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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