Image Processing Reference
In-Depth Information
Fig. 3.2.
The Cartesian coordinate system and polar coordinate system. Cartesian
coordinates (
x, y, z
)
↔
polar coordinates (
r, θ, φ
).
(f) Laplacian
f
=(
∂
2
f/∂x
2
)+(
∂
2
f/∂y
2
)+(
∂
2
f/∂z
2
)
.
∇
(3.26)
(g) Derivatives in the direction of coordinate axis
(first-order derivatives)
f
x
=(
∂f/∂x
)
,f
y
=(
∂f/∂y
)
,f
z
=(
∂f/∂z
)
(3.27)
(second-order derivatives)
f
xx
=(
∂
2
f/∂x
2
)
,f
yy
=(
∂
2
f/∂y
2
)
,f
zz
=(
∂
2
f/∂z
2
)
f
xy
=(
∂
2
f/∂x∂y
)
,f
yz
=(
∂
2
f/∂y∂z
)
,f
zx
=(
∂
2
f/∂z∂x
)
(3.28)
The set of these partial derivatives are often denoted in the form of matrix
called
Hessian
,thatis,
f
xx
f
xy
f
xz
f
yx
f
yy
f
yz
f
zx
f
zy
f
zz
Hessian
(3.29)
3.3.3 Derivatives in digitized space
Derivatives are approximated by differences on a digitized image. Several ex-
amples are shown below.
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