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be pruned. Moreover, this provides a mean for categorizing transforms to either
C
K
or
D
K
.
We analyze each correspondence map
C
i
, by applying a normalized DLT algo-
rithm [HZ04], and fitting a projective motion model
T
i
⎡
⎤
⎡
⎤
⎡
⎤
t
11
t
12
t
13
t
21
t
22
t
23
t
31
t
32
x
1
y
1
1
x
2
y
2
1
⎣
⎦
⎣
⎦
=
⎣
⎦
,
T
i
X
1
=
X
2
=
(15)
1
X
1
and
X
2
being the spatial coordinates of corresponding points in
C
i
.
Equation 15 can also be solved for an affine motion model, where the choice
of the model (projective vs. affine) depends on the expected distortion within the
image. The correspondence map
C
i
can be pruned for erroneous point matchings by
applying a robust Least-Squares scheme, such as RANSAC [FB81].
Given the transform
T
i
, we can now apply Eqs. 3 and 7, and classify a transform
T
i
as a cyclic symmetry
C
K
if det
1,
or discard it as an erroneous self-alignment otherwise. Algorithm 1 summarizes the
first two steps of the SSA.
We emphasize that the spectral matching and pruning schemes can also be ap-
plied to sets of points in higher dimensions. An example of symmetry analysis of a
three-dimensional object is provided in Section 5.
(
T
i
)
≈
1, a reflectional symmetry
D
K
if det
(
T
i
)
≈−
4.2.3
Computing the Geometrical Properties of the Symmetry
The center of symmetry and axis of rotation can be computed by complementary
geometrical
and
analytical
approaches. The axis of reflection can be computed an-
alytically given the corresponding transform
T
D
K
, by applying Eq. 8. The reflection
axis is the line connecting the two points, corresponding to the two eigenvectors of
T
D
K
with an eigenvalue of
=
1. We denote this the
analytical
solution. In addition,
one can apply a
geometrical
solution, where we connect the corresponding points
in
D
K
found by the spectral matching. These are the points, which were used to
estimate
T
D
K
in the previous section. The reflection axis is the line that fits through
the middle point of each such line segment.
Given the transform
T
C
K
corresponding to some rotational symmetry
C
K
, the cen-
ter of rotation can be recovered by applying Eq. 4 and computing the eigenvector
of
T
C
K
, corresponding to
λ
i
1. The center of rotation can also be computed
geo-
metrically
, by connecting matching points. For each such line, consider the normal
passing through its middle, as all such normals intersect at the center of rotation.
Thus, the center of rotation is derived by solving an overdetermined set of equations
in the least squares sense, similar to the robust fitting of the reflection axes.
Theorems 1 and 2 provide the foundation for inferring the complete set of sym-
metries, given a subset of them, detected by the spectral analysis. Given two reflec-
tional symmetry transforms
λ
i
=
, the order of symmetry can be derived by
computing the angle between the reflection axis
{
D
K
1
,
D
K
2
}
Δα
. The order of symmetry is then
give by solving:
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