Image Processing Reference
In-Depth Information
thickness. A closed digital curve (surface) S is irreducible (i.e., of unit thick-
ness) if and only if exclusion of any point p from S (and its inclusion in S)
gives rise to a k-connected path from each point of H to each point of S H.
Let L be a real line that partitions an irreducible and closed digital curve
S into two or more (k-connected) components such that all points in each
component are on the same side of L or lying on L, and no two components
share a common point. Then each such component of S is an open digital
curve (irreducible segment). Partitioning of an open digital curve segment, in
turn, gives rise to several open segments. Clearly, if the closed digital curve
S is such that any horizontal (vertical) line L
y
(L
x
) always partitions S into
at most two components, then S is a y-monotone (x-monotone) digital curve.
Similarly, partitioning an irreducible and closed digital surface S by a plane Π
produces two or more components such that all points in each component are
on the same side of Π or lying on Π , and no two components share a common
point. Each component obtained by partitioning S with Π is an open digital
surface (irreducible). If the closed digital surface S is such that any plane Π
y
(Π
x
,Π
z
) orthogonal to the y-axis (x-, z-axis) always partitions S into at most
two components, then S is a y-monotone (x-, z-monotone) digital surface. It
may be mentioned here that, if the generatrix is an open, irreducible, and
y-monotone digital curve segment, C, then the digital surface of revolution,
S, produced by the algorithm is also a y-monotone digital surface.
Figure 7.9 illustrates the notions of connectivity and irreducibility of a dig-
ital curve [115], which, in turn, produces an irreducible digital surface. Since
there exists no k(= 8)-connected path from the topmost point to the bottom-
most point of the digital curve in Fig. 7.9a, the concerned curve is disconnected.
Again, in Fig. 7.9b, there exist multiple paths from the topmost point to the
bottommost point, which implies that there are some redundant points in
the corresponding digital curve, wherefore it is reducible. The set of reducible
points, once eliminated, produces an irreducible digital curve (Fig. 7.9c). In
Fig. 7.9d, a closed and irreducible digital surface S is shown, which contains a
hole H inside it. The surface S is connected in the k(= 26)-neighborhood as
mentioned earlier. Irreducibility of S implies that no voxel p ∈ H is k(= 6)-
connected to any voxel p lying outside S (i.e., p ∈ Z
3
(S ∪H)); further, if
we remove any voxel q ∈ S from S, then each voxel p ∈ H has a k-connected
path to each voxel p ∈ Z
3
(S ∪ H). Figure 7.9e shows an open and irre-
ducible digital surface, namely S
′
, which is obtained by partitioning S with
a horizontal (integer) plane Π
′
: z = c
′
(∈ Z). Since S itself is irreducible,
S
′
is also irreducible, as explained earlier. If we again partition S by some
other orthogonal integer plane, then we again get a smaller irreducible digital
surface, S
′′
, as shown in Fig. 7.9f.
7.2.3 Algorithm to Wheel-Throw a Single Piece
We generate a wheel-thrown piece by revolving the generatrix G := {p
i
:
i = 1,2,...,n} about an axis of revolution given by α : z = −c,x = a, where
