Biomedical Engineering Reference
In-Depth Information
When an edge map is extracted from the original image and then presented
to a snake model, the boundary-finding process in color images is quite similar to
that in graylevel images. Dumitras and Venetsanopoulos [82] computed an angular
map to identify major color changes (edges) in an image, and hence provided a
method for object shape description in color images. Similarly, in [83] an affinity
map that in nature was a bi-level image was computed and employed in estimation
of object boundaries, which were then presented to a parametric deformable model.
In our study we directly apply one of the color edge detection techniques, i.e., the
Di Zenzo operator, to achieve an edge map of a color image, as in the work of
Sapiro [84], where a color geodesic deformable model was established.
3.2.1. Di Zenzo color gradient
In graylevel images the gradient is defined as the first derivative of the image
luminance values. It has a high value in those regions that exhibit a high luminance
contrast. However, this strategy is not suitable for color images. Both changes
in luminance and color between neighboring pixels should be exploited by more
efficient color gradient definition. Many robust and complex color gradient opera-
tors have been proposed [85-88]. Without loss of generality, we adopt Di Zenzo's
definition of gradients for a color image [87] in our study.
Let Φ(
x, y
):
R
2
→
R
3
be a color image with components Φ
i
(
x, y
):
R
2
→
R
,
i
=1
,
2
,
3. We look for the local variation
d
Φ
2
, which can be given in matrix
form as follows:
dx
dy
T
g
11
dx
dy
,
g
12
d
Φ
2
=
(20)
g
21
g
22
∂
∂x
·
∂
∂x
,
g
12
=
g
21
=
∂
∂x
·
∂
∂y
,
g
22
=
∂
∂y
·
∂
∂y
.
where
g
11
=
G
=(
g
i,j
), also called the
structure tensor
in [89],
is symmetric as well as positive definite. The interesting point about
Note that the matrix
is that
its positive eigenvalues
λ
+
/−
are the maximum and minimum of
d
Φ
2
, while the
orthogonal eigenvectors
θ
+
/−
are the corresponding variation orientations, and
are formally given by
G
g
22
)
2
+4
g
12
g
11
+
g
22
±
(
g
11
−
λ
+
/−
=
,
(21)
2
and the eigenvectors are (cos
θ
±
,
sin
θ
±
), where the angles
θ
+
=1
/
2
·
arctan
(2
g
12
/
(
g
11
−
g
22
)),
θ
−
=
θ
+
+
π/
2. Note that for graylevel images, i.e., Φ(
x, y
):
R
2
→
R
,
λ
+
≡|∇
Φ
|
2
,
λ
−
≡
0 and (cos
θ
+
,
sin
θ
−
)=
∇
Φ
/
.
In pr
actice, the
re are t
hree diffe
rent choices of vector gradient norms [84,89]:
|∇
Φ
|
λ
+
,
λ
+
−
λ
−
and
λ
+
+
λ
−
. Here we select the additive form to define the
color gradient in an RGB color space, as shown in Eq. (22). The reason for this is
