tural state and
vity, e; and e» are
s described as the
Fa) 0
e discontinuities
m is a measure of
fectiveness). The
of stretching and
tion of the set of
ergy of the curve
y
/ersion we get is
^B (u)) ]
(4)
(5)
" (9)Bi" (uj)]
1.2 nnt; 1.40)
used to solve the
(7)
ed matrix (mxm).
e inversion is m
e).
es, the following
the desired road
Iman filter using
T; indicates the
the trajectory.
lines in the first
delineated by the
, à corresponding
nt is denoted by
‚een the positions
ector TiTi +1 from
on one by one. A
te centerline is
| N |
| > g |
3 (N " |
d 0 tés x dass |
| i
| |
Vehicle trajectory
T;: the position of exposure station
Figure 1. Generation of a 3D approximate shape model
— Projection between the
stereo pair T; d
op M model and the image
Si
Sia
€ sampling point E] control vertex
o extracted point from image pair (T;.1)
>
extracted point from image pair (T;)
X
extracted point from image pair (Ti+1)
Figure 2(b). External force field
4. Apply an algorithm of B-splines approximation to these data
points S; (i=0,1,2,...) and then an approximate 3D shape model
of road centerlines is set up (denoting in dotted line, figure 1).
5. Since the orientation parameters of the camera stations are
known, the shape model can be back projected on the
sequential image (figure 2a). At least two consecutive stereo
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
image pairs covering the same segment along the model are
available.
6. The extraction and matching of road centerline features on
these images are conducted based on the image projection of
the model. Section 4 will describe an algorithm on model
driven extraction and matching of road centerline features. At
the end of this step, 3D points associated with image features
of road centerlines are obtained. In figure 2b, The symbols
marked by “o“, “A“ and “x“ represent the 3D points obtained
by the feature extraction and matching from images. As can be
seen, it looks like a deformable curve placed in a force field.
Each point in the field acts a force on the curve and deforms
the curve. We will define these forces (external forces) in
section 3.5
7. The internal and external energy for the curve segments
from S; to S;+3 are calculated. The final shape of these
segments is determined by solving the motion equations
described in section 3.3.
8. Move to the next station (e.g., 5; > S;-;). Repeat the process
from the step 5 to 7. The shape model is thus deformed
incrementally, driven by the successive images. The final result
of the deformation of the model is a 3D centerline shape when
the last sequential image pair 1s processed.
3.5. Definition of External Energy
The external energy plays a key role in pushing the model into
the desired position. As shown in figure 2b (2D illustration),
there are forces between extracted points P and the curve
points Q(u) . In order to quantify this kind of energy, we apply
a gravity-type field to describe 1t. The reason 1s that the closer
the distance between the points, the greater the force. To avoid
the singulanty when the distance is approaching zero, we
employ the following function in quantifving the external
energy:
Eext = f(Dp/ ry) (8)
and the above f(x) 1s defined as:
[x?. rı> De
feodum r<Dp<r (9)
| 0, r2< Dp
where D, ||Q(u)P||, the distance between Q(u) and P, r; and rz
are the coefficients representing the range of the influence of
the point to the curve. r;-0.I(meter) and ry/r;=3 are chosen
according to the system accuracy. Thus f(x) acts like a spring
when the point is close to the curve (D,<r;) and makes no
effect when D,-r;. The corresponding external force can be
derived:
Feu = VEen = f'(Dp/r1)/ ri (10)
Considering the historic effect of forces, the total Fe becomes
Fa oW (Tii) + OFT) is @3Fext(T +1) (1 1)
Again, &;, 0? and «s are weighting coefficients. During the
process of the iterative solution for equations (7), the Feu of
each point Q(u) along the model should be calculated
repeatedly until the motion equations reach a converge state.
