Image Processing Reference
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Proof. From Corollary 2.9, we get A V (m,n;r) =
−v n )r n n−1
V n (r) −vol(m,n;r)
v i r i
−v n )r n |,r →∞
=
(L n
≈ |(L n
i=0
Hence, the absolute error is unbounded.
R V (m,n;r) = A V (m,n;r)
V n (r)
,r ≥ 1
n
n
n
v i
v i
L n r
v i
L n
L n r i−n −1
−i −1
−i + 1
=
=
<
r
i=0
i=0
i=0
n
|v i
|
L n + 1
<
i=0
Hence, the relative error is bounded. Note that v i 's are dependent on m
and this upper bound is used to select an optimal m for a given n to get the
best m-Neighbor distance in the volumetric sense.
Similar results follow for errors A S (m,n;r) and R S (m,n;r) in surface area.
2.5.3.3
Hyperspheres of Real m-Neighbor Distance
Next we extend the notion of H(m,n;r) for real m-Neighbor distance
δ m . Besides the expression of volume like before, the explorations in the real
(continuous) space offer nice results in terms of the vertices of the hyperspheres
and their inscribed and circumscribed Euclidean hyperspheres.
We start with the definitions of hypersphere and its volume in the contin-
uous case.
Definition 2.35. An m-hypersphere H(m,n;r) in n-D, of radius r and
center 0, is defined as a subset of R n as follows:
H(m,n;r) = {x : x ∈ R n , δ m (x) ≤r}.
H being 2 n -symmetric, it is su cient to restrict our attention to the all-
positive hyperoctant R + where x i
≥ 0,∀i,1 ≤ i≤ n.
Definition 2.36. The volume ||H(m,n;r)|| of an m-hypersphere is defined
as: ||H(m,n;r)|| =
dx, where dx = dx 1 dx 2 ...dx n and x = (x 1 ,x 2 ,··· ,x n ).
δ m (x)≤r
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