Image Processing Reference
In-Depth Information
In the digital space surf(N(·);r) is often a polynomial in r of degree n−1
with rational coe
cients.
Definition 2.30. H(N(·);r) is the hypersphere of radius r in n-D for
neighborhood Set N(·). It is the set of n-D grid points that lie within at a
distance r, r ≥ 0, from the origin when d(N(·)) is used as the distance.
H(N(·);r) = {x : x ∈ Z
n
,0 ≤d(x;N(·)) ≤ r}
The volume vol(N(·);r) = ||H(N(·);r)|| of a hypersphere H(N(·);r) is
defined as the number of points in H(N(·);r).
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In the digital space vol(N(·);r) is often a polynomial in r of degree n with
rational coe
cients.
Note that
r
r
H(N(·);r) =
S(N(·);s) and vol(N(·);r) =
surf(N(·);r).
s=0
s=0
Definition 2.31. The shape feature ψ
n
(N(·)) of a hypersphere in n-D is
defined as:
(surf(N(·);r)
n
(vol(N(·);r))
n−1
ψ
n
(N(·)) = lim
r→∞
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vol(N(·);r) and surf(N(·);r) being polynomials in r of degree n and n−1,
respectively, ψ
n
(N(·)) is a dimension-less quantity characterized by the param-
eters of the neighborhood set N(·). Further, if vol(N(·);r) and surf(N(·);r)
contain the term for the highest power of r alone, the limit in the above
definition becomes redundant.
Though the above definition of the shape feature is provided in general for
n-D, it finds major use up to three dimensions, only where the surfaces and
volumes (and therefore shapes) can be physically visualized.
Definition 2.32. An x ∈ Z
n
is said to be a vertex or corner of H(N(·);r)
if x ∈H(N(·);r) and
x
i
= max{y : d((x
1
,x
2
,··· ,x
i−1
,y,0,0,··· ,0);r) = r,1 ≤ i ≤n.
In other words, ∀1 ≤i ≤ n,
d((x
1
,x
2
,··· ,x
i−1
,x
i
,0,0,··· ,0);r) = r and
d((x
1
,x
2
,··· ,x
i−1
,x
i
+ 1,0,0,··· ,0);r) > r.
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Note that, by symmetry, if x is a vertex of H(N(·);r), then all points
belonging to φ(x) are vertices of H(N(·);r).
