| Though the most common sampling strategy used in image processing is the rectangular lattice, the hexagonal lattice offers many advantages. In this thesis, we study techniques for processing hexagonally sampled data, with particular interest given to the design and implementation of filters and filterbanks.; A large variety of analysis, visualization, and processing techniques unique to the hexagonal lattice are collected from scattered sources and analyzed in a unified framework. In many cases, these techniques are presented in a more general form than their source.; We introduce the theory of generalized separability, which addresses the analysis of filters that can be implemented with a cascade of one-dimensional filtering operations along arbitrary orientations. We then show that the polyphase filters of a generalized separable filter can be implemented using a parallel sum of generalized separable filters.; Hexagonal separability is defined and examples are provided that show how hexagonal separability can be used to design a variety of important filters in a simple manner. Furthermore, efficient implementations for these hexagonally separable filters are provided.; The use of hexagonally separable filters within a three-band, a four-band, and a novel directional filterbank are presented and analyzed.; A MATLAB toolbox is discussed that can assist researchers studying topics in lattice-sampled systems. |
