Biomedical Engineering Reference
In-Depth Information
where
Z
d
is the volume of the unit
d
-dimensional sphere. Using
K
(
x
) and window
radius
h
, the
multivariate kernel density estimate
on the point
x
is
K
x
−
x
i
h
n
1
nh
d
f
(
x
)
=
.
(10.35)
i
=
1
The estimate of the density gradient can be defined as the gradient of the kernel
density estimate since a differentiable kernel is used:
i
=
1
∇
K
x
−
x
i
n
1
nh
d
ˆ
f
(
x
)
=
∇
f
(
x
)
≡∇
.
(10.36)
h
Applying (10.34) to (10.36), we obtain
1
n
x
[
x
i
−
x
]
n
x
n
(
h
d
Z
d
)
d
+
2
h
2
ˆ
∇
f
(
x
)
=
,
(10.37)
x
i
∈
H
h
(
x
)
where the region
H
h
(
x
) is a hypersphere of radius
h
and volume
h
d
Z
d
, centered
on
x
, and containing
n
x
data points. The
sample mean shift
is the last term in
(10.37)
1
n
x
M
h
(
x
)
≡
[
x
i
−
x
]
.
(10.38)
x
i
∈
H
h
(
x
)
n
(
h
d
Z
d
)
is the kernel density estimate
f
(
x
) computed with the hy-
persphere
H
h
(
x
), and thus (10.37) can be rewritten as
n
x
The quantity
f
(
x
)
d
+
2
ˆ
∇
f
(
x
)
=
h
2
M
h
(
x
)
,
(10.39)
which can be rearranged as
ˆ
h
2
d
+
2
∇
f
(
x
)
f
(
x
)
.
M
h
(
x
)
=
(10.40)
Using (10.40), the mean shift vector provides the direction of the gradient of the
density estimate at
x
which always points toward the direction of the maximum
increase (in the density). Hence, it converges along a path leading to a mode of
the density.
In [13], Comaniciu
et al.
performed the mean shift procedure for image seg-
mentation in a joint domain, the image (
spatial
) domain, and color space (
range
)
domain. The spatial constraints were then inherent in the mode searching proce-
dure. The window radius is the only significant parameter in their segmentation
scheme. A small window radius results in oversegmentation (i.e. larger number
of clusters), and a large radius produces undersegmentation (yielding a smaller
