Image Processing Reference
In-Depth Information
3
1
2
0.8
1
0.6
0
0.4
−1
−2
0.2
−3
3π/2
5π/2
π/2
0
2
3
4
5
6
7
2
3
4
5
6
7
Fig. 10.31. The graphs in the
left image
represent the estimated directions (arg(
I
20
),
solid
)
as well as the ideal direction angles (
dashed
) on the circle passing through 1 to 4 in Fig.
10.15 when using 18 tune-on directions. The graphs on the
right
show
|I
20
|
(
green
)and
I
11
(
magenta
) on the same ring
Every parameter pair (
φ, b
) represents a unique and infinitely long line. Like-
wise, every (
x, y
) pair defines a curve in the (
φ, b
) space.
2. The local edge strengths along with their directions are extracted by a gradient
filtering,
|∇
f
(
x, y
)
|
=
|
(
D
x
+
iD
y
)
f
(
x, y
)
|
(10.92)
θ
(
x, y
)=arg(
∇
f
)=arg[(
D
x
+
iD
y
)
f
]
(10.93)
. The
result is an edge image,
α
(
x, y
), which is quantized to either 0 (background
point), or 1 (edge point) with the local direction of the edge being
θ
(
x, y
).
3. Given an edge point (
x
0
,y
0
) in the edge image
α
(
x, y
), it defines uniquely one
curve in the (
φ, b
)-plane:
and a binary image is obtained by thresholding the edge strength,
|∇
f
|
cos(
φ
)
· x
0
+sin(
φ
)
· y
0
=
b
(10.94)
because for every imaginable
φ
there is a uniquely defined
b
. However, only a
discrete version of (
φ, b
)-plane is investigated in practice. A grid
A
(
φ
k
,b
k
) is
defined by having an appropriate level of quantization for
φ
and
b
parameters.
Each point of the grid is assigned initially the value zero,
A
(
φ
k
,b
k
). This arti-
ficially constructed grid is called the accumulator, because it will be used for a
voting procedure storing the votes at each cell node,
A
(
φ
k
,b
k
).
4. Assume that (
x
0
,y
0
) is an edge point of the image found in
α
(
x, y
). Because
every point in the (
x, y
) plane defines a curve in the (
φ, b
) plane, one votes for
all points of the curve
C
corresponding to (
x
0
,y
0
) in the (
φ, b
) plane. Adding
the value 1 (a vote) in each of the grid cells
A
(
φ
m
,b
m
) through which the curve
C
passes, one accomplishes the voting by repeating the procedure for all edge
points of
α
(
x, y
). To know which cells the curve
C
passes in the (
φ, b
) plane,
one only needs to substitute
φ
=
φ
k
in Eq. (10.94), where
φ
k
is incremented
