[R] particle count probability

[Jim Lemon]

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 using 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 using gmail.com>
> > Sent: Thursday, February 21, 2019 12:24 AM
> > To: Rolf Turner <r.turner using auckland.ac.nz>
> > Cc: PIKAL Petr <petr.pikal using precheza.cz>; r-help using 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 using 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
> > >
> > > ______________________________________________
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