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Noise Discrimination and Tracking in Depth

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

  equation77

and

  equation81

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 tex2html_wrap_inline805 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 tex2html_wrap_inline805 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 tex2html_wrap_inline805 cluster analysis, such as those shown in Fig. 1(d), constitute the measured particle locations tex2html_wrap_inline859 in the snap-shot at time t.

The distribution of data in the tex2html_wrap_inline805 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 tex2html_wrap_inline803 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 tex2html_wrap_inline805 plane. The mean and standard deviations of tex2html_wrap_inline949 and tex2html_wrap_inline951 for each discrete step tex2html_wrap_inline953 in z then can be collected into a probability distribution tex2html_wrap_inline957 for a given particle to be within dz of tex2html_wrap_inline953 given its descriptors tex2html_wrap_inline949 and tex2html_wrap_inline951 . This probability distribution is then used to estimate particles' vertical positions through

  equation90

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 tex2html_wrap_inline973 -th frame fall in a Gaussian distribution about tex2html_wrap_inline953 . We adopt the width of this Gaussian as an estimate of the error in the location estimate for spheres near tex2html_wrap_inline953 . In practice, we find this value to be at best 10 times larger than the in-plane location error.


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Next: Linking Locations into Trajectories Up: Five Stages of Colloidal Previous: Refining Location Estimates

David G. Grier
Mon Mar 11 23:01:27 CST 1996