Posted by
Kenneth Sloan-2 on
May 25, 2007; 8:27pm
URL: http://imagej.273.s1.nabble.com/particle-distribution-randomness-uniformity-tp3699301p3699305.html
On May 25, 2007, at 10:54 AM, Albert Cardona wrote:
> Statistics books suggest that a random spread of particles would
> follow a Poisson distribution.
This is not realistic for points which are representative centers of
finite-sized particles. There are two common twiddles: a HARD center
(where a given point absolutely forbids another point with a small
epsilon, and a SOFT center (where the observed density rises as you
move away from a particular central point. Variations on these
schemes were all the rage 20 years ago. They are based on real
physical considerations on where a "point" can actually be.
Just about the most complicated distribution worth worrying about has
a HARD region (R0) where other points are excluded, followed by a
smooth transition from 0 density to a Maximum density (from R0 to
R1), followed by an absolutely uniform distribution at distances
greater than R1.
> For example, consider as many cells in a grid over the image as
> particles per image. Then, following a Poisson, determine the
> expected number of cells with zero, one, two, etc. particles.
> Finally compare the distribution of your spread with the expected,
> Poisson-following one. You can do this with a Chi-square.
>
> The above gives you a qualitative answer: are your particles
> randomly distributed or not. If they are, then you can do a k-means
> clustering analysis to find out how many clusters arise from your
> particle spread.
>
Quite correct - just be careful of physical considerations that make
0 < R0 < R1. As a sanity check, if you set R1 = R0 = 0, then you
should get the pure Poisson Distribution (assuming, with complete
loss of generality, that no two points are in the IDENTICAL position.
> Albert
--
Kenneth Sloan
[hidden email]
Computer and Information Sciences +1-205-934-2213
University of Alabama at Birmingham FAX +1-205-934-5473
Birmingham, AL 35294-1170
http://www.cis.uab.edu/sloan/