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3 TRACKING TARGET PATHS IN
IMAGES
Tracking the motion of an object using video data en-
tails identifying images of the same points (targets)
in a sequence of successive frames. When dealing
with small numbers of easily distinguishable rigid ob-
jects (e.g., when colour and shape are clearly distinct
or when obvious feature points are available), associ-
ating targets from frame to frame is straightforward.
However, when trying to track large collections of
very similar objects, more sophisticated measures are
needed to assess the credibility of the reconstructed
motion of individual targets.
3.1 Limitations of Tracking Targets
Strictly Frame-by-frame
To state the tracking problem in concrete terms, re-
call that T
(k)
denotes the set of all targets (images of
droplets) detected in frame k. Each target in t
(k)
∈
T
(k)
can be thought of as a pair t
(k)
=
u
(k)
,v
(k)
of pixel coordinates (given by the target’s centre of
mass). Targets have additional properties other than
their centroids (e.g., areas, moments, etc.) that can be
used in tracking. Many tracking algorithms (e.g., in
(Balch et al., 2001)) are based on linking targets in
T
(k)
(i.e., from frame k) directly to targets in T
(k+1)
(i.e., from frame k + 1).
Such approaches work well when the cardinality
of the target sets does not vary significantly between
frames. For the droplet tracking problem, this as-
sumption breaks down in a number of ways:
1. a false target t
(k)
is spuriously detected in frame
k, i.e., pixels contaminated by noise are mistaken
for a legitimate physical object in the scene;
2. a true target t
(k)
is not detected by the camera in
frame k + 1;
3. a true target t
(k)
is temporarily occluded by an-
other object in frame k + 1;
4. a target t
(k)
temporarily leaves the field of view
between frames k and k + 1;
5. a target t
(k+1)
returns to the field of view between
frames k and k + 1;
6. a target t
(k)
permanently leaves the field of view
between frames k and k + 1; and
7. a target t
(k+1)
initially enters the field of view be-
tween frames k and k + 1.
These modes of failure are the norm rather than ex-
ceptions. In case (1), it is necessary to handle spuri-
ous detections gracefully. The cases (2), (3), and (4)
are temporary failures; it is best not to match the tar-
get t
(k)
∈ T
(k)
with any target in T
(k+1)
. Similarly,
in case (5) when the target t
(k+1)
is detected again in
frame k + 1, it should be paired with a candidate from
a frame prior to frame k (i.e., the most recent frame
in which the corresponding droplet was detected). Fi-
nally, the situations (6) and (7) should be considered
only when there is no suitable candidate target in any
later or earlier frames respectively.
3.2 Assignment Problems for Tracking
We modify the formulation of the tracking problem
to allow for the difficulties outlined in Section 3.1.
The key problem that needs to be solved at each stage
is an assignment problem. Assignment or matching
problems constitute a fundamental class of problems
in combinatorial optimization. In specific language,
let A and B be finite sets (i.e., vertices of a bipartite
graph), let cost : A×B → R be a cost function associ-
ated edges between A and B, and let M ⊂ P (A × B)
be a proper subset of the power set P (A × B). The
Assignment Problem (AP) is to construct a matching
M ⊂ A × B such that the total cost summed over all
edges in M is minimized, i.e.,
min
M∈M
∑
(a,b)∈M
cost(a,b). (3)
If
|
A
|
=
|
B
|
are equal and M consists of all pos-
sible matchings that cover both A and B, the problem
(AP) is called a Linear Assignment Problem (LAP). If
|
A
|
6=
|
B
|
, and M consists of all matchings that cover
the smaller set, the problem (AP) is called a Rectan-
gular Linear Assignment Problem (RLAP). Observe
that the constraint M on the set of possible match-
ings is necessary; otherwise, for many cases, the min-
imizer would simply be the empty matching M =
/
0.
Assuming for the moment that the constraint M ⊂
P (A × B) is known, we shall assume henceforth that
“M = SolveAP(A,B,cost, M )” (4)
means “Solve (AP) by computing the matching
M ∈ M that minimizes the sum in (3).” We implicitly
assume that a solution of (AP) exists as does a rea-
sonable algorithm for computing it. For all practical
purposes, we use the techniques in (Dasgupta et al.,
2008) to compute M in (4).
In some cases of (AP), not all edges are present
in the bipartite graph, i.e., some of the elements of
A are not connected to elements of B. This as-
signment problem can still be solved using a variant
of the Kuhn-Munkres algorithm that assigns (prac-
tically) infinite cost to absent edges (see (Dasgupta
et al., 2008)). The missing edges also introduce the
constraint M ⊂ P (A × B).
EXTRACTION OF BLOOD DROPLET FLIGHT TRAJECTORIES FROM VIDEOS FOR FORENSIC ANALYSIS
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