DHOND
AND
AGGARWAL: STRUCTURE
FROM
STEREO-A
REVIEW
1493
the filtered images are found by scanning them along lines
perpendicular to the orientation of the mask.
3)
For each
mask size, matching takes place between the zero-crossing
segments extracted from each filtered image output that
are of the same sign and roughly the same orientation.
Local matching ambiguities are resolved by considering
the disparity sign of nearby unambiguous matches.
4) Matches obtained from wider masks control vergence
movements aiding matches among output of smaller masks;
5)
The correspondence results are stored in a dynamic
buffer called the 2.5-D sketch.
Marr and Poggio [41] formulate two basic rules for
matching left- and right-image descriptions. Each item in
an image can be assigned to one and only one disparity
value (uniqueness). Secondly, matter is cohesive. Hence
disparity varies smoothly almost everywhere, except where
depth discontinuities occur at surface boundaries (continu-
ity).
B.
Grimson’s Implementation
Grimson
[
191 implemented the computational theory of
Marr and Poggio [41] and addressed certain implementa-
tion details that were not covered earlier by the Marr-
Poggio theory.
1)
Feature Extraction:
Marr and Hildreth [39] have
shown theoretically that, provided two simple conditions
on the image intensity function in the neighborhood of an
edge are satisfied, intensity changes occurring at a particu-
lar scale may be detected by locating the zero-crossings in
the output of the
v2G
(Laplacian of Gaussian) filter.
Instead of convolving each image with
12
directional
DOG
operators, each of which yield an approximation to
the second directional derivative, Grimson
[
191 used the
Laplacian of Gaussian
(v2G)
operator and grouped the
zero-crossing points in 12 directional bins. The precise
form of the operator is given in polar coordinates
(r,
0)
by
where
U
is the Gaussian space-constant. This is a rotation-
ally symmetric function shaped like an inverted Mexican
hat (Fig.
3).
The width of the central negative region is
given by
w2-D
=
2au. Grimson used three [20] or four
[19] different sizes
of
filters for his images.
2)
Matching:
The algorithm begins with images filtered
by the largest filters because the reduced density of zero-
crossings makes matching easier. The overall matching
strategy of Grimson [19] uses a coarse-to-fine iterative
approach with disparities found at coarser resolutions used
to guide match-point search at finer resolutions. Marr [38],
[41] studied the probability distribution of the interval
between adjacent zero-crossings of the same sign obtained
from the convolution of random dot stereograms with the
Laplacian of Gaussian filter. The results indicated that
if
the disparity between the images is less than
+(w/2),
a
search for matches within the range
,(w/2)
will yield
only the correct match with probability
0.95.
However the
Fig.
3.
2-D
Laplacian
of
Gaussian.
alternate strategy of using a search space with range
f
o
is
used by Grimson [19] since it allows one to search for
matches over a larger disparity range and yet get unam-
biguous and correct matches with probability
0.5.
In
Grimson’s implementation [19] for each zero crossing
PL(x, y)
in the left image, possible candidate matches
P;(x’,
y)
are searched for along the epipolar line in the
right image such that,
x
+
d,
-
<
x’<
x
+
d,
+
(2)
as shown in Fig. 4(a), where
d,
is the estimated disparity
and
w
(
=
2au) is the width of the
LOG
filter. Zero-cross-
ings in the left and right images having the same contrast
sign and approximately the same orientation (within
k
30’)
are matched. If only one match is found within the
+w
region, then that match is accepted as unambiguous, and
the disparity is recorded.
3)
Disambiguation
of
multiple matches:
If more than one
match is found within the
+
w
region, then the one having
disparity of the same type (convergent, divergent, or zero)
as the dominant disparity in the neighborhood is accepted.
Otherwise the match at that point is left ambiguous. This
can be regarded as the pulling effect which is described in
the psychophysical experiments of Julesz and Chang [32].
Each 2-D array of matched results is scanned and if the
percentage of matched points is
<
0.7 then all matches in
that region are discarded.
C. Grimson’s Modified Implementation
of
Marr
-
Poggio Theory
Grimson’s earlier implementation [19] of the Marr-
Poggio theory [41] imposes a regional continuity check on
disparity. Later, Grimson
[20]
highlights some of the prob-
lems associated with the earlier implementation of the
Marr-Poggio theory and presents a modified implementa-
tion.
1)
Figural Continuity:
Grimson’s implementation
[
191 of
the Marr-Poggio theory [41] used a regional continuity
check on disparity in order to validate the matches.
Grimson
[20]
observed that this caused difficulties in prop-
agation of disparity at occluding boundaries between ob-
jects and along thin elongated surfaces. Elsewhere the
matched feature points tended to form extended contours.
Hence the figural continuity constraint of Mayhew and
Frisby [44] that required continuity of disparity along
contours was deemed more appropriate.
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