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Algorithm 1.
Spectral Symmetry Analysis of Image
1: Create an image model
M
of the input image. Suppose it contains
N
points.
2: Set
P
, the number of eigenvectors to analyze and
ε , the norm error of the symmetric
transform.
3: Progressively sample the interest regions
4: Reflect all the local descriptors
{
D
i
}
1
in the image while preserving the originals
5: Compute an affinity matrix based on putative correspondences drawn by similar descrip-
tors
{
D
i
}
1
and the dominant scale pruning measure
6: Add random noise to the affinity matrix.
7: Solve Eq. 13 and compute the eigendecomposition
{
ψ
i
,
λ
i
}
.
8:
while
i
<
P
do
9:
Derive correspondence map
C
i
from ψ
i
10:
Estimate transformation matrix
T
i
11:
if
|
det
(
T
i
)
−
1
| <
ε
then
12:
Rotation detected - use Eq. 4 to find the center
| <
ε
then
14: Reflection detected - use Eq. 8 to find the reflection axis
15:
end if
16:
end while
else if
|
det
(
T
i
)+
1
13:
π
Δα
2
Z
=
K
,
Z
,
K
∈
Z
.
Δα
=
2
, implies that
the object has at least two reflectional symmetry axes. But, there might also be four
or even eight symmetry axes. That would imply that the spectral scheme identified
only a subset of the axes.
Hence, the symmetry order can be estimated up to scale factor
Z
.Givenmore
than two reflection axes, one can form a set of equations over the integer variables
{
For instance, given two reflectional axes with a relative angle
Z
i
}
π
Δα
1
Z
1
=
2
K
(16)
.
π
Δα
n
Z
n
=
2
K
As
K
<
8 for most natural objects, Eq. 16 can be solved by iterating over
K
=
1
..
8
and looking for the value of
K
for which all of the
{
Z
i
}
are integers.
5
Experimental Results
In this section we experimentally verify the proposed Spectral Symmetry Analy-
sis scheme by applying it to real images and volumes. In Section 5.1 we apply
the SSA to a set of real images, where the detection of the symmetries becomes
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