Digital Geometry in Image Processing

62 Digital Geometry in Image Processing

2.5.2 Euclidean Hyperspheres

To compare the digital hyperspheres of various neighbor hood sets with the

hyperspheres H

n

E

(r), r ≥ 0, of Euclidean norm in n-D, we note the following:

Definition 2.33. The Euclide an surface area, volume, and shape fea-

ture measures are given as follows:

Volume: V

E

n

(r) = {x : x ∈ Z

n

, E

n

(x) ≤ r} = L

n

r

n

Surface Area: S

E

n

(r) = {x : x ∈ Z

n

, E

n

(x) = r} =

d

dr

V

E

n

(r) = nL

n

r

n−1

Shape Feature: ψ

E

n

=

(S

E

n

(r))

n

(V

E

n

(r))

n−1

= n

n

L

n

where

L

n

=

π

⌊n/2⌋

Q

⌈n/2⌉−1

i=0

(

n

2

−i

)



The comparison is quantified in terms of the following error measure s:

Definition 2.34. The error measures between Euclidean and digital hyp ...