Thus, it is important to realize that even the ``random'' samples are deterministic. They are designed to optimize performance on statistical tests. Many sophisticated statistical tests of uniform randomness are used. One of the simplest, the chi-square test, is described here. This test measures how far computed statistics are from their expected value. As a simple example, suppose and is partitioned into a by array of square boxes. If a set of samples is chosen at random, then intuitively each box should receive roughly of the samples. An error function can be defined to measure how far from true this intuition is:
This irregularity can be observed in terms of Voronoi diagrams, as shown in Figure 5.3. The Voronoi diagram partitions into regions based on the samples. Each sample has an associated Voronoi region . For any point , is the closest sample to using Euclidean distance. The different sizes and shapes of these regions give some indication of the required irregularity of random sampling. This irregularity may be undesirable for sampling-based motion planning and is somewhat repaired by the deterministic sampling methods of Sections 5.2.3 and 5.2.4 (however, these methods also have drawbacks).
Steven M LaValle 2020-08-14