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given the pruned valid transforms we show in Section 4.2.3 how to recover the
intrinsic geometrical properties (center of rotation and axis of reflection for
C
K
or
D
K
, respectively) of each detected symmetry.
4.2.1
Image Representation by Local Features
Given an input image
I
, we compute an image model
M
for each type of local de-
tector/descriptor. A reflected replica of the features is then added to
M
[LE06]. This
allows us to handle reflections, recalling that the local features are rotationally, but
not reflectionally invariant. The features are then progressively sampled [ELPZ97]
to reduce their number to a few thousands. The progressive sampling spreads the
points evenly over the image, thus reducing the number of image regions with high
numbers of features. These are prone to produce local partial matches.
We also utilize the dominant scale property of the local features [Low03] to prune
false pairwise assignments and further sparsify the affinity matrix
H
.Asweanalyze
self-alignments within the same image, corresponding features will have similar
dominant scales. Hence, we prune the pairwise affinities in Eq. 12 by
⎧
⎨
Sc
x
j
k
1
> |
log
Sc
x
i
k
1
/
log
(
Δ
S
)
|
0
or
H
log
Sc
x
i
k
2
Sc
x
j
k
2
> |
(
k
1
,
k
1
)=
(14)
⎩
)
|
/
log
(
Δ
S
H
(
k
1
,
k
1
)
else
(
)
where
Sc
x
is the dominant scale of the point
x
,and
Δ
S
is a predefined scale differ-
ential. We use the
to symmetrize the scale discrepancy with respect to both
points. This implies that if a pair of corresponding points is scale-inconsistent, all
of its pairwise affinities are pruned.
In order to effectively utilize the sparsity of the affinity matrix
H
, we threshold
it by
T
|
log
(
.
)
|
10
−
5
. Namely, the affinity in Eq. 12 is always nonzero, but for geometri-
=
cally inconsistent pairs,the affinity is of
O
10
−
7
, while the consistent pairs are of
O
10
−
1
.
4.2.2
Symmetry Categorization and Pruning
Given the image model
M
we apply the spectral matching scheme described in
Section 3.3, and derive
P
tentative self alignments of the image denoted
P
1
.
{
C
i
}
Typically
P
K
,
K
being a typical order of image symmetry. In practice
K
<
10
and
P
15. The reason being that in real images, the symmetric patterns might be
embedded in clutter, resulting in the recovery of spurious self-alignments (eigen-
vectors) unrelated to the symmetry.
To address this issue we propose an assignment pruning scheme, that is based
on the norm property of symmetry transforms in
≈
2
. Namely, by computing the
projective transform corresponding to the recovered matching, and recalling that
the norm of a symmetry transform is
R
±
1 (Section 2), erroneous matchings can
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