Re: distance between adjacent particles

[Noel BONNET]

Hi all,

I do not completely agree with these statements, especially the one
considering that "the user need to decide how many of the closest neighbors
he want to include in the average".
Instead, I think "the image is as it is" and "the closest neighbors are as
they are" ...
In the framework of Computational Geometry (the framework in which the
contributions of Gabriel Landini and Karsten Rodenacker are), the number of
neighbors is perfectly defined (at least in an Euclidean framework): it is
the number of Voronoï zones that are adjacent to the Voronoï zone associated
to the current object. As a consequence (one algorithm for computing the
Delaunay triangulation starts from the Voronoï partition), the number of
neighbors is equal to the number of segments that start from a given object
to join other objects. This number is thus the result of a computation
instead of a choice of the user. As such, it is ONE parameter which can
serve to discriminate different distributions of objects (the hexagonal
close packing is certainly not the distribution we are interested in!).

But many other parameters can also be used. In the discussion generated by
the question from France Girault, the mean of the distances from an object
to its neighbors was mainly considered. From my experience (and from
theoretical considerations), I have to mention that the variance is at least
as important as the mean if one wants to discriminate different
distributions of objects. In fact, the "best" solution (in this framework!)
is to consider the mean and the variance simultaneously. For example,
plotting each image (characterized by the mean and standard deviation of the
Delaunay triangulation or the Euclidean Minimum Spanning Tree (EMST)) in a
2D parameter space (mean, standard deviation) allows to recognize
immediately the type of spatial distribution of the objects (periodic,
random, clustered, with gradient, etc). In addition, some normalized
parameters involving the mean and the standard deviation can be computed and
compared.
See for instance:
Dussert et al. J. theor. Biol (1987) 125, 317.
Marcelpoil & Usson. J. theor. Biol (1992) 154, 359.
Nawrocki Raby et al. Int. J. Cancer (2001) 93, 644.

Another point:
With these methods of Computational Geometry, it is not necessary to reduce
the objects (in binary images) to their center points. Provided the objects
are not touching, the Voronoï partition, and hence the Delaunay
triangulation and the EMST can be computed (and are computed more precisely)
by keeping the original shape of the objects instead of reducing them to
their center of mass.

Now I recognize that this methodology is not the only one that can solve
this type of pattern analysis problem. Unfortunately, comparative studies
involving different methodologies are missing.
See, however:
Wallet & Dussert. J. theor. Biol. (1997) 187, 437.

I hope this can help.

Noel

[Kenneth Sloan-2]

On May 29, 2007, at 7:59 AM, Noel BONNET wrote:

> Hi all,
>
> I do not completely agree with these statements, especially the one
> considering that "the user need to decide how many of the closest  
> neighbors
> he want to include in the average".
> Instead, I think "the image is as it is" and "the closest neighbors  
> are as
> they are" ...
> In the framework of Computational Geometry (the framework in which the
> contributions of Gabriel Landini and Karsten Rodenacker are), the  
> number of
> neighbors is perfectly defined (at least in an Euclidean  
> framework): it is
> the number of Voronoï zones that are adjacent to the Voronoï zone  
> associated
> to the current object.

But...but...while this approach is the currently predominant one, not  
EVERY problem fits the mold.  If adjacent particles interact with  
each other in ways *other* than common borders, then neighbors which  
are not nearest-neighbors in the Delaunay/Voronoi sense may still be  
relevant.

The current combinatorial-oriented version of "computational  
geometry" is near and dear to my heart, but I still think it is  
relevant to lookd at older methods.  I agree with a previous poster's  
recommendation of Ripley's book on spatial statistics.

and...just for the record...the Delaunay/Voronoi definitions of  
nearest-neighbor is not the *only* view (although it is certainly the  
most popular).

--
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/

[Dimiter Prodanov]

May be this is a late input, but I
think that computationally simplest is
simultaneous consideration of Ripley's K-function
and the inter-point distance distribution (H-function).
See my article in J. Neurosci. Methods:  Prodanov, D.; Nagelkerke, N.  
and Marani, E. "Spatial clustering analysis in neuroanatomy:
applications of different approaches to motor nerve fiber
distribution",J Neurosci Methods, 2007, 160, 93-108

Cheers
Dimiter Prodanov