Information Technology Reference
In-Depth Information
What we described above is the “classic” kernel regression framework, which, as
we just mentioned, yields a pointwise estimator that is always a local
linear
combi-
nation of the neighboring samples. As such, it suffers from an inherent limitation.
In the next sections, we describe the framework of
steering
KR in two and three
dimensions, in which the kernel weights themselves are computed from the local
window, and therefore we arrive at filters with more complex (nonlinear) action on
the data.
2.2
Steering Kernel Function
The steering kernel framework is based on the idea of robustly obtaining local sig-
nal structures (e.g. discontinuities in 2-D and planes in 3-D) by analyzing the radio-
metric (pixel value) variations locally, and feeding this structure information to the
kernel function in order to affect its shape and size.
Consider the (2
2) smoothing matrix
H
in (5). As explained in the previous
section, in the generic “classical” case, this matrix is a scalar multiple of the iden-
tity. This results in kernel weights which have equal effect along the
x
1
-and
x
2
-
directions. However, if we properly choose this matrix locally (i.e.
H
×
H
i
for each
y
i
), the kernel function can capture local structures. More precisely, we define the
smoothing matrix as a symmetric matrix
→
2
H
i
=
h
C
−
,
(14)
i
which we call the
steering
matrix and where, for each given sample
y
i
, the matrix
C
i
is estimated as the local covariance matrix of the neighborhood spatial gradient
vectors. A naive estimate of this covariance matrix may be obtained by
C
naive
=
J
i
J
i
,
(15)
i
with
⎡
⎤
z
x
1
(
x
1
)
z
x
2
(
x
1
)
⎣
⎦
.
.
J
i
=
,
(16)
z
x
1
(
x
P
)
z
x
2
(
x
P
)
·
·
where
z
x
1
(
) are the first derivatives along
x
1
-and
x
2
-axes, and
P
is the
number of samples in the local analysis window around a sampling position
x
i
.
However, the naive estimate may in general be rank deficient or unstable. Therefore,
instead of using the naive estimate, we can obtain the covariance matrix by using
the (compact) singular value decomposition (SVD) of
J
i
:
) and
z
x
2
(
J
i
=
U
i
S
i
V
i
,
(17)
where
S
i
= diag[
s
1
,
s
2
],and
V
i
=[
v
1
,
v
2
]. The singular vectors contain direct infor-
mation about the local orientation structure, and the corresponding singular values
represent the energy (strength) in these respective orientation directions. Using the
