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The purpose of this work is to broaden the theoretical foundations of interpolation
of spatial data, by showing how ideas and methods from information theory and signal
processing are applicable to the 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. |
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