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Ta b l e 1 A compact lookup table for covariance matrices
c 11
c 12
c 22
ρ j
θ j
1.0000
0
1.0000
1.0
0
2.5000
0
0.4000
2.5
0
1
8 π
2.1925
1.0253
0.7075
2.5
8 π
1.4500
1.4500
1.4500
2.5
3
8 π
0.7075
1.0253
2.1925
2.5
8 π
0.4000
0
2.5000
2.5
5
8 π
0.7075
-1.0253
2.1925
2.5
8 π
1.4500
-1.4500
2.1925
2.5
7
8 π
2.1925
-1.0253
0.7075
2.5
4.4
Adaptive Regression Order
As mentioned earlier, although the kernel estimator with a higher regression order
preserves high frequency components, the higher order requires more computation.
In this section, we discuss how we can reduce the computational complexity, while
enabling adaptation of the regression order. According to [26], the second order
equivalent kernel, W 2 , can be obtained approximately from the zeroth order one, W 0 ,
as follows. First, we know that the general kernel estimator (12) can be expressed
as:
z ( x )= e 1 X T KX 1 X T Ky = W N y
(40)
×
where again W N
P vector containing the filter coefficients and which we call
the equivalent kernel. The zeroth order equivalent kernel can be modified into W 2
by
is a 1
W 2 = W 0 κ
LW 0 ,
(41)
where L is Laplacian kernel in matrix form (we use [1 , 1 , 1; 1 ,
8 , 1; 1 , 1 , 1] as a
discrete Laplacian kernel) and
is a regression order adaptation parameter. This
operation can be seen to “sharpen” the equivalent kernel, and is equivalent to sharp-
ening the reconstructed image. Fig. 9 shows the comparison between the actual
second order equivalent kernel, W 2 , and the equivalent kernel, W 2 ,givenby(41).
In the comparison, we use the Gaussian function for K , and compute the zeroth or-
der and the second order equivalent kernels shown in Fig. 9(a) and (b) respectively.
The equivalent kernel, W 2 , is shown in Fig. 9(c), and Fig. 9(d) shows the horizon-
tal cross section of W 0 , W 2 ,and W 2 . As seen in Fig. 9(d), W 2 is close to the exact
second order kernel W 2 .
There are two advantages brought by (41): (i) The formula simplifies the com-
putation of the second order equivalent kernels, i.e. there is no need to generate the
basis matrix, X , or take inversion of matrices. (ii) Since the effect of the second
κ
 
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