While looping over candidate particle locations to calculate centroid refinements, we calculate other moments of each sphere image's brightness distribution:

and

where (*x*,*y*) are the coordinates of the sphere's centroid.
These additional moments are useful for distinguishing spheres
from noise and for estimating their displacements from
the focal plane.

Colloidal spheres tend to fall into broad yet well-separated
clusters in the plane as can be seen in Fig. 2.
Non-particle identifications, including colloidal aggregates,
misidentified noise, and imperfections in the optical system,
generally fall well outside the target cluster.
The breadth of the cluster of valid points arises from the
changing appearance of spheres as they move out of the microscope's
focal plane.
The exact nature of the broadening depends on whether spheres
are being imaged in transmitted or reflected light.
In the absence of a convenient formulation for the anticipated
distribution of sphere images in the plane, we find that
statistical cluster analysis[14] is
effective for categorizing candidate identifications as either
particles or noise and at distinguishing different classes of particles
in bi- and polydisperse suspensions.
Consequently,
the spatial coordinates of features selected in the cluster
analysis, such as those shown in Fig. 1(d),
constitute the measured
particle locations in the snap-shot at time *t*.

The distribution of data in the
plane reflects the spheres' positions along the direction normal to the
imaging plane.
This dependence is difficult to calculate, but straightforward to
calibrate.
We obtain calibration data by preparing a single layer
sample of each of the monodisperse colloidal suspensions in our study,
either by confining the spheres between parallel glass walls,
or by allowing spheres to aggregate onto a glass substrate.
The first method more closely mimics
the configuration in our investigations.
The second has the advantage
of being easier to prepare.
The calibration sample is mounted on the microscope
and aligned so that the plane containing the particles is parallel
to the focal plane.
An electric motor then is coupled to
the microscope's focusing knob
so that the layer of particles moves through the usable depth of focus
at a rate of about 1 m per second.
When such a focus scan is digitized at 30 frames per second,
each frame is displaced vertically by about *dz* = 33 nm
relative to the one before.

Since all the particles in a given frame of a focus scan
are at the same displacement *z* from the focal
plane, their images form a compact,
roughly elliptical cluster in the plane.
The mean and standard
deviations of and for
each discrete step in *z*
then can be collected into a probability distribution
for a given particle to be within *dz* of
given its descriptors and .
This probability distribution is then used to estimate particles' vertical
positions through

where the sum runs over the frames from the focus scan.

Applying eqn. (8) to the original calibration data
provides an estimate for the error in *z* as a function of *z*.
The values of *z* measured for the -th frame fall in a
Gaussian distribution about .
We adopt the width of this Gaussian as an estimate of the
error in the location estimate for spheres near .
In practice, we find this value to be at best 10 times larger
than the in-plane location error.

Mon Mar 11 23:01:27 CST 1996