Page 69 - Dynamic Vision for Perception and Control of Motion

2.3 Points of Discontinuity in Time 53

which yields the discrete model
˜
x Ax ˜ b v . (2.37)
k 1 k k k
2
2
The value of the expectation is E[ ] 0v , and the variance is E[ ]v V (essen-
q
k
k
tial for filter tuning).The covariance matrix Q for process noise is given by
§ T 4 4 T 3 2 T 2 2·
¨ 2 T 3 2 ¸ 2
Q b ˜ ı b ˜ T 2 T T ˜ ı . (2.38)
k q k ¨ ¸ q
¨ T 2 2 T 1 ¸
© ¹
This model may be used independently in all six degrees of freedom as a default
model if no more specific knowledge is given.
2.3 Points of Discontinuity in Time

The aspects discussed above for smooth parts of a mission with nice continuity
conditions alleviate perception; however, sudden changes in behavior are possible,
and sticking to the previous mode of interpretation would lead to disaster.
Efficient dynamic vision systems have to take advantage of continuity condi-
tions as long as they prevail; however, they always have to watch out for disconti-
nuities in object motion observed to adjust readily. For example, a ball flying on an
approximately parabolic trajectory through the air can be tracked efficiently using
a simple motion model. However, when the ball hits a wall or the ground, elastic
reflection yields an instantaneous discontinuity of some trajectory parameters,
which can nonetheless be predicted by a different model for the motion event of re-
flection. So the vision process for tracking the ball has two distinctive phases
which should be discovered in parallel to the primary vision task.

2.3.1 Smooth Evolution of a Trajectory

Flight phases (or in the more general case, smooth phases of a dynamic process) in
a homogeneous medium without special events can be tracked by continuity mod-
els and low-pass filtering components (like Section 2.2.5.3). Measurement values
with oscillations of high frequency are considered to be due to noise; they have to
be eliminated in the interpretation process. The natural sciences and engineering
have compiled a wealth of models for different domains. The least-squares error
model fit has proven very efficient both for batch processing and for recursive es-
timation. Gauss [1809] opened up a new era in understanding and fitting motion
processes when he introduced this approach in astronomy. He first did this with the
solution curves (ellipses) for the differential equations describing planetary motion.
Kalman [1960] derived a recursive formulation using differential models for the
motion process when the statistical properties of error distributions are known.
These algorithms have proven very efficient in space flight and many other appli-
cations. Meissner, Dickmanns [1983]; Wuensche [1987] and Dickmanns [1987] extended
this approach to perspective projection of motion processes described in physical