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This is a very interesting question and you really make me aware that
I always neglect the measurement error in my experiments. Anyway, I
think we can correct it in the speed meansure as follows :
Suppose that the particle situates at the position p1=(x1, y1, z1) at
the time t and it situates at p2 = (x2, y2, z2) at the time t+dt. Now
as we have measurement errors, the observed positions are actually
q1=(x1+e1, y1+e2, z1+e3) and q2=(x2+f1, y2+f2, z2+f3). The (e1, e2,
e3) and (f1, f2, f3) are vectors due to the measurement error. Now the
speed that we measure is written as v(t) = sqrt( (x1-x2+(e1-f1))^2 +
(y1-y2+(e2-f2))^2 + (z1-z2+(e3-f3))^2) / dt. We can suppose that the
measurement error is independently zero-mean gaussian distributed in
each axis direction with the same variance (isotropic error), and that
the measurement errors are independent from one image frame to
another. Consequently w(t) := v(t) * dt / (sqrt(2) sigma) is
noncentral chi distributed where sigma is the measurement error
standard deviation in one axis. Now the mean of a w(t), which can be
analytically written out, is an explicit function of the true speed
i.e. sqrt ( (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 ) / dt. Using this
relation, by measuring the observed average speed, you can solve the
underlying true speed. See
http://en.wikipedia.org/wiki/Noncentral_chi_distribution for more
details about the noncentral chi distribution. Finally, the
measurement standard deviation can be precalculated by measuring
locations for fixed testing objects.
From your question, I just think of another thing. As the MSD is a
more appropriate way to measure "speed", how would the MSD plot be
influenced by a measurement error ?
On 6/26/07, Eric Olson <[log in to unmask]> wrote:
> Search the CONFOCAL archive at
> http://listserv.acsu.buffalo.edu/cgi-bin/wa?S1=confocal
>
> I am curious how people have dealt with the issue of variance in
> positional measurement ( using the measuring tool in ImageJ, for
> example) and its contribution to particle speed measurements. The
> positional variance produces paradoxical "speed" for non moving
> particles - if particle displacement is determined between successive
> images. This effect directly increases with sampling frequency.
>
> I have looked at a number of papers on cell migration and not found a
> correction or mention of this effect - which can be large for high
> sampling frequencies.
> One idea would be to define a minimum detectable displacement based
> on the standard deviation of the measurement. Every value less than
> that minimum would be set to zero???
>
> Thanks,
> Eric
>
>
> Eric C. Olson, PhD
> Assistant Professor
> Department of Neuroscience and Physiology
> SUNY Upstate Medical
> 3295 Weiskotten Hall
> 766 Irving St.
> Syracuse, NY 13210
>
> office: 315-464-7776
> lab : 315-464-8157
>
--
Bo ZHANG
Ph.D. Student
Quantitative Image Analysis Group
Institut Pasteur
25-28, Rue du Docteur Roux
75724 Paris Cedex 15, France
Tel : +33 (0)1 40 61 39 74
Fax: +33 (0)1 40 61 33 30
http://www.freewebs.com/bozhang/
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