Image Processing Reference
In-Depth Information
2.5.2 Euclidean Hyperspheres
To compare the digital hyperspheres of various neighborhood sets with the
hyperspheres H
E
(r), r ≥ 0, of Euclidean norm in n-D, we note the following:
Definition 2.33. The Euclidean surface area, volume, and shape fea-
ture measures are given as follows:
V
n
(r)
{x : x ∈ Z
n
,E
n
(x) ≤ r}
= L
n
r
n
Volume:
=
d
dr
V
n
(r) = nL
n
r
n−1
S
n
(r)
{x : x ∈ Z
n
,E
n
(x) = r}
Surface Area:
=
=
(S
n
(r))
n
(V
n
(r))
n−
1
Shape Feature: ψ
n
= n
n
L
n
=
where
π
⌊n/
2
⌋
L
n
=
Q
⌈n/
2
⌉−
1
i
=0
(
2
−i
)
€
The comparison is quantified in terms of the following error measures:
Definition 2.34. The error measures between Euclidean and digital hyper-
spheres are defined as follows:
S
n
(r)−surf(N(·);r)
r
n−
1
E
S
(N(·))
Surface Error:
=
lim
r→∞
V
n
(r)−vol(N(·);r)
r
n
Volume Error:
E
V
(N(·))
=
lim
r→∞
Shape Feature Error:
E
ψ
(N(·))
=
ψ
n
−ψ
n
(N(·))
V
n
(r) −vol(N(·);r)
Absolute Volumetric Error: A
V
(N(·);r)
=
A
V
(N(·);r)
V
n
(r)
Relative Volumetric Error: R
V
(N(·);r)
=
S
n
(r) −surf(N(·);r)
Absolute Surface Error:
A
S
(N(·);r)
=
A
S
(N(·);r)
S
n
(r)
Relative Surface Error:
R
S
(N(·);r)
=
€
Now we present expressions for surf(N(·);r) and vol(N(·);r) for various
neighborhood sets and compare these quantities against the Euclidean hyper-
spheres in the sense of the above error measures.
2.5.3 Hyperspheres of m-Neighbor Distance
First we explore the properties of hypersurfaces and hyperspheres of O(m)-
neighbor distance d
n
from [70]. We use the following notation:
S(m,n;r) =⇒ S(N(·);r), surf(m,n;r) =⇒ surf(N(·);r) and
H(m,n;r) =⇒ H(N(·);r), vol(m,n;r) =⇒ vol(N(·);r).
