[R] particle count probability

[PIKAL Petr]

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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