Image Processing Reference
In-Depth Information
From the above theorems, we state the measures of disks of inverse square
root weighted t-cost norm in 2-D and 3-D in the following corollaries.
Corollary 2.14. In 2-D, the perimeter and area of H
isr
(r) are given by
6.6274r and 3.313708r
2
, respectively. H
isr
(r) is a regular octagon.
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Corollary 2.15. In 3-D, the surface area and volume of H
isr
(r) are given by
14.3319r
2
and 4.7773r
3
, respectively.
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For inverse square root weighted t-cost norm the n-D hypersphere H
isr
(r)
encloses the n-D Euclidean hypersphere H
E
(r). Also H
E
(r) touches all the
center points of the faces (hyperplane of dimension n − 1) of H
isr
(r). For
example, in 2-D the regular octagon of radius r encloses the Euclidean circle
of the same radius (refer to Fig. 2.14), which touches each side of the regular
octagon at its midpoints, that is, the signed permutation sets φ((r,0)) and
φ((
√
2
,
√
2
)).
Reprinted from
Pattern Recognition Letters
, 32(2011), J. Mukherjee, On approximating Euclidean metrics by
weighted t-cost distances in arbitrary dimension, 824-831, Copyright (2011), with permission from Elsevier.
FIGURE 2.14: Circle of inverse square root weighted t-cost norm in 2-D
(that is, a regular octagon) enclosing the Euclidean circle of the same radius.
This containment of the Euclidean hypersphere within that of the inverse
square root weighted t-cost norm of the same radius is stated in the following
theorem [150].
Theorem 2.33. H
E
(r) is enclosed within H
isr
(r). Also H
E
(r) touches
H
isr
(r) exactly at 3
n
−1 points, which are centers (that is, φ(v
r,t
), 1 ≤t ≤ n)
of the hyperfaces (of dimension n−1) of the hypersphere.
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For example, in 2-D and 3-D, the Euclidean sphere touches the circle and
sphere of the inverse square root weighted t-cost norms at 8 and 26 points,
respectively. The points of contact in 2-D are already described before and
demonstrated in Fig. 2.14. In 3-D, for a radius of r, these points belong to the
sets φ((r,0,0)), φ(
√
2
,
√
2
,0)), and φ((
√
3
,
√
3
,
√
3
)).
