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0.1
0.1
0.05
0.05
0
0
5
5
5
5
0
0
0
0
−5
−5
x 1
−5
x 1
−5
x 2
x 2
(a) The 0 th
(b) The 2 nd
order equivalent kernel ( W 0 )
order equivalent kernel ( W 2 )
0.16
The zeroth order
The second order
The sharpened zeroth order
0.14
0.12
0.1
0.1
0.08
0.05
0.06
0.04
0
0.02
5
5
0
0
0
−0.02
−5
−5
x 1
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 2
x 2
order equivalent kernel ( W 2 )
(c) A sharpened 0 th
(d) Horizontal cross sections of the equivalent kernels
Fig. 9 Equivalent kernels given by classic kernel regression: (a) the 0 th
order equivalent
kernel with the global smoothing parameter h = 0 . 75, (b) the 2 nd
order equivalent kernel
order equivalent kernel ( W 2 ) with a 3
( W 2 ) with h = 0 . 75, (c) a sharpened 0 th
3 Laplacian
kernel ( L =[1 , 1 , 1; 1 , − 8 , 1; 1 , 1 , 1])andκ = 0 . 045, and (d) Horizontal cross sections
of the equivalent kernels W 0 , W 2 ,and W 2 . For this example, we used a Gaussian function
for K ( · ).
×
order regression is now explicitly expressed by
LW 0 in (41), the formulation al-
lows for adjustment of the regression order across the image, but also it allows for
“fractional” regression orders, providing fine control over the amount of sharpening
applied locally.
We propose a technique to automatically select the regression order parameter
κ
(
near zero in flat regions and to a large value
in edge and texture regions, we can expect a reduction of computational complexity,
prevent amplifying noise component in flat regions, and preserve or even enhance
texture regions and edges. In order to select spatially adapted regression factors, we
can make use of the scaling parameter
κ
) adaptively as follows. By setting
κ
γ i , which we earlier used to normalize the
covariance matrix in (37). This makes practical sense since
γ i is high in texture and
edge areas and low in flat area as shown in Fig. 10. Because
γ i is already computed
 
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