Biomedical Engineering Reference
In-Depth Information
The first level of the DoG hyperpyramid is obtained by subtracting
L
0
from
L
1
.
Then,
L
1
is sub-sampled to a smaller scale (
2
of
L
1
is used). The convolution and
subtraction process is repeated for
L
1
to generate the second level of the hyper-
pyramid. The whole procedure is repeated recursively to generated the consecutive
levels.
The interest points are detected at the local extrema of the DoG hyperpyramid.
This is performed by checking every voxel in the current level. If the checked voxel
is a local extremum, then it is compared with its neighbors in the upper and the
lower levels. The location of the voxel is selected as an interest point if it is also an
extremum with respect to its local neighborhood in the upper and the lower levels
of the DoG hyperpyramid.
3.4.2. Descriptor building and matching
We build our descriptor in 3D space using gradient orientations histograms
with 2D polar-coordinate bins for neighboring cells that consist of voxels in the
current level-neighborhood of every interest point. This method was previously
used in [72] for 2Dmedical applications and was proven to be efficient with respect
to rotation and affine transformations in other applications [74, 73]. The gradient
magnitude of each voxel in the neighborhood of an interest point is calculated as
G
x
+
G
y
+
G
z
,
r
=
(4)
where
G
x
,G
y
and
G
z
are the gradient components in the
x
,
y
and
z
directions,
respectively. The gradient orientations in the polar coordinate system are given by
G
x
+
G
y
r
G
y
G
x
G
z
r
θ
= tan
−
1
,
and
φ
= sin
−
1
= cos
−
1
.
(5)
To successfully describe the neighborhood of an interest point, the closer voxel
should have a larger influence on the descriptor's entries. Therefore, Gaussian
weights are assigned to every voxel in the neighborhood of the interest point, and
are used with mean at the interest point itself to guarantee a distance-weighted
contribution to the gradient orientation histogram. Moreover, this gives the built
descriptor a robustness with respect to skew distortions [70].
According to [70, 74], one way to achieve rotation invariance is to describe
all the descriptor entries relative to a canonical orientation. This orientation can
be set to the dominant gradient orientation in the interest point neighborhood,
which corresponds to the histogram bin with the maximum value. Therefore,
the 2D histogram bin (
φ
i
,θ
i
) for the
i
th feature is updated by adding the term
r
(
x
i
)
·
g
(
x
i
,σ
2
). The considered bin for update is calculated as
θ
r
=
θ
−
θ
c
, and
φ
r
=
φ
φ
c
, where
θ
c
,φ
c
are the components of the canonical orientation of the
interest point, and
θ, φ
are the components of the gradient orientation referred to
−
