Digital Signal Processing Reference
In-Depth Information
T
HEOREM
C
,
the VOS values on the Voronoi Pivots are mapped onto a tree where the
ancestor-descendant relationship is monotonically decreasing w.r. t.
3.9
(M
ULTI-RESOLUTION
OF
χ
V
)
Given a contour C
χ
V
.
Proof.
Consider the Voronoi Ordered Skeleton value of the two pivots
a
and
b
in child-parent relationship where a is the parent and b is the
child. Since a is a child, all members of Φ(
b
), must be contained by a
member of Φ(
a
),
→
→
→
→
Â.
.
Using theorem 3.8,
→
a
→
Voronoi - Contains
b
→
max
(α(
C, B
)) ≤ α (
C, Â
)
(3.8)
→
B
Φ(
C,b
)
Loosening the upper bound,
max
(α(
C,B
)) ≤ α (Â)
≤
max
(α(
C,A
))
(3.9)
→
Φ(
C,b
→
This maximum operation is equivalent to our definition of χ
V
.
A
€Φ (
C,a
)
B
→
(3.10)
χ
V
(
C, a
) ≤ χ
V
→
(
C, b
)
By taking the contrapositive of Eq. 3.8, we can relate χ
V
the Voronoi-
containment between pivots.
→
→
⇒
(χ
V
(
C, a
)>χ
V
→
(
C, b
))
b
does not Voronoi-contain
a
→
(3.11)
→
i.e., if the VOS value of
a
→
is greater than the VOS value of
b,
then the
→
pivot
a
cannot be a parent of
b
pivot in the tree induced by the VOS.
Coupled with the fact that the non-zero points of the VOS (the MAT),
form a tree
→
structure, χ
V
can be used to derive a tree
structure that
reflects the Voronoi-containment relation of a given shape.
In summary, these theorems show that the Voronoi Order Space has
a deep connection to shape structure. For all practical purposes, we can
derive a singly rooted tree structure from the non-zero χ
V
values of a
given shape contour. Furthermore, the depth and branching of this tree
structure are directly related to an implicit containment relation that can
be directly derived from the shape of contour. Much like each parent in
the tree that can represent all its descendants, the Voronoi areas from
Voronoi Pivots contain all the areas derived from pivots within those
areas. Each point in the tree structure can become a representation of all
its children; likewise, the Voronoi Pivots related by Voronoi containment
can become a multiresolution representation of shape. Using the VOS
values, χ
V
,
we can derive this multiresolution representation and use it
as a robust shape representation.
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