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singular vectors and values, we compute a more stable estimate of our covariance
matrix as:
γ
i
V
i
ρ
i
V
i
=
γ
i
ρ
i
v
2
v
2
,
1
C
i
=
ρ
i
v
1
v
1
+
(18)
1
ρ
i
where
γ
i
=
s
1
s
2
+
α
.
λ
λ
ρ
i
=
s
1
+
,
(19)
λ
s
2
+
P
The parameters
ρ
i
and
γ
i
are the
elongation
and
scaling
parameter, respectively, and
λ
λ
are “regularization” parameters, respectively, which dampen the effect
of the noise and restrict
and
γ
i
and the denominator of
ρ
i
from becoming zero. The
λ
= 0
.
1,
λ
= 0
.
1,
parameter
α
is called the
structure sensitivity
parameter. We fix
and
= 0
.
2 in this work. More details about the effectiveness and the choice of the
parameters can be found in [14]. With the above choice of the smoothing matrix and
a Gaussian kernel, we now have the steering kernel function as
α
x
)=
det(
C
i
)
2
exp
.
x
)
T
C
i
(
x
i
−
−
(
x
i
x
)
K
H
i
(
x
i
−
−
(20)
h
2
2
h
2
π
Fig. 2 illustrates a schematic representation of the estimate of local covariance ma-
trices and the computation of steering kernel weights. First we estimate the gradients
and compute the local covariance matrix
C
i
by (18) for each pixel. Then, for exam-
ple, when denoising
y
13
, we compute the steering kernel weights for each neighbor-
ing pixel with its
C
i
. In this case, even though the spatial distances from
y
13
to
y
1
and
y
21
are equal, the steering kernel weight for
y
21
(i.e.
K
H
21
(
x
21
−
x
13
))islarger
(a) Covariance matrices from local gradients with 3
×
3 analysis window
(b) Steering kernel weights
Fig. 2
A schematic representation of the estimates of local covariance metrics and the steer-
ing kernel weights at a local region with one dominant orientation: (a) First, we estimate the
gradients and compute the local covariance matrix
C
i
by (18) for each pixel, and (b) Next,
when denoising
y
13
, we compute the steering kernel weights with
C
i
for neighboring pixels.
Even though, in this case, the spatial distances between
y
13
and
y
1
and between
y
13
y
21
are
equal, the steering kernel weight for
y
21
(i.e.
K
H
21
(
x
21
−
x
13
)) is larger than the one for
y
1
(i.e.
K
H
1
(
x
1
−
x
13
)). This is because
y
13
and
y
21
are located along the same edge.
