Independent Quadtrees

This dissertation deals with the problem of manipulating and storing an image using quadtrees. A quadtree is a tree in which each node has four ordered children or is a leaf. It can be used to represent an image via hierarchical decomposition. The image is broken into four regions. A region can be a solid color (homogeneous) or a mixture of colors (heterogeneous). If a region is heterogeneous it is broken into four subregions, and the process continues recursively until all subregions are homogeneous. The traditional quadtree suffers from dependence on the underlying grid. The grid coordinate system is implicit, and therefore fixed. The fixed coordinate system implies a rigid tree. A rigid tree cannot be translated, scaled, or rotated. Instead, a new tree must be built which is the result of one of these transformations. This dissertation introduces the independent quadtree. The independent quadtree is free of any underlying coordinate system. The tree is no longer rigid and can be easily translated, scaled, or rotated. Algorithms to perform these operations axe presented. The translation and rotation algorithms take constant time. The scaling algorithm has linear time in the number nodes in the tree. The disadvantage of independent quadtrees is the longer generation and display time. This dissertation also introduces an alternate method of hierarchical decomposition. This new method finds the largest homogeneous block with respect to the corners of the image. This block defines the division point for the decomposition. If the size of the block is below some cutoff point, it is deemed to be to small to make the overhead worthwhile and the traditional method is used instead. This new method is compared to the traditional method on randomly generated rectangles, triangles, and circles. The new method is shown to use significantly less space for all three test sets. The generation and display times are ambiguous. More time is taken for each node, but there are, on average, fewer nodes. The worst case is significantly worse