Master Thesis - Department of Computer Science

(i) For each block centered at (i, j),
Ω ′ ⎧
⎪⎨
(i, j) =
⎪⎩
where
Ω(i, j) if Ω(i, j) �= −1
�KΩ /2 �KΩ /2
Kg(u,v)µ(Ω(i−uW,j−vW ))
u=−KΩ /2 v=−KΩ /2
�KΩ /2 �KΩ /2
Kg(u,v)ν(Ω(i−uW,j−vW )+1)
u=−KΩ /2 v=−KΩ /2
µ(x) =
ν(x) =
⎧
⎪⎨
0 if x ≤ 0
⎫
⎪⎬
⎪⎩ x
⎪⎭
Otherwise
⎧
⎪⎨ 0 if x ≤ 0
⎫
⎪⎬
⎪⎩ 1
⎪⎭
Otherwise
⎫
⎪⎬
Otherwise ⎪⎭
(A.15)
(A.16)
(A.17)
Kg is a discrete Gaussian kernel with mean and variance of zero and nine,
respectively, and KΩ = 7 is the size of the kernel.
(ii) If there exists at least one block with the frequency value -1, then swap Ω
and Ω ′ and go to step (i).
6. Finally, a low-pass filter is used to remove the outliers,
F (i, j) =
LΩ/2
�
LΩ/2
�
u=−LΩ/2 v=−LΩ/2
L(u, v)Ω ′ (i − uW, j − vW ), (A.18)
where L is a two-dimensional low-pass filter with unit integral and LΩ = 7 is
the size of the filter.
The frequency image after low-pass filtering is shown in Fig. A.6.
A.1.5 Filtering
The derived orientation and frequency images are used for noise removal. A bandpass
filter tuned to the corresponding frequency and orientation can remove undesired
noise. Gabor filters have both frequency-selective and orientation-selective properties
and have optimal joint resolution in both spatial and frequency domains [32, 53]. The
even-symmetric Gabor filter has the general form [53],
�
h(x, y : φ, f) = exp − 1
� ��
2 xφ cos (2πfxφ) , (A.19)
2
δ 2 x
+ y2 φ
δ 2 y
xφ = x cos φ + x sin φ, (A.20)
yφ = −x sin φ + y cos φ. (A.21)
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