Progressive Minimum-Link Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. To facilitate exploration of these simplifications, we wish to seamlessly switch between different levels of detail. These simplifications must thus be consistent with one another. We call such a set of inter-consistent simplifications a progressive simplification. Existing geometric optimization algorithms used for curve simplification minimize the complexity of the simplified curve for each level of detail independently. These algorithms therefore do not produce progressive simplifications. All currently known progressive simplification algorithms are based on heuristics and therefore do not strictly minimize the simplification complexity. In this thesis, we wish to bridge the gap between these two types of algorithms. More specifically, we wish to produce progressive simplifications for which the sum or integral of the simplification complexity over all levels of detail is minimized. We present an algorithm that solves this problem in O(nm) time, where n is the length of the input curve and m the number of different levels of detail. This algorithm is compatible with any distance measure such as Hausdorff or Fréchet, and can be used to compute an optimal simplification for continuous scaling in O(n) time. We further explore two greedy algorithms with a running time of O(nm). These algorithms greedily construct each simplification, and may therefore yield a set of simplifications for which the cumulative complexity is non-minimal. All algorithms proposed in this thesis are based on an existing computational framework that employs finding shortest paths in so-called shortcut graphs. These are graphs in which each edge represents a single line segment that is a valid simplification for a contiguous subsequence of the input curve. Each such line segment is called a shortcut. To speed up the simplification algorithms, we present multiple supporting techniques pertaining to these shortcut graphs. First of all, we propose a space-efficient representation of the shortcut graph that can be used to find shortest paths in O(n log n) time in practice. This is an improvement over using breadth-first search in O(n) time. Furthermore, we present an algorithm for efficiently constructing shortcut graphs under the Hausdorff distance for many different levels of detail. This algorithm computes the error of all shortcuts in O(n log n) time, improving over O(n) time using existing techniques. Experimental evaluation of the developed techniques on animal movement data reveals that the new representation of the shortcut graph drastically lowers the memory usage while significantly improving the running time of using shortcut graphs for curve simplification when compared to existing techniques. The optimal progressive simplification algorithm produces simplifications that are comparable in size to minimal non-progressive simplifications, but was found to be limited to small input curves. We showed that one of the greedy algorithms is a good alternative which produces near-minimal progressive simplifications.