point whose deletion is a topology-preserving reduc-
tion (Kong and Rosenfeld, 1989). Gau and Kong
gave the following characterization of simple points
in (18, 12) pictures:
Theorem 1. (Gau and Kong, 1999) A point p ∈ B is
simple for B if and only if the following conditions
hold:
1. N
18
(p) ∩B contains exactly one 18-component.
2. N
12
(p) \B contains exactly one 12-component.
Theorem 1 implies that only non-isolated border
points may be simple, and simple points can be lo-
cally characterized (i.e., the simpleness of a point p
can be decided by examining the points in N
18
(p)).
Figure 3 givesfour illustrative examples of simple and
non-simple points.
Figure 3: Examples of simple and non-simple points in
(18, 12) pictures. The positions denoted by ‘•’ and ‘◦’ refer
to black and white points, respectively. Black point p is sim-
ple only in the top left configuration. In the top right case, p
is an isolated black point, while in the bottom left example,
N
18
(p) ∩B contains two 18-components, hence both cases
violate condition 1 of Theorem 1. In the bottom right figure,
there are two 12-components in N
12
(p) \B, thus condition
2 of Theorem 1 does not hold.
It is obvious that a sequential reduction (or a thin-
ning algorithm composed of sequential reductions)
preserves the topology if and only if it deletes only
simple points. Unlike the sequential case, parallel
reductions can delete a set of points simultaneously.
Thus we need to consider what is meant by topology
preservation when more than one point is deleted at a
time.
We are to define the concepts of a simple set, a
simple sequence, and a minimal non-simple set.
Definition 1. (Kong, 1995) Let P be an arbitrary pic-
ture. A set of k black points Q is a simple set in P if it
is possible to arrange the elements of Q in a sequence
hq
1
, . . . , q
k
i such that q
1
is simple in P and each q
i
is
simple after the set of points {q
1
, . . . , q
i−1
} is deleted
(i = 2, . . . , k). Such a sequence is called a simple se-
quence. (And let the empty set be simple.)
Definition 2. (Ronse, 1988) A set of black points is
a minimal non-simple (MNS) set in an arbitrary pic-
ture if it is not simple, but any of its proper subsets is
simple.
Figure 4 presents examples of simple, non-simple,
and MNS sets in an (18, 12) picture.
Figure 4: Examples of simple and non-simple sets. The set
of black points {a,b, c, d} is simple since the 16 sequences
(of the possible 24 ones) ha, b, c,di, ha, b, d, ci, ha, c, b, di,
ha, c, d, bi, ha, d, b, ci, ha, d, c, bi, hb, a, c, di, hb, a, d, ci,
hb, d, a, ci, hc, a, b, di, hc, a, d, bi, hc, d, a, bi, hd, a, b, ci,
hd, a, c, bi, hd, b, a, ci, hd, c, a, bi are simple. The set {b, c}
is minimal non-simple, since both sequences hb, ci and
hc, bi are non-simple and both proper subsets {b} and {c}
of {b, c} are simple. The set {b, c, d} is non-simple but not
minimal non-simple, since {b,c} is its proper non-simple
subset. Note that points a, b, d, e, and f are all 18-adjacent
to point c.
We state the following proposition that is a
straightforward consequence of Definition 1:
Proposition 1. Let Q ⊂ B be a simple set for B. If
p ∈(B\Q) is a simple point for B\Q, Q∪{p}is also
a simple set for B.
Here, we recall a general lemma stated by Kardos
and Pal´agyi:
Lemma 1. (Kardos and Pal´agyi, 2015) Let p and q
be two black simple points in an arbitrary picture. If
p remains simple after the deletion of q, q remains
simple after the deletion of p.
In other words, the simpleness of a set of two sim-
ple points can be decided by examining just one se-
quence of its elements.
The following theorem gives a universal sufficient
condition for topology-preservingparallel reductions:
Theorem 2. (Ronse, 1988) A reduction preserves the
topology for an arbitrary picture if it does not delete
any MNS set.