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Peter Martin's M.S. Degree, 2004
Spatial Interpolation in Other Dimensions
Master of Science, Geography, Oregon State University, Summer 2004
Emphasis in Geographic Information Science, Integrated Minor in Oceanography
Graduate committee: D. Wright, C. Goldfinger, A.J. Kimerling, Court Smith
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The purpose of this work is to expand the theoretical foundations of interpolation of spatial data, by showing how ideas and methods from information theory and signal processing are applicable to the work of geographers. Attention is drawn to the distinction between what we study and how we represent it as a sum of components; hence mathematical transforms are introduced as rearrangements of information that result in alternative representations of the signals of interest. A spatial model is developed for understanding transforms as the means to obtain different views of function space, and the question of interpolation is recast in geometric terms, as a matter of placing an approximation within the bounds of likelihood according to context, using data to eliminate possibilities and estimating coefficients in an appropriate base.
With an emphasis on terrain elevation- and bathymetry signals, applications of the theory are illustrated in the second part, with particular attention to 1/f spectral characteristics and scale-wise self-similarity as precepts for algorithms to generate "expected detail". Methods from other fields as well as original methods are tested for scientific visualization and cartographic application to geographic data. These include fractal image super-resolution by pyramid decomposition, wavelet-based interpolation schemes, principal component analysis, and Fourier-base schemes, both iterative and non-iterative. Computation methods are developed explicitly, with discussion of computation time.
Finally, the quest to simulate "detailed" data is justified by challenging the traditional measure of interpolation accuracy in the standard base, proposing instead an alternative measure in a space whose transform reflects the relative importance of the components to communication of information.