[R] particle count probability

[Rolf Turner]

On 2/21/19 12:16 AM, PIKAL Petr wrote:
> Dear all
> 
> Sorry, this is probably the most off-topic mail I have ever sent to
> this help list. However maybe somebody could point me to right
> direction or give some advice.
> 
> In microscopy particle counting you have finite viewing field and
> some particles could be partly outside of this field. My
> problem/question is:
> 
> Do bigger particles have also bigger probability that they will be
> partly outside this viewing field than smaller ones?
> 
> Saying it differently, although there is equal count of bigger
> (white) and smaller (black) particles in enclosed picture (8), due to
> the fact that more bigger particles are on the edge I count more
> small particles (6) than big (4).
> 
> Is it possible to evaluate this feature exactly i.e. calculate some
> bias towards smaller particles based on particle size distribution,
> mean particle size and/or image magnification?

This is fundamentally a stereology problem (or so it seems to me) and as 
such twists my head.  Stereology is tricky and can be full of apparent 
paradoxes.

"Generally speaking" it surely must be the case that larger particles 
have a larger probability of intersecting the complement of the window,
but to say something solid, some assumptions would have to be made.  I'm 
not sure what.

To take a simple case:  If the particles are discs whose centres are 
uniformly distributed on the window W which is an (a x b) rectangle,
the probability that a particle, whose radius is R, intersects the 
complement of W is

    1 - (a-R)(b-R)/ab

for R <= min{a,b}, and is 1 otherwise.  I think!  (I could be muddling 
things up, as I so often do; check my reasoning.)

This is an increasing function of R for R in [0,min{a,b}].

I hope this helps a bit.

Should you wish to learn more about stereology, may I recommend:

> @Book{baddvede05,
>   author =       {A. Baddeley and E.B. Vedel Jensen},
>   title =        {Stereology for Statisticians},
>   publisher =    {Chapman and Hall/CRC},
>   year =         2005,
>   address =      {Boca Raton},
>   note =         {{ISBN} 1-58488-405-3}
> }

cheers,

Rolf

-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276