Image Processing Reference
In-Depth Information
treated an ideal form of a line figure in 3D continuous space [Kauffman87].
Reports on the processing of the knot and the link by computer were very
few.
Past research of the knot can be roughly classified into two kinds. One
is to transform a knotted line figure into a kind of symbolic representation,
and replace the shape transformation of a continuous line figure by the symbol
manipulation. This symbol manipulation is performed by computer. Examples
are determination of the kind of a knot and decision on whether a line figure
is knotted or not. This is performed by utilizing the transformation called
Reidemeister transformation
which is well known in mathematics.
The other is to study properties of a digitized knotted figure that corre-
spond to those features that have been examined for a continuous knotted
figure in mathematics. This type of study has been performed very little ex-
cept for [Nakamura00, Saito90]. Let us introduce here basic properties of a
digitized knot according to [Saito90].
Definition 5.13 (Digital knot).
If a connected component
C
consists of
four or more 1-voxels, and any of those 1-voxels has just two 1-voxels in its
k
-
neighborhood (
k
=6
,
18
,
26), that is, if a figure consists of connecting voxels
only, a connected component (
k
-connected) is called a
digital simple closed
curve
or (
k
-connected
)
digital knot
. The number of 1-voxels contained in
C
is
called
length
of this digital knot.
Consider then a line figure obtained by connecting center points of 1-voxels
mutually adjacent on a digital knot with line segments. By doing this a polyg-
onal knot in 3D continuous space is obtained. We call this a
continuous knot
C
C
corresponding to a digital knot
C
. A continuous knot
C
C
corresponding to
a digital knot
C
becomes a continuous knot in the continuous space. Results
in the knot theory of mathematics can be applied to the above
C
C
.
First a knot of a continuous figure is defined as follows in mathematics.
Let us denote by
C
p
a figure derived by projecting the above continuous
knot (a polygonal knot) onto a suitable plane
P
(more strictly an orthogonal
projection of
C
C
presented in Chapter 7). Then, if
n
points of
C
C
are projected
to the same point of
C
p
, this point is called
n
-fold point
(
multiple point
)(
n
≥
2
). If the number of
n
-fold points is definite and all of them are generated from
only projections of points of
C
C
, the projection is called a
regular projection
.
In particular, a double-fold point is called a
crossing point
.Thenumberof
a crossing point varies according to a projected plane. It may change by the
transformation of a figure, even if the topological properties of a figure do not
change. The minimum number of crossing points in all regular projections of
all knots topologically equivalent to a knot
C
C
is called a
minimum crossing
point number
of a knot
C
C
. For a stricter description, see the monograph and
text of mathematics [Kauffman87].
Thus, properties of a continuous figure
C
C
as a knot are frequently dis-
cussed by using its regular projection on to a 2D plane. We also discuss char-
acteristics of a digital knot
C
using the regular projection of a polygon
C
C
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