Image Processing Reference
In-Depth Information
Corollary 2.13. The vertices of H
isr
(r) are φ(u), where u ∈ Σ
n
and
r
for i = 1.
√
i−
√
u
i
=
(
i−1)r 2 ≤ i≤ n
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Shapes of a digital circle and sphere in 2-D and 3-D for inverse square root
weighted t-cost norms are shown in Figs. 2.13 (a) and (b), respectively.
(a)
(b)
FIGURE 2.13: (a) A digital circle and (b) a digital sphere of inverse square
root weighted t-cost norm.
Using the results from [154], different measures related to the shape of the
disks of the well-behaved weighted t-cost norm in 2-D and 3-D are stated now.
We restrict to the (normalized) ordered set of weights with w
1
= 1.
Theorem 2.31. The perimeter and area of the circle of radius r for a weighted
t-cost norm in 2-D with the ordered set of weights as {1,w
2
}, such that 0 <
w
2
≤ 1, are 4rP(β) and r
2
F(β), where β =
1
w
2
−1, and
P (β) = (2−
√
√
2,
2)β +
and,
F (β) = 2 + 4β −2β
2
.
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Theorem 2.32. The surface area and volume of the sphere of radius r for
a weighted t-cost norm in 3-D with the ordered set of weights as {1,w
2
,w
3
},
≤ 1, are 4r
2
G(β,γ) and
3
r
3
T(β,γ), where β =
1
w
2
such that 0 < w
3
≤ w
2
−1,
−
w
2
, and,
G(β, γ) = (3−2
1
w
3
γ =
√
3)β
2
+(
√
√
3−6
√
√
√
√
√
3−3)γ
2
+(2
2+6)βγ+2
3β+(6
2−4
3)γ+
3,
and,
T (β, γ) = 1 + 3β + 3γ + 6βγ + 3β
2
−6γ
2
+ 3βγ
2
−6β
2
γ −2β
3
+ γ
3
.
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