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