International Archives of the Photogrammetry, Remote Sensing 
5. HELIX GENERATION 
The spatiotemporal helix model is constructed in a four-step 
process, which will be detailed in this section: 1) find the center 
of mass for the object at each time instance, 2) detect changes 
in the object's outline in each cardinality quadrant, 3) construct 
a self-organizing map (SOM) that picks out only those nodes 
which are necessary to generalize the object’s behavior and 
forms a “spine” for the helix, and 4) add information about 
outline changes to the spine with “prongs” that show expansion 
or contraction. 
A 400x400 pixel grid has been utilized to create a synthetic 
dataset of five polygons, one in each frame. The polygons in 
these frames represent snapshots in the evolution of an object or 
phenomenon over time. Before reaching this stage, an object 
extraction procedure would need to be performed on our real- 
world data, but this is outside the scope of the current paper. 
For more information on our relevant activities in object 
extraction, the reader is referred to (Agouris, Beard et al. 2000; 
Agouris, Stefanidis et al. 2001; Doucette, Agouris ct al. 2001). 
Frame #1 Frame #2 Frame 83 Frame 84 Frame 85 
Figure 4: Five sample frames used for input in helix extraction 
5.1 Center of Mass Extraction 
In the first stage of helix construction, the object's center of 
mass is extracted and plotted on a three-dimensional grid. Each 
asterisk indicates the location of the object at a given time 
instance. In this example dataset, we assume that each of the 
frames used in this example was taken after a ten-minute delay. 
The first frame is thus linked to time t=10 and the fifth frame is 
linked to t=50. In this first stage, a trajectory is also 
constructed by linking the centers of mass for each frame 
(Figure 5). 
=, A 
i" Ë 
=; bd 
1 1 
Xs 2; 
x x 
12i f 
X en 
HN 3 
oe get rr A 
3 L Zu x n * Mi 25 xm se 
Figure 5: Object's center of mass and rough trajectory 
In order to test the robustness of this method under more 
realistic conditions, we added random noise to our images using 
Matlab’s “randerr” function, which introduces a user-selected 
number of nonzero elements into each row of a matrix. We 
multiplied our original frames by these new matrices in order to 
create new noisy images (Figure 6 left). 
zd 
i 
! é 
reverse sites 
  
Figure 6: Frame #5 before and after noise removal 
and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
160 + 
120 + 
100 + 
  
Figure 7: Object trajectories from initial (solid) 
and cleaned (dotted linc) frames 
When a 9x9 median filter was applied to the noisy image, the 
number of erroneous DN-0 pixels was reduced dramatically 
(Figure 6 right). Most of the pixels that arc left are located 
around the edges of the frame, due to algorithm limitations. In 
order to determine how the few remaining noisy pixels will 
effect the center of mass calculations, we mapped the initial 
trajectory of the objects center of mass, and compared it to the 
center of mass after the noise removal procedure. We found 
that there is nearly a one-to-one correspondence between these 
trajectories (Figure 7). These results indicate that the procedure 
has been successful in removing noise. 
5.2 Cardinality Changes 
The second stage divides the object into four quadrants, based 
on the cardinal directions of north, south, cast, and west 
(assuming an orientation where north is towards the top of the 
frame). This is done for each frame, and the center of mass 
found in the first step is used as the origin for cach division 
(Figure 8). The object in frame n is then compared to the same 
object in frame n+1 in order to discover whether there has been 
an expansion or contraction during cach time interval. For 
instance, the object grows significantly between frames 2 and 3, 
and this leads to an increase in area for all four quadrants. This 
change will be quantified in the final step of helix construction, 
which is discussed later in this section. 
Figure 8: Object divided into cardinality quadrants 
UA 
.3 Self-Organizing Map Construction 
The third stage is concerned with construction of a Self- 
Organizing Map (SOM) that generalizes the trajectory of the 
helix by picking out locations where changes such as rapid 
acceleration, deceleration, or rotation occur and marking them 
with nodes. A SOM is a neural network solution that organizes 
nodes into an ordered sequence through competitive learning 
(Kohonen 1997). In this example, the object is moving at a 
fairly uniform pace, so it does not experience much acceleration 
or deceleration. The major change is rotation, occurring most 
notably at frame 3, the apex of the object's trajectory. 
International Arch 
When we ask for 2 
we can see that th 
merged into a sing 
When we ask for 3 
This decreases tl 
polygon’s locatior 
space needed to 
important charac 
behavior. For moi 
(Kohonen 1982; K 
  
Figure 9: SON 
5.4 Node Placem 
The final stage in 
node to the closes 
prong information. 
in the SOM proc 
object's position i 
located between th 
closer to that of fr 
the helix, our algor 
and 5 (Figure 10 le 
When examining t 
node placement in 
example, this woul 
use only them to de 
It is more accurat: 
third step, because 
uses locations that 
  
Figure 10: Com 
In addition to selec 
should be recorded 
construction proces 
or contraction to a 
threshold has been 
found in our exam 
where there is a lar; 
a smaller reduction 
our helixes by a lor 
node at frame 4 and