Image Processing Reference
In-Depth Information
By definition, f(0) = 0.
Definition 2.22. A sorted N-Sequence S(B) of an N-Sequence B is a
reordering of the elements of B in non-decreasing order. That is,
S(B) = {s(1),s(2),··· ,s(p)}
where
s(i) ≤s(i + 1), ∀i, 1 ≤ i≤ p−1.
If an N-Sequence is intrinsically ordered, that is, B = S(B), then it is
alternately represented as:
B = [α 1 2 ,··· ,α n ] = {1 α 1 ,2 α 2 ,··· ,n α n }
where α i denotes the number of consecutive i's in the sequence. Clearly,∀i,1 ≤
i ≤n, 0 ≤ α i ≤p, and
n
i=1 α i = p.
Definition 2.23. A subsequence B(i,j) of an N-Sequence B is formed by
taking j elements from B starting from the i-th element. That is,
∀i∀j,1 ≤ i,j ≤ p, B(i,j) = {b(i),b(i + 1),··· ,b(p),b(1),b(2),··· ,b(k)}
where
(i + j)−(p + 1), i + j > p,
i + j −1,
k =
i + j ≤ p.
Definition 2.24. The wave front set WF(B) of an N-Sequence B is defined
recursively as:
W F (B(1, i))
=
W (i), 1 ≤ i ≤ p
W (0)
= {0}
W (i)
= {w : w = u + v∧w ∈ Σ n
P
n
j=1 u(j) = b(i)∧∀j, 0 ≤ u(j) ≤ 1∧v ∈ W (i−1)}.
Finally, WF(B) = WF(p).
Definition 2.25. For two sequences u,v ∈ Z n , u component-wise domi-
nates v iff:
u ≥
c v or u(i) ≥ v(i), ∀i,1 ≤i ≤ n.
Definition 2.26. X and Y are two N-Sequences with |X| = |Y|. X dom-
inates Y (written as X ≻ Y ) iff ∀u ∈ WF(Y ),∃v ∈ WF(X) such that
v ≥
c u.
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