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edge rather than across it. This last point is indeed the motivation behind the steer-
ing KR framework [14] which we will review in Section 2.2.
Returning to the optimization problem (4), regardless of the regression order and
the dimensionality of the regression function, we can rewrite it as a weighted least
squares problem:
( y
Xb ) ,
b = arg min
b
Xb ) T K ( y
(7)
where
b =
T
, y P ] T ,
T
2 ,
T
N
y =[ y 1 , y 2 ,
···
β 0 ,
β
···
,
β
,
(8)
K = diag K H ( x 1
x ) ,
x ) , K H ( x 2
x ) ,
···
, K H ( x P
(9)
and
x ) T , vech T ( x 1
x ) T ,
1 , ( x 1
x )( x 1
···
x ) T , vech T ( x 2
x ) T ,
1 , ( x 2
···
x )( x 2
X =
(10)
.
.
.
.
x ) T , vech T ( x P
x ) T ,
1 , ( x P
x )( x P
···
with “diag” defining a diagonal matrix. Using the notation above, the optimization
(4) provides the weighted least square estimator
W N
W N , x 1
W N , x 2
.
b = X T KX 1 X T Ky =
y ,
(11)
where W N
P vector that contains filter coefficients, which we call the equiv-
alent kernel weights, and W N , x 1
is a 1
×
P vectors that compute the
gradients along the x 1 -and x 2 -directions at the position of interest x . The estimate of
the signal (i.e. pixel) value of interest
and W N , x 2
are also 1
×
β 0 is given by a weighted linear combination
of the nearby samples:
P
i =1 W i ( K , H , N , x i x ) y i ,
P
i =1 W i ( · )=1 ,
z ( x )=β 0 = e 1 b = W N y =
(12)
where e 1 is a column vector with the first element equal to one and the rest equal to
zero, and we call W i the equivalent kernel weight function for y i (q.v.[14]or[21]
for more detail). For example, for zero-th order regression (i.e. N = 0), the estimator
(12) becomes
i =1 K H ( x i
x ) y i
β 0 =
z ( x )= ˆ
,
(13)
P
i =1 K H ( x i
x )
which is the so-called Nadaraya-Watson estimator (NWE) [22].
 
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