Topics in multifractal analysis of two- and three-dimensional structures in spaces of constant curvature.

摘要

This thesis will investigate the application of fractal and multifractal scaling indices as a discriminator of patterns and pattern formation mechanisms in spaces of constant curvature (both flat and curved manifolds). It is found that curvature of the space in which the fractal is built does not influence the resulting dimensionality, and that the associated dimension is an artifact of the construction paradigm itself. For the full multifractal analysis, two distinct data sets will be studied, including cosmological structure models, as well as non-representational images. The multifractal spectrum of various three-dimensional representations of Packed Swiss Cheese cosmologies in open, closed, and flat spaces are measured, and it is similarly determined that the curvature of the space does not alter the associated fractal structure. These results are compared to observational data and simulated models of large scale galaxy clustering, to assess the viability of the PSC as a candidate for such structure formation. It is found that the PSC dimension spectra do not match those of observation, and possible solutions to this discrepancy are offered, including accounting for potential luminosity biasing effects. Various random and uniform sets are also analyzed to provide insight into the meaning of the multifractal spectrum as it relates to the observed scaling behaviors. The method will also be tested as a tool for nonrepresentational image analysis and classification. For the images considered, while the associated dimensions are perhaps useful in generalized classifications of patterns, in most cases the associated fractal dimension and multifractal spectra do not yield any uniquely-identifying structural characteristics. The procedure is much more effective at detecting specific structural formation signatures in the case of the three dimensional distributions. The images are also analyzed for potential fractal structural signatures in their associated luminance gradients, as a toy model for visual discrimination. Apparent structural differences are found, but as with the former case, the exact meaning of the statistics are vague.