International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
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Figure 1: A spatiotemporal helix (left) and a detail showing the
azimuth of a prong (right).
As a spatiotemporal trajectory, a spine is a sequence of (x,y,t)
coordinates. It is expressed in a concise manner as a sequence
of spatiotemporal nodes Sx’, ...n"). These nodes correspond to
breakpoints along this trajectory, namely points where the
object accelerated/decelerated and/or changed its orientation.
Thus, each node n! is modelled as n (x, y, t,q), where:
— (X,y.t) are the node spatiotemporal coordinates, and
— q is a qualifier classifying the node as an acceleration
(4°), deceleration (g^), or rotation (q') node.
Each prong models the local expansion or collapse of the
outline at the specific temporal instance when this event is
detected, and is a horizontal arrow pointing away from or
towards the spine. It is modelled as p'(t,r,aj,a;) where:
— tis the corresponding temporal instance (intersection of
the prong and the spine in Fig. 2 left),
— ris the magnitude of this outline modification, expressed
as a percentage of the distance between the center of the
object and the outline, with positive numbers expressing
expansion (arrows pointing away from the spine) and negative
numbers indicating collapse (arrows pointing /owards the
spine),
— a, a, is the range of azimuths where this modification
occurs; with each azimuth measured as a left-handle angle
from the North (y) axis (Figure 1 right).
3. SIMILARITY METRICS
While a single helix conveys valuable information on the
behavior of an object, it is the comparison of object behaviors
that typically leads to knowledge discovery in typical geospatial
applications. For example, comparing how two phenomena
evolve may lead to the establishment of causality relationships.
[n order to support such applications we have developed metrics
that support the comparison of ST helixes, to support the
discovery of similarities or differences in object behaviours
(Stefanidis et al., 2003).
We have developed abstract and qualitative metrics for helix
comparisons. In such comparisons, one helix serves as reference
and the second is a matching candidate (Figure 2). When
examining a node on the reference helix, we do not simply look
for a match at the same time instance on the candidate helix, but
expand our time window to account for variations that may
have occurred while obtaining the dataset. Thus we are not
looking solely at time t2 for a match, but rather in an interval
(11,03) (Fig. 2).
46
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Figure 2: Comparison of reference helix (right) to candidate
(left)
3.1 Abstract Comparisons
In abstract comparisons we are only considering the presence or
absence of specific node and prong qualifiers. Two helixes are
compared at each node in order to evaluate whether they exhibit
similar behaviors. We proceed by assigning cost values to each
nodal comparison. If both the reference and candidate helixes
accelerate or decelerate at a given instant, then a cost value of 0
is given to that pair. If one is accelerating and the other is
decelerating, a value of 2 is assigned (Table 1). Thus the
comparison of highly dissimilar helixes produces high
comparison results, while the comparison of two perfectly
similar helixes produces a result of 0.
Helix 14 Helix 2 Accel. Cons. Decel
Acceleration 0 1 2
Constant 1 0 1
Deceleration 2 1 0
Table 1. MST Cost metrics for comparing qualifier attributes of
acceleration and deceleration
Similarly we can produce cost metrics for rotation attributes. Of
interest in this abstract comparison is whether rotations are
clockwise or counterclockwise. The corresponding mobility
states in this context are clockwise rotation, counterclockwise
rotation, and no rotation. The latter is indicated by the lack ofa
rotation node over a search interval. The corresponding metrics
are shown in table 2, and follow a rationale similar to the one
used in the cost metrics of table 1.
Helix 1 4 Helix 2 — Clockw. No Counter. |
Clockwise rotation 0 1 2 |
No rotation ] 0 1
Counterclockwise rotat. 2 1 0
Table 2. MST Cost metrics for comparing qualifier attributes of
rotation
Regarding prong information we have a similar situation, where
there may be expansion, contraction, or no change in an
object's outline. This last option is indicated by the lack of à
prong over a search interval. The corresponding metrics are
shown in table 3, and follow a rationale similar to the ones used
to form the cost metrics of tables ] and 2.
Helix 14 — Helix 22 Expand No Contract |
Expansion 0 1 2
No change 1 0 : 1
Contraction 2 1 0
Table 3. MST Cost metrics for comparing prong magnitudes
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