Image Processing Reference
In-Depth Information
norm) terms to elucidate various good approximations of the Euclidean norm
for digital geometry. We also highlight a number of interesting special cases
in 2-D and 3-D.
2.1 Mathematical Definitions and Notation
We start with a few basic definitions that are frequently used.
Definition 2.1. All-positive monotone hyperoctant Σ
n
is defined as
follows:
Σ
n
= {x : x ∈ Z
n
and∀i,1 ≤ i < n,x
i
≥x
i+1
≥ 0}
That is, x belongs to the all-positive hyperoctant and its components are sorted
in non-increasing order. By definition, 0 ∈ Σ
n
.
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Definition 2.2. φ(·) is defined as the 2
n
Symmetry Function of an
n-D point. That is, given x ∈ Z
n
, and x
i
≥ 0,∀i, φ(x) gives the set
of points in Z
n
obtained by reflections and permutations of x. For ex-
ample, if x = (4,1,1), we get the set of symmetric points as φ(x) =
{(±4,±1,±1),(±1,±4,±1),(±1,±1,±4)}.
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n
i=1
[b(i).2
n−i
]
is denoted by [b(1),b(2),...,b(n)]
2
, where b(i) ∈ {0,1},∀i,1 ≤ i ≤ n,n ≥ 1.
In particular, we write x as [1
r
0
n−r
]
2
when
Definition 2.3. Binary representation of a number x =
b(i)
=
1,
1 ≤ i≤ r
=
0,
r < i ≤n,1 ≤ r ≤ n;
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Definition 2.4. f
i
: Z
n
→ P,1 ≤ i ≤ n is the i
th
maximum absolute
component function for a vector u, ∀u ∈ Z
n
. That is, if k
1
,k
2
,...,k
n
are
n distinct indices, 1 ≤ k
j
≤ n, 1 ≤ j ≤ n, then |u(k
1
)|≥|u(k
2
)|≥|u(k
3
)|≥
···≥|u(k
n
)| and f
i
(u) = |u(k
i
)|, 1 ≤ i≤ n. By definition, f
0
(u) = 0. Clearly,
max
i=1
f
1
(u)
=
{|u
i
|}
Largest component
f
2
(u)
=
max
1≤i≤j≤n
{min(|u
i
|,|u
j
|)} Second largest component
min
i=1
f
n
(u)
=
{|u
i
|}
Smallest component
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Definition 2.5. h
i
: Z
n
→ P,1 ≤i ≤ n is the sum of i maximum absolute
components function for a vector u, ∀u ∈ Z
n
. That is, ∀u ∈ Z
n
, h
i
(u) =
i
j=1
f
j
(u). Clearly, h
0
(u) = 0 and h
n
(u) =
n
i=1
|u(i)|.
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