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