Image Processing Reference
In-Depth Information
corresponding to
C
. For example, the number of crossing points of
C
C
is
regarded as that of a digital knot
C
.
Definition 5.14.
A digital knot of which the number of crossing points is
0
is called a
trivial knot
or
unknot
.
Intuitively speaking, the
knot
is a property of a single 3D simple curve or
a
string
. If it is knotted when we pull both ends of it, then a string is said to
be
knotted
or called a
knot
in the mathematical sense. Otherwise such string
is called a
trivial knot
or said to be
unknotted
. Here (in this section) a knot
means a
knotted
knot (Fig. 5.19).
Remark 5.21.
A set of more than one closed curve is called a
link
.Thereare
two types of links, too. One is the case that two closed curves are linked each
other, and the other is a pair of two curves that are separate from each other.
5.7.2 Reduction of a digital knot
In order to study properties of a digital knot by the same method as in math-
ematics, we need to generate a large number of continuous polygonal knots
corresponding to a given digital knot and derive their regular projections. It
is not always easy. Large parts of drawing regular projections closely relate
to our intuition or heuristic decisions as a human. Algorithmic generation of
regular projections is never a trivial procedure. Thus it becomes significant to
transform directly a digital knot contained in a digitized line figure and study
its features.
Let us present here an algorithm to shrink a digital knot.
Algorithm 5.19 (Shrinking of a knot - arc elimination).
Assume a dig-
ital knot
C
is given. At an arbitrary pair of voxels P and Q on
C
, if the length
(= the number of voxels) from P to Q along the curve
C
is longer than the
26-neighbor distance between P and Q, then the part of the knot
C
between
P and Q is translated so that the length of
C
may become shorter. Unless
the topology of the curve
C
is preserved, the translation is not performed. If
a voxel becomes redundant in a result of translation, that voxel is eliminated
at that moment (Figs. 5.20, 5.21).
Algorithm 5.20 (Shrinking of a knot - corner elimination).
A bulge of
a curve is deformed to be straight as shown in Fig. 5.22. The deformation is
not applied if topological properties of a curve
C
change by the deformation.
The following property holds true:
Property 5.10.
Any trivial knot (unknot) is reduced to any one of unknots
of the length
4
shown in Fig. 5.23 by repetitive application of Algorithms 5.19
and 5.20. There does not exist a knotted knot of the length
4
.
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