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satisfied. 
SOM generalization of a single S-T trajectory is illustrated in 
fig. 2, in which an multi-node neural chain is used to abstract 
the movement fluctuations of a moving object that corresponds 
to the center of the region we model. The reader is referred to 
[Partsinevelos et al., 2001] for a detailed presentation of the use 
SOM to generalize S-T point trajectories. 
  
Figure 2: SOM nodes describing a S-T trajectory. 
Among the advantages of the SOM technique is that the product 
output space is indicative of intrinsic statistical features 
contained in the input patterns. In addition, according to the 
selection of the number of the nodes in the algorithm multiple 
resolutions can be achieved. 
4. TRACKING OUTLINE DEFORMATIONS 
WITH DIFFERENTIAL SNAKES 
Tracking changes in the outline of an event involves the 
comparison of the event’s outline at an instance to the same 
outline at another instance, to identify the points where this 
outline has expanded or collapsed. The model of deformable 
contour models (a.k.a. snakes) provides a theoretical foundation 
for the delineation of an outline [Kass et al., 1987]. It proceeds 
by establishing a theoretical model of an ideal outline as it is 
expressed by radiometric and geometric criteria. These criteria 
are formulated as energy functions that describe geometric and 
radiometric constraints to be satisfied by the extracted outline. 
During solution iterations they act as forces that deform an 
outline to comply with the local image content while optimizing 
the total energy metrics. 
In a traditional snake model the total energy of each snake point 
is expressed as: 
E sake = @-Econt + f E uy * Y Ecdee (1) 
where : Econt » Ej, are energy terms expressing first and second 
order continuity constraints (forcing the outline to be as smooth 
as possible); Eq, is an energy term expressing edge strength 
(forcing the outline to describe locations where gray values 
differ significantly from their neighbors); and a, f, y are 
(relative) positive weights of each energy term, describing 
Which part we emphasize most. 
In order to detect changes in an object's outline we proceed by 
comparing the content of an image (time Td) to the last record 
of this outline (time 7). The outline extracted through Eq. 1 is 
Stochastic in nature, as varying image conditions affect our 
ability to differentiate between an object and its neighbors. To 
take this information into account we extended the traditional 
snake model by introducing an additional energy term (and 
corresponding weight coefficient) to express a buffer zone in 
the area of each snake node, expressing the local fuzziness 
effect of uncertainty. The new model of differential snakes is 
expressed as: 
E dk = OE ent T [EE vt YE edo: T Y: Eedge + SE 2 
Furthermore, as our objective is to detect major trends in an 
outline’s variations, we mark as such the locations where the 
outline has moved the beyond the buffer zone more than a pre- 
specified threshold (typically selected to be 3 times the buffer 
zone). Prongs (as they were described in Section 2 of this paper) 
correspond to such instances. A detailed description of our 
differential snakes model may be found in [Agouris et al., 
2001]. 
Fig. 3 shows an example of the application of our technique. 
The two outlines show an event’s evolution from instance 7 to 
T+dt. The arrows mark the points where we detected major 
variations with our differential snakes method. These results 
produce 4 prongs for the vent’s helix at time T+dt, with their 
size and azimuth corresponding to the size and azimuths of the 
arrows of Fig. 3. 
  
Figure 3: Major object outline variations detected with a 
differential snake. 
S. SPATIOTEMPORAL ANALYSIS USING HELIXES 
A description of the integration of SOM and snakes techniques 
to automatically generate spatiotemporal helixes is shown in 
Fig. 4. 
   
Figure 4: Overview of helix modeling 
It should be noted that for practical purposes SOM-based 
trajectory generalization takes place first. Using this information 
we register all outlines to the same center and proceed with 
differential snakes. This cycle of processes allows us to extract 
all necessary information to construct a spatiotemporal helix 
describing a spatiotemporal event. 
Spatiotemporal analysis commonly involves the comparison of 
events, to identify similarities, establish causalities, and thus 
identify interrelationships among them. Spatiotemporal helixes 
are valuable tools towards this goal. They serve as signatures of 
spatiotemporal events and, drawing upon the obvious 
parallelism, helix similarity analysis resembles DNA 
—517-