i
i
i
i
i
i
i
i
158 8 Directional DCT Transform
For a given direction angle θ unequal to zero as shown in Figure 8.2b, the first
type of directional primary operation with a parameter α from one pixel to another
is defined in Z domain as
O(X [n
i
],X [n
j
],θ ,α) =
X ←−
1
1
1
1
1
αF(θ) 1
1
1
X
, (8.6)
where n
i
= n
5
and n
j
= n
2
in the case illustrated in Figure 8.2b. F(θ ) is the function
of shifting and interpolation in Z domain and locates at the n
i
-th row and the n
j
-th
column of operation matrix. It is defined as
F (θ ) = Z
−|n
i
−n
j
|tan(θ)
. (8.7)
If |n
i
−n
j
|tan(θ) is an integer, F(θ) only means a left shifting of |n
i
−n
j
|tan(θ)
pixels; otherwise the interpolation is needed and F(θ) is redefined as
F(θ ) =
∑
k
w
k
z
−k