7.4.2
Figure 7.7: An exemplary configuration (top) for mapping from a scene (middle)
to a model (bottom).
From (Li 1995c) with permission; © 1995 Kluwer.
This experiment performs the recognition of jigsaw objects under 2D rotation, translation and uniform scaling. There are eight model jigsaw objects which was shown in Fig.5.10. A scene contains rotated, translated and scaled parts of the model jigsaws some of which can be considerably occluded (Fig.7.7(middle)).
Boundaries are computed from the image of the scene using the canny
detector followed by hysteresis and edge linking. After that, corners
of the boundaries are located as 
. No unary relations
are used (
). The following five types of bilateral relations are
used (
):
: ratio of curve arc-length 
and chord-length 
,
: ratio of curvature at 
and 
,
: invariant coordinates vector,
: invariant radius vector, and
: invariant angle vector
(
) of the same types. Therefore, there are six (one for
the NULL
and five for the invariant features) components
(
) in each x and 
.
Figure 7.8: The learned estimate is used to recognize other scenes and models.
The matched model jigsaw objects are aligned with the scene.
From (Li 1995c) with permission; © 1995 Kluwer.
Fig.7.7 shows an exemplary instance 
in which 
is computed from the scene in the middle and
from the model at the bottom. The exemplary configuration 
is shown on the top; the alignment of the model jigsaw (the
highlighted curve) and the scene indicates the exemplary correspondences
of corners (the corners are not shown there); un-aligned parts of the
scene are actually labeled the NULL
.
The 
-optimal parameters are computed as
{0.95540, 0.00034,
0.00000, 0.06045, 0.03057, 0.28743} which satisfies 
. It
takes a few seconds on the HP workstation. Note that the weight
for 
and 
(ratio
of curvature) is zero. This means that this type of feature is not
reliable enough to be used. Because our recognition system has a fixed
value of 
, 
is multiplied by a factor of
, yielding the final weights
{0.70000, 0.00025, 0.00000, 0.04429, 0.02240, 0.21060}.
The 
is used to define the energy function for recognizing
other objects and scenes as shown in Fig.7.8. There
are two scenes, one in the left column and the other in right column.
The scene on the left was the one used in the exemplar and on the right
is a new scene. The optimally matched object lines are shown as
the highlighted curves aligned with the scenes. The same results can
also be obtained using the 
-optimal estimate of {0.7,
0.00366, 0.00000, 0.09466, 0.00251}.
Figure 7.9: Convergence of the non-parametric learning algorithm. Left:
trajectories of 
. Right: trajectories of
.
From (Li 1995c) with permission; © 1995 Kluwer.
The algorithm is very stable. Fig.7.9 shows
convergence properties of the learning algorithm where each curve is
obtained with a different random initialization. The plots show that
values for the global instability become stabilized after a few hundred
iterations and the parameter values of 
take more
than 1000 iterations to converge to the value of about 0.60.