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) may remain unspecified, we can develop a
generic local expansion of the function about a sampling point
x
i
. Specifically, if
the position of interest
x
is near the sample at
x
i
,wehavethe
N
-th order Taylor
series
While the particular form of
z
(
·
x
)+
1
T
(
x
i
−
x
)
T
z
(
x
i
)
≈
z
(
x
)+
{
∇
z
(
x
)
}
2
(
x
i
−
{
H
z
(
x
)
}
(
x
i
−
x
)+
···
2
vech
(
x
i
x
)
T
+
T
1
(
x
i
T
β
β
=
β
0
+
−
x
)+
−
x
)(
x
i
−
···
(2)
∇
H
×
×
where
and
are the gradient (2
1) and Hessian (2
2) operators, respectively,
·
and vech(
) is the half-vectorization operator that lexicographically orders the lower
triangular portion of a symmetric matrix into a column-stacked vector. Furthermore,
β
0
is
z
(
x
), which is the signal (or pixel) value of interest, and the vectors
β
1
and
β
2
are
β
1
=
∂
T
z
(
x
)
∂
,
∂
z
(
x
)
∂
,
x
1
x
2
∂
T
2
z
(
x
)
∂
2
z
(
x
)
2
z
(
x
)
∂
β
2
=
1
2
2
∂
x
2
,
∂
,
.
(3)
x
1
∂
x
1
∂
x
2
Since this approach is based on
local
signal representations, a logical step to take is
to estimate the parameters
P
i
=1
while
giving the nearby samples higher weights than samples farther away. A (weighted)
least-square formulation of the fitting problem capturing this idea is
N
n
=0
from all the neighboring samples
{
β
n
}
{
}
y
i
i
=1
y
i
−
β
0
−
β
2
P
2
vech
(
x
i
−
x
)
T
−···
T
1
(
x
i
−
T
min
{
β
n
}
x
)
−
β
x
)(
x
i
−
K
H
(
x
i
−
x
)
n
=0
(4)
with
det(
H
)
K
H
−
1
(
x
i
−
x
)
,
1
K
H
(
x
i
−
x
)=
(5)
where
N
is the regression order,
K
(
) is the kernel function (a radially symmetric
function such as a Gaussian), and
H
is the smoothing (2
·
2) matrix which dictates
the “footprint” of the kernel function. In the classical approach, when the pixels (
y
i
)
are equally spaced, the smoothing matrix is defined as
×
H
=
h
I
(6)
for every sample, where
h
is called the
global smoothing parameter
.Theshapeof
the kernel footprint is perhaps the most important factor in determining the quality
of estimated signals. For example, it is desirable to use kernels with large footprints
in the smooth local regions to reduce the noise effects, while relatively smaller foot-
prints are suitable in the edge and textured regions to preserve the signal disconti-
nuity. Furthermore, it is desirable to have kernels that adapt themselves to the local
structure of the measured signal, providing, for instance, strong filtering along an
