Image Processing Reference
In-Depth Information
2.5.6.1
Proximity to Euclidean Hyperspheres
In [154], a geometric approach is adopted to measure the proximity of
the octagonal distance functions to Euclidean norms in 2-D and 3-D. In this
method, measures related to the perimeter, area, and shape of octagonal disks
in 2-D are compared with those of disks in the Euclidean space. In 3-D, sur-
face area, volume, and shape of the spheres are compared. For comparing
the degree of approximation to the Euclidean norm, the error measures from
Definition 2.34 are used [154].
Error measures in 2-D: For w
1
= 1, and β =
1
w
2
−1,
2πr−4rP(β)
r
Perimeter Error = E
S
(β)
=
= 2|π−2P(β)|,
πr
2
−r
2
F(β)
r
2
Area Error = E
V
(β)
=
= |π−F(β)|, and
√
√
2)
2
π−
(β(2 −
2) +
Shape Feature Error = E
ψ
(β)
=
= |π−S(β)|.
2 + 4β−2β
2
where S(β) =
(β(2−
√
√
2)
2
2)+
.
2+4β−2β
2
1
w
2
1
w
3
−
w
2
,
Error measures in 3-D: For w
1
= 1, β =
−1, and γ =
Surface Area Error = E
S
(β,γ) = |π−(G(β,γ))|,
Volumetric Error = E
V
(β,γ) = |π−T(β,γ)|, and
π−
G(β,γ)
3
T(β,γ)
2
Shape Feature Error = E
ψ
(β,γ) =
.
The functional forms of P(β) and F(β) are described in Theorem 2.31,
whereas T(β,γ) and G(β,γ) are defined in Theorem 2.32. In Tables 2.10 and
2.11 we present values of corresponding error measures for some of the repre-
sentative octagonal distances approximated by weighted t-cost distance func-
tions. This set includes distance functions, which are reported [154] to be of
good approximation of Euclidean norm. For examples, in 2-D the distance
functions defined by N-Sequences of {1,1,2} and {1,1,1,2} provide very low
values of error measures. In Table 2.10, we also report the corresponding er-
ror measures for inverse square root weighted t-cost norm (WD
isr
(·)) in 2-D.
It has been observed that all the error measures have low values with this
norm. Similar observations are also drawn from the results presented in Table
2.11. In particular, in both tables, the error values related to the shape of the
circle and sphere are the smallest among the distance functions chosen for
comparison.
