particle distribution - randomness/uniformity

[Annapragada, Sriram K.]

I had the following question. Any help will be really appreciated
 
I have a stack of binary-8bit images with a random distribution of particles; also each image has a different number of particles. I was trying to find out how random the distribution was i.e. how uniformly the particles are distributed across each image.
 
One of the previous mails to this list suggested averaging the shortest distance between all the particles - but that does not show any significant difference between images.
 
Is there a another way to do this?
 
thanks,
Kiran.

[Anast, John]

Kiran,

Have you looked at the std dev of the shortest distances...should have a
very low stdev for uniform distribution. With clusters, you may want to
look at three or N- nearest neighbors for each particle.

Another approach is to grid the image and count particles in each box
and look at that distribution vs box size

...john

-----Original Message-----
From: ImageJ Interest Group [mailto:[hidden email]] On Behalf Of
Annapragada, Sriram K.
Sent: Thursday, May 24, 2007 12:30 PM
To: [hidden email]
Subject: particle distribution - randomness/uniformity

I had the following question. Any help will be really appreciated
 
I have a stack of binary-8bit images with a random distribution of
particles; also each image has a different number of particles. I was
trying to find out how random the distribution was i.e. how uniformly
the particles are distributed across each image.
 
One of the previous mails to this list suggested averaging the shortest
distance between all the particles - but that does not show any
significant difference between images.
 
Is there a another way to do this?
 
thanks,
Kiran.

[Kenneth Sloan-2]

On May 24, 2007, at 11:29 AM, Annapragada, Sriram K. wrote:

> I had the following question. Any help will be really appreciated
>
> I have a stack of binary-8bit images with a random distribution of  
> particles; also each image has a different number of particles. I  
> was trying to find out how random the distribution was i.e. how  
> uniformly the particles are distributed across each image.
>
> One of the previous mails to this list suggested averaging the  
> shortest distance between all the particles - but that does not  
> show any significant difference between images.
>
> Is there a another way to do this?
>
> thanks,
> Kiran.

One traditional method is to plot the Cumulative Distribution of  
point-point distances.  There is some literature (I would have to dig  
to find it, sorry) on interpreting the results.  I used this a long  
time ago with some success to analyze the distribution of cones in  
human retina.  (search in PubMed for "Sloan KR" and you'll find the  
papers).

--
Kenneth Sloan                                          
[hidden email]
Computer and Information Sciences                        [phone]
University of Alabama at Birmingham              FAX [phone]
Birmingham, AL 35294-1170                http://www.cis.uab.edu/sloan/

[Gabriel Landini]

> I have a stack of binary-8bit images with a random distribution of
> particles; also each image has a different number of particles. I
> was trying to find out how random the distribution was i.e. how
> uniformly the particles are distributed across each image.

Hi,
There is also Ripley's book on Spatial Statistics.
There are implementations of some of those methods in R.

> One of the previous mails to this list suggested averaging the
> shortest distance between all the particles - but that does not
> show any significant difference between images.

Isn't that useful? Perhaps the differences in the number of particles is not
large enough to show any differences in the mean shortest distances.

Regards,

Gabriel

[Aloysius Phillips]

Kiran,

The open source stat package R has plugins called "splancs" and "spatstat".
These will tell you whether your data points are self-avoiding, clustered or
distributed in a poisson fashion. A google search for "point pattern
analysis" will be helpful towards how these statistical methods operate. I
have a similar dataset but havent gotten as far as the nuts and bolts of
these programs. The programs can input the image but its probably easier to
export the (x,y) coordinates from imageJ and import the matrix. Thats as far
as I've gotten. Good luck.

Aloysius Phillips
Division of Invertebrates
American Museum of Natural History
79th street and Central Park West
10024

Voice [phone]
FAX [phone]
----- Original Message -----
From: "Anast, John" <[hidden email]>
To: <[hidden email]>
Sent: Thursday, May 24, 2007 1:50 PM
Subject: Re: particle distribution - randomness/uniformity

> Kiran,
>
> Have you looked at the std dev of the shortest distances...should have a
> very low stdev for uniform distribution. With clusters, you may want to
> look at three or N- nearest neighbors for each particle.
>
> Another approach is to grid the image and count particles in each box
> and look at that distribution vs box size
>
> ...john
>
> -----Original Message-----
> From: ImageJ Interest Group [mailto:[hidden email]] On Behalf Of
> Annapragada, Sriram K.
> Sent: Thursday, May 24, 2007 12:30 PM
> To: [hidden email]
> Subject: particle distribution - randomness/uniformity
>
> I had the following question. Any help will be really appreciated
>
> I have a stack of binary-8bit images with a random distribution of
> particles; also each image has a different number of particles. I was
> trying to find out how random the distribution was i.e. how uniformly
> the particles are distributed across each image.
>
> One of the previous mails to this list suggested averaging the shortest
> distance between all the particles - but that does not show any
> significant difference between images.
>
> Is there a another way to do this?
>
> thanks,
> Kiran.
>

[Dimiter Prodanov]

Hi,

You can uses for example Chi-square test or nearest neighbour distance
distribution or Ripley's K-function.
None of them are available as plugins.
Best regards

Dimiter Prodanov

[Albert Cardona]

Statistics books suggest that a random spread of particles would follow
a Poisson distribution.
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.

Albert

[Kenneth Sloan-2]

On May 24, 2007, at 2:24 PM, Gabriel Landini wrote:

>> I have a stack of binary-8bit images with a random distribution of
>> particles; also each image has a different number of particles. I
>> was trying to find out how random the distribution was i.e. how
>> uniformly the particles are distributed across each image.
>
> Hi,
> There is also Ripley's book on Spatial Statistics.
> There are implementations of some of those methods in R.

Yes.  Ripley's book was good 20 years ago (although I found it  
necessary to upgrade some of the code.  You could do just fine with a  
close reading of Ripley and a modern implementation.

--
Kenneth Sloan                                          
[hidden email]
Computer and Information Sciences                        [phone]
University of Alabama at Birmingham              FAX [phone]
Birmingham, AL 35294-1170                http://www.cis.uab.edu/sloan/

[Kenneth Sloan-2]

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                        [phone]
University of Alabama at Birmingham              FAX [phone]
Birmingham, AL 35294-1170                http://www.cis.uab.edu/sloan/

[Noel BONNET]

You can also use some parameters from computational geometry, i.e.
parameters deduced from the Voronoi partition, the Delaunay triangulation,
the Euclidean minimum spanning tree (emst), etc.

These (normalized) parameters can tell you if your distribution of objects
is random, clustered, periodic, with gradient, etc.
An example of application (from our group) can be found in : Nawrocki Raby
et al. Int. J. Cancer (2001), 93, 644-652.

There is a plugin available for ImageJ that computes the Voronoi partition,
the Delaunay triangulation, etc.

Noel Bonnet