Image Processing Reference
In-Depth Information
The expression for ||H(m,n;r)|| as presented in Theorem 2.22 is derived
in [66] with the use of the inclusion-exclusion principle from combinatorics.
Theorem 2.22. ∀n ∈ N,0 < m ≤ n,r ≥ 0, the volume of the m-hypersphere
H(m,n;r) is given by ||H(m,n;r)|| = v
n
(m)·r
n
, where
⌈m⌉−1
n
j
v
n
(m) = (2
n
/n!)·
·(m−j)
n
·(−1)
j
.
j=0
[66] also establishes the vertices of the hypersphere as:
Theorem 2.23. The vertices of the polytope H(m,n;r) are φ(x), where
x = (r,r,··· ,r
⌊m⌋
,(m−⌊m⌋)r,0,0,··· ,0
n−⌈m⌉
)
and φ(·) is the 2
n
symmetry function of n-D point. Given x ∈ (R
+
)
n
, φ(x)
gives the set of points in R
n
obtained by the reflection and permutation of x.
The Euclidean hyperspheres that can be inscribed within or circum-
scribed around these m-hyperspheres demonstrate interesting properties.
While the former is called an insphere, the latter is termed as a circum-
sphere. We define the inradius r
I
and the circumradius r
C
, respectively, as:
′
: H
E
(r
′
r
I
=
max{r
) ⊆ H(m,n;r)} and
′
: H
E
(r
′
r
C
=
min{r
) ⊇H(m,n;r)}
where H
E
(r
with the center at the
origin. Note that r
I
and r
C
both are functions of m,n, and r.
Clearly, H
E
(r
I
) and H
E
(r
C
) are the insphere and circumsphere of
H(m,n;r), respectively. They touch the H(m,n;r) at the furthest inner points
t
I
and nearest outer points t
C
, where δ
m
(t
I
) = δ
m
(t
C
) = r and E
n
(t
I
) = r
I
and E
n
(t
C
) = r
C
. In the next theorem we present the expressions for these
quantities.
′
) is a Euclidean hypersphere of radius r
′
Theorem 2.24. ∀n,n ≥ 1,0 < m ≤ n we have
√
1. r
I
= min(1,m/
n)·r and r
C
=
(⌊m⌋+ (m−⌊m⌋)
2
)·r
∈ φ(r,0,0,··· ,0) for m ≤
√
t
I
n
∈ φ(mr/n,mr/n,··· ,mr/n) for m ≥
√
n and
2.
t
C
∈ φ(r,r,··· ,r
⌊m⌋
,(m−⌊m⌋)r,0,0,··· ,0
n−⌈m⌉
)
where φ(·) is the 2
n
symmetry function of an n-D point.
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