Biomedical Engineering Reference
In-Depth Information
3.4.1. 3D local invariant features for voxel-based similarity measure
Building a good invariant feature descriptor starts from the selection of the
points that are less affected by geometrical variations. Hence, distinct character-
istics, which also should be invariant to different imaging changes, are carefully
collected to build the feature descriptor. Finally, matching these feature descriptors
is performed to find the correspondent pairs of control points.
Interest point detection
: Interest points are usually selected in highly informa-
tive locations such as edges, corners, or textured regions. In the context of feature
invariance, interest points should be selected so that they achieve the maximum
possible repeatability under different imaging conditions.
The most challenging point is the invariance with respect to scale changes.
Scale-space theory offers the main tools for selecting the most robust feature
locations, or the interest points, against scale variations. Indeed, given a signal
f
n
:
R
→
R
, where
n
=3in the case of volumetric data, the scale-space
n
×
R
+
→
R
representation
L
:
R
is defined as the following convolution:
L
(
x
,t
)=
g
(
x
,t
)
∗
f
(
x
)
,
(2)
n
, and
g
(
x
where
L
(
x
,t
) denote the scale-space kernel that is
proven to be Gaussian with
t
=
σ
2
[70]. Note that as
t
increases, the scale-space
representation
L
(
x
,
0) =
f
(
x
),
∀
x
∈
R
,t
) of the signal tends to coarser scales [70].
Normalization of the Laplacian of Gaussian
∇
2
g
with factor
σ
2
=
t
is nec-
essary for true scale invariance, as proven by Lindeberg [70]. Later, Mikolajczyk
and Schmid [71] proved experimentally that the extrema (maxima and minima) of
σ
2
∇
2
g
produce the most stable image features:
σ
g
(
x, kσ
)
−
g
(
x, σ
)
σ
2
∇
2
g
≈
,
kσ
−
σ
−
1)
σ
2
∇
2
g,
⇒
g
(
x, kσ
)
−
g
(
x, σ
)
≈
(
k
which shows that the
σ
2
normalization of the Laplacian of Gaussian can be ap-
proximated by the Difference-of-Gaussian (DoG). In other words, the locations of
the extrema in the DoG hyperpyramid, i.e., scale-space levels, correspond to the
most stable features with respect to scale changes.
In this work, the scale-space representation of an input 3D signal
f
is generated
as follows. First, let's define
L
0
=
g
(
x
,t
0
)
∗
f
(
x
)
,
and
L
1
=
g
(
x
,t
1
)
∗
f
(
x
)
,t
1
=
C.t
0
,
(3)
where
C>
1 is a real number, and
T
1
(2
πt
)
exp(
−
x
x
g
(
x
,t
)=
)
.
3
2
2
t
