Having located colloidal particles in
a sequence of video images, we match up locations
in each image with corresponding locations in later images to
produce the trajectories in 
.
This requires determining which particle in a given
image most likely corresponds to one in the preceding
image.
Tracking more than one particle requires care since
any particle can be identified with only one particle
in each of the successive and preceding frames.
Thus, we seek the most probable set of N identifications
between N locations in two consecutive images.
If the particles
are indistinguishable, as for monodisperse colloidal spheres, this
likelihood can be estimated only by proximity in the two images.
The probability that a single Brownian particle will diffuse a distance
in the plane in time 
is
where D is the particle's self-diffusion coefficient. For an ensemble of N noninteracting identical particles, the corresponding probability distribution is the product of single particle results:
The most likely assignment of particle labels from one image to the
next is the one which maximizes 
,
or, equivalently, minimizing 
.
Calculating for all possible combinations
of label assignments, however, would require O(N!) computations,
which is impractical for all but the most trivial distributions.
Each label assignment
can be thought of as a bond drawn between a pair of particles in
consecutive frames.
The set of all possible label assignments forms a network in the plane.
To reduce the complexity of assigning labels,
we select only those bonds shorter than a characteristic
length scale L to be possible
candidates for the assignment of particle labels between
two frames.
This is equivalent to truncating the single-particle probability
distribution 
at 
.
If L is somewhat smaller
than the typical interparticle spacing in a snapshot, the remaining network
usually is reduced to a collection of disconnected subnetworks.
We then can solve for the optimal set of identifications
within each of these subnetworks separately.
For small enough values of L, most
subnetworks contain only single bonds for which the
assignment of labels is trivial.
Trajectory linking proceeds in 
time for such trivial cases.
The least difficult nontrivial subnetworks to resolve consist of bonds
linking M particles in one frame to M in the next, with 
.
The assignment of labels in this case proceeds according to
eqn. (10) with no more than O(M!) operations
and often much fewer.
Subnetworks connecting unequal numbers of sites in two frames pose
more of a problem since some labeling assignments then cannot
be resolved.
Such subnetworks are not unusual because particles often wander into
and out of the observable sample volume, particularly near the edges.
To proceed with labeling, we add as many ``missing'' bonds
as are needed to complete trial labeling assignments.
These missing bonds are assigned the length 
for the purpose of evaluating eqn. (10).
The most probable set of identifications therefore requires labeling
some particles as ``missing'' in individual time steps.
The last known locations of missing particles are retained in case
unassigned particles reappear sufficiently nearby to resume
the trajectory.
This process is repeated for the particle locations in each frame
until is completely determined.
Trajectories for monodisperse colloidal spheres at a crystal-fluid
interface appear in Fig. 3.
Linking particle distributions into trajectories is only feasible
if the typical single particle displacement in one
time step is sufficiently smaller than the typical interparticle
spacing, a.
Otherwise, particle positions will become inextricably confused
between snapshots.
The optimal cutoff parameter falls in the range
.
