilarity measures derived from different genes between a pair of species and using the mean of these scaled dissimilarity measures as a summary statistic to measure the taxonomic distances across multiple genes. We introduced two criteria for computing scale coefficients which can then be used to combine information across genes, namely the minimum variance (MinVar) criterion and the minimum coefficient of variation squared (MinCV) criterion. The scale coefficients obtained with the MinVar and MinCV criteria can then be used to derive a combined-gene tree from the weighted average of the distance or dissimilarity matrices of multiple genes. The second method is based on the singular value decomposition of a matrix made up of the p-vectors of pairwise distances for k genes. By decomposing such a matrix, we extract the common signal present in multiple genes to obtain a single tree representation of the relationship between a given set of taxa. Influence functions for the components of the singular value decomposition are derived to determine which genes are most influential in determining the combined-gene tree.

Thesis (Ph.D.) -- McMaster University, 2002. / Includes bibliographical references (leaves 145-157). Also available via World Wide Web.

x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )-graded equivariant cohomology of G -spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. In addition, the cohomology of Rep( G )-complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes. / Adviser: Daniel Dugger

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The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems. In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations. We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of v-functions. In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.

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