-----Original Message-----
From: Jim Lemon <drjimlemon at gmail.com>
Sent: Thursday, February 21, 2019 11:36 AM
To: PIKAL Petr <petr.pikal at precheza.cz>
Cc: Rolf Turner <r.turner at auckland.ac.nz>; r-help at r-project.org
Subject: Re: [R] particle count probability
Hi Petr,
My second message was to show that if you take the limiting cases of "just
inside" and "just outside" - which should have been:
just inside the field:
R0 = sqrt((x1+R1-x0)^2 + (y1+R1-y0)^2)
just outside the field:
R0 = sqrt((x2-R1-x0)^2 + (y2-R1-y0)^2)
the two differences are equal along any radius, supporting the averaging
strategy.
Jim
On Thu, Feb 21, 2019 at 7:53 PM PIKAL Petr <petr.pikal at precheza.cz> wrote:
Hallo
Thanks all for valuable suggestions. As always, people here are generous and
clever. I will try to think through all your suggestions, including recommended
literature.
Jim. Standard practice in particle measurement is to count (and
mesure) only particles which are fully inside viewing area. So using
your equation I could compare probability for let say particles with
R1 = c(0.1, 1). But I probably misunderstand something. Having x0, y0
= 0 and x1 =10 and y1 = 0 I get
sqrt((10+c(0.1, 1)-0)^2 + (0+c(0.1,1)-0)^2)
[1] 10.10050 11.04536
which gives in contrary higher value for bigger particle.
OTOH, if I take your first reasoning I get quite satisfactory values.
1-(10-c(0.1, 1))* (10-c(0.1,1))/(10^2)
[1] 0.0199 0.1900
Cheers.
Petr
-----Original Message-----
From: Jim Lemon <drjimlemon at gmail.com>
Sent: Thursday, February 21, 2019 12:24 AM
To: Rolf Turner <r.turner at auckland.ac.nz>
Cc: PIKAL Petr <petr.pikal at precheza.cz>; r-help at r-project.org
Subject: Re: [R] particle count probability
Okay, suppose the viewing field is circular and we consider two
particles as in the attached image.
Probability of being within the field:
R0 > sqrt((x1+R1-x0)^2 + (y1+R1-y0)^2) Probability of being outside
the field:
R0 < sqrt((x2-R1-x0)^2 + (y2-R1-y0)^2)
Since these are the limiting cases, it looks like the averaging I
suggested will work.
Jim
On Thu, Feb 21, 2019 at 9:23 AM Rolf Turner
<r.turner at auckland.ac.nz>
wrote:
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