most recent 30 from http://mathoverflow.net 2013-06-19T17:15:01Z http://mathoverflow.net/feeds/question/51335 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51335/a-point-process-for-modeling-location-of-trees-in-an-infinite-forest Scott Armstrong 2011-01-06T19:28:31Z 2011-01-07T00:03:11Z

<p>I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process with these features, but I don't know the field very well and have had no success in my search thus far.</p> <p>The most important property I want is for the points to be <em>repulsive</em> and in the sense that there is a characteristic distance $r> 0$ between any two nearby points, and <em>attractive</em> in the sense that there is zero probability of finding a ball of radius say, $10r$, in which there are no points. Finally, the process should be stationary so that the distribution is unchanged by translation. Isotropy (invariance under rotations) would be nice, but I don't really care. It is crucial for my purposes that it be an infinite process, defined on all of $\mathbb R^n$, and in dimension $n\geq 2$. I believe that in one dimension it is easy enough to construct such an example.</p> <p>The idea is to model, for example, the location of trees in a forest. </p> <p>Is there some well-known point process I am informally describing (or is it easy enough to construct one?), or is there some good reason I am having trouble finding one?</p>

http://mathoverflow.net/questions/51335/a-point-process-for-modeling-location-of-trees-in-an-infinite-forest/51360#51360 Carl Feynman 2011-01-07T00:03:11Z 2011-01-07T00:03:11Z

<p>All your requirements are satisfied by the Poisson-Disk process. It's the limit of a uniform sampling process with a minimum-distance rejection criterion. The easiest way to describe it is as the limit of the following process: uniformly sample points in the area of interest, rejecting any points that are less than $r$ from an existing point. Keep sampling points until there is no area not within $r$ of a point, and you're done. </p> <p>This process is popular in computer graphics and image processing because its Fourier Transform has some nice properties.</p> <p>This process generates a distribution without any big holes, so it may not be the right thing for a forest where small clearings are allowed but big ones are not.</p> <p>The process as described is very slow to converge. Some Googling for "Poisson Disk" suggests there are far more efficient modern algorithms than this for generating a Poisson-Disk distribution. But I can't guide you to that literature; I last generated a Poisson Disk distribution in 1982, and we did it the hard way.</p>