Image Processing Reference
In-Depth Information
(iii) connections of common features.
Human vision is sucient when working with a 2D image, making the
above statements valid for this type of image. For a 3D image, however, we
tend to develop methods for computer processing. At first we tend to imagine
that an area of nearly uniform density suggests the existence of a 3D object.
Secondly we expect that borders or edges of 3D objects may exist. A sudden
change in density values will be detected by calculating the spatial difference
of a 3D image.
3.3.2 Differentials in continuous space
Before proceeding to the explanation of a difference filter, we will summarize
the basics of differentials of a continuous function. Let us consider a func-
tion of three variables
f
(
x, y, z
). We presently assume
f
(
x, y, z
)atleasttwice
differentiable.
(a) Gradient
∇
f
(
x, y, z
)=(
∂f/∂x
)
·
i
+(
∂f/∂y
)
·
j
+(
∂f/∂z
)
·
k
,
(3.19)
where
i
,
j
,and
k
are unit vectors in
x
,
y
,and
z
directions, respectively.
(b) Maximum gradient
=
(
∂f/∂x
)
2
+(
∂f/∂y
)
2
+(
∂f/∂z
)
2
1
/
2
.
∇
f
(3.20)
(c) Direction of the maximum gradient
θ
∗
=tan
−
1
[(
∂f/∂y
)
/
(
∂f/∂x
)]
.
(3.21)
(
∂f/∂z
)
/
(
∂f/∂y
)
2
+(
∂f/∂x
)
2
1
/
2
ϕ
∗
=tan
−
1
{
}
.
(3.22)
(d) Coordinate transform (Fig. 3.2)
(
r, θ, φ
)(polar coordinate system)
(
x, y, z
)(Cartesian coordinate system)
x
=
r
cos
θ
cos
φ, y
=
r
sin
θ
cos
φ, z
=
r
sin
φ.
→
(3.23)
(
x, y, z
)(Cartesian coordinate system)
(
r, θ, φ
)(polar coordinate system)
r
=(
x
2
+
y
2
+
z
2
)
1
/
2
,θ
=tan
−
1
(
y/x
)
,φ
=tan
−
1
(
z/
(
x
2
+
y
2
)
1
/
2
)
.
(3.24)
→
θ
,ϕ
)
(e) Derivative in angular direction (
(
∂f/∂x
)cos
θ
cos
φ
+(
∂f/∂y
)sin
θ
cos
φ
+(
∂f/∂z
)sin
φ
.
(3.25)
Search WWH ::
Custom Search