Master Thesis - Department of Computer Science

2. Compute the gradients ϱx(i, j) and ϱy(i, j) at each pixel (i, j). The gradient
operation in our case is done by sobel operator.
3. Estimate the local orientation of each block centered at pixel (i, j) using the
following equation,
γx(i, j) =
γy(i, j) =
i+ W
2
�
u=i− W
2
i− W
2
�
u=i− W
2
θ(i, j) = 1
2 tan−1
j+ W
2
�
v=j− W
2
j− W
2
�
v=j− W
2
� γy(i, j)
γx(i, j)
2ϱx(u, v)ϱy(u, v), (A.4)
�
ϱ 2 x (u, v)ϱ2y (u, v)� , (A.5)
�
. (A.6)
where θ(i, j) is the least square estimate of the local ridge orientation at the
block centered at pixel (i, j) and represents the direction which is orthogonal
to the dominant direction of the Fourier spectrum of the W × W window.
4. Noise, corrupted ridge and valley structures may result incorrect estimation
of local ridge orientation. As local ridge orientation varies slowly in a local
neighborhood low-pass filtering can be used to correct θ(i, j). For low-pass
filtering the orientation image is converted to a continuous vector field and is
defined as,
Φx(i, j) = cos(2θ(i, j)), (A.7)
Φy(i, j) = sin(2θ(i, j)). (A.8)
where Φx(i, j) and Φy(i, j) are the x and y components of the vector field,
respectively. The low-pass filtering can be done as follows,
Φ ′ x (i, j) =
Φ ′ y (i, j) =
D Φ
2�
u=− D Φ
2
D Φ
2�
u=− D Φ
2
D Φ
2�
v=− D Φ
2
D Φ
2�
v=− D Φ
2
D(u, v)Φx(i − uW, j − vW ), (A.9)
D(u, v)Φy(i − uW, j − vW ). (A.10)
where D is a two-dimensional low-pass filter with unit integral and DΦ × DΦ
specifies the size of the filter.
5. Compute the local ridge orientation at (i, j) using,
O(i, j) = 1
2 tan−1
� �
′ Φ y(i, j)
126
Φ ′ x (i, j)
(A.11)