COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 32, 221-243 (1985)
A Theory of Binary Digital Pictures
T. Y. KONG AND A. W. ROSCOE
Oxford University Computing Laboratory, Programming Research Group,
11 Keble Roa~ Oxford, United Kingdom, OXI 3QD
Received December 4, 1984; revised February 21, 1985
We study 2- and 3-dimensional digital geometry in the context of almost arbitrary adjacency
relations. (Previous authors have based their work on particular adjacency relations.) We define
a binary digital picture to be a pair whose components are a set of lattice-points and an
adjacency relation on the whole lattice. We show how a wide class of digital pictures have
natural "continuous analogs." This enables us to use methods of continuous topology in
studying digital pictures. We are able to prove general results on the connectivity of digital
borders, which generalize results that have appeared in the literature. In the 3-dimensional case
we consider the possibility of using a uniform relation on the whole lattice. (In the past authors
have used different types of adjacency for "object" and "background.") 9 1985 Academic Press,
Inc.
PREREQUISITES
Familiarity with the content of [15] is probably essential for understanding some
of the remarks in the introduction; the rest of the paper is more or less self-con-
tained, but familiarity with [15] might still be helpful. The graph-theoretic terminol-
ogy we use is defined in the first chapter of [3]. A httle elementary topology is
assumed in our discussion of continuous analogs--the relevant concepts are covered
in the third and fourth chapters of [1].
INTRODUCTION
Digital images are, of course, arrays of non-negative numbers (gray values);
binary images are obtained when the array elements are partitioned ("segmented")
into two subsets by thresholding. The array elements are called pixels in 2D and
voxels in 3D. Pixels are sometimes thought of as little squares and voxels as small
cubes (the
cuberille
approach--see [5]), but we shall not think of pixels and voxels in
this way; instead we shall identify each pixel or voxel with a lattice-point in the
plane or in 3-space.
Rosenfeld's 1981 paper [15] provides a very clear exposition of the fundamental
concepts of 3-dimensional digital topology. However, only two kinds of adjacency
relation are considered in [15]: either 6-adjacency is used for the "objects" and
26-adjacency for the "background," or 26-adjacency is used for the "objects" and
6-adjacency for the "background." Yet it is clear that the concepts like digital paths
and digital components make sense for a wide variety of adjacency relations. The
first goal of this paper is to present a simple unified approach to digital pictures
which places no artificial restrictions on the adjacency relations used.
Morgenthaler and Rosenfeld suggest in [10] that we might sometimes wish to
define adjacency between points "which are not even 'near' each other." It might be
possible to extend our new theory so as to allow this. In another direction
Mylopoulos and Pavlidis showed in [11] that many of the basic concepts of digital
geometry remain valid when the conventional rectangular grid is replaced by the
221
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Copyright 9 1985 by Academic Press, Inc.
All rights of reproduction in any form reserved.
222 KONG AND ROSCOE
Cayley diagram (of. [3, Chap. 8]) of any finite presentation of an abelian group--they
called such group presentations discrete spaces. It is likely that the ideas introduced
in our present article can be applied to many of these discrete spaces.
Further motivation for the present paper comes from our investigation of the
"surface-points" introduced by Morgenthaler, Reed, and Rosenfeld [10, 13, 12]. In
our paper [7] we prove non-trivial results by transforming problems of digital
topology into problems of polyhedral topology. The transformation is done by
constructing "continuous analogs" of digital pictures. However, the full potential of
this approach is not realized in [7]: although continuous analogs exist for "most"
digital pictures, their existence is proved only in very special cases. The second goal
of this paper is to develop a general theory of continuous analogs and to give good
necessary and sufficient conditions for their existence.
Our work on general binary digital pictures has interesting corollaries for particu-
lar adjacency relations. Not only can we obtain Propositions 5 and 6 of [15] but we
also find that similar results hold for every pair of "pure" adjacency relations (6-,
18-, or 26-adjacency) other than (6, 6). We regard a proof of these results as our third
goal. That these results are fairly deep becomes apparent when we note that the
2-dimensional analogs of both results fail on the surface of a cylinder (when Z 2 is
replaced by Z, • Z, where Z n denotes the integers modulo n). The point is that if n
is large then Z n • Z is locally indistinguishable from Z 2, which shows that the
two propositions express
global
properties of Euclidean space. (Geometric topol-
ogists will recognize that the propositions express discrete versions of two
"Phragmen-Brouwer properties" [18]). Many of the propositions proved in our
earlier paper [7] are also global results in this sense.
To establish the validity of such results we must use a "global proof method":
purely local methods (such as straightforward induction on the number of points in
S, or simple graph-theoretic arguments) are unlikely to suffice. Our method of attack
involves applying techniques from continuous topology to the continuous analogs of
binary digital pictures. We do this in the "IV implies I" part of the proof of
Theorem 2.
Finally we point out some of the drawbacks of using any single elementary
adjacency relation (4-, 6-, 26-, etc.) on the conventional square and cubic grids. In
order to overcome these difficulties many other workers have resorted to the use of
different adjacency relations for object and background points. Our investigations
suggest that for some purposes it may be a better choice to use a 2-dimensional
hexagonal lattice or a 3-dimensional face-centered cubic lattice equipped with the
corresponding "nearest neighbor" adjacency relations.
This paper is structured in such a way that it is possible to omit the sections
relating to continuous analogs, and the proofs of Theorems 2 and 2', provided that
the "I is equivalent to II" part of these theorems is assumed without proof. The
statement and proof of Proposition 3 may also be omitted on a first reading;
however, the corollary to Proposition 3 is used in the proof of Proposition 4.
SIMPLY-CONNECTED SETS
It will emerge from our paper that Propositions 5 and 6 in [15] are valid because
R 2 and R 3 are both simply-connected.
A connected subset Y of R 2 or R 3 is said to be simply-connected if it has no
"holes." (A solid cube is simply-connected but a solid torus is not.) Equivalently, Y
A THEORY OF BINARY DIGITAL PICTURES
223
is simply-connected if given any two curves in Y with the same endpoints we can
transform one curve into the other by means of a "continuous deformation" during
which both endpoints remain fixed and the rest of the curve remains in Y. The
precise definition is as follows:
A curve in a subset Yof R 1 is a continuous map `/: [0,1] --, Y. The curve `/is said
to be a curve in Y from the point `/(0) to the point ,/(1). The
trace
of a curve `/ is
another name for the image of `/. A connected subset Y of R" is said to be
simply-connected
if given any two points p and q in Y and any two curves `/0 and `/1
each of which is a curve in Y from p to q, we can find a continuous map h:
[0,1] • [0,1] --, Y such that for all s and t in [0,1] we have
(i)
h(s,O)= Vo(S)
(ii) h(s,
1) = `/l(s)
(iii) h (0, t) = p
(iv) h (1, t) = q.
The "continuous deformation" h is called a fixed endpoint homotopy of `/0 onto `/1.
It is customary to think of t as "' time."
The definition we have just given is the one we shall use; but it is not the standard
definition. We show in the Appendix that the two definitions are equivalent. We
asserted above that a solid torus is not simply-connected. This is intuitively clear but
not so easy to prove. However, it is a corollary of Theorem 2 below.
ELEMENTARY TERMINOLOGY
In this paper Z denotes the set of integers and R denotes the set of real numbers;
Z is regarded as a subset of R. Thus R" denotes Euclidean n-space and Z" is the set
of all lattice-points in Euclidean n-space.
Recall that two points in 13 are said to be 26-adjacent if they are distinct and
each coordinate of one differs from the corresponding coordinate of the other by at
most 1; two points are 18-adjacent if they are 26-adjacent and differ in at most two
of their coordinates; two points are 6-adjacent if they are 26-adjacent and differ in at
most one coordinate. Two points (x, y) and (x', y') in Z 2 are said to be 8-adjacent
or 4-adjacent according as
(x, y,O)
and (x',
y',O)
are 26-adjacent or 6-adjacent.
Unless otherwise stated the greek letters a, p, `/, and 8 will denote integers from the
set (4, 8, 6, 18, 26}. We use the term "lattice-point" to denote a point in Z 2 or Z 3. In
this paper we identify Z n with the set of points in R n that have integer coordinates
(in the obvious way). If W _c Z" then W c denotes the complementary set Z'\ W
(here n = 3 in the sections relating to 3-dimensional digital pictures and n = 2 in
the sections on 2-dimensional pictures). A unit cell is a closed unit cube (in 3D) or a
closed unit square (2D) whose corners are all lattice-points. (Note that a unit cell is a
connected subset of R 3 or R 2 and not a set of lattice-points.) A window is any union
of unit cells.
If K is a unit cell and S_Z 3 then KNS can be mapped by rotation or
reflection onto one of the 22 sets shown in Fig. 1. (A proof is given in the Appendix,
but in any event it is readily confirmed that Fig. 1 does exhaust all possibilities.) We
shall say that the pair (K, S) is of type n if K (3 S can be mapped by rotation or
reflection on the nth set in Fig. 1.
224
KONG AND ROSCOE
v v -- v
FIG. 1.
The twenty-two types of unit cell.
BINARY DIGITAL PICTURES
The following definition is the basis of our new approach to digital topology:
A 3- (2-)dimensional binary digital picture is a pair (A, S), where S is any subset
of Z 3 (Z 2) and A is any symmetric binary relation on Z 3 (Z 2) that satisfies the
axioms (i) and (ii) below. We shall say that x is A-adjacent to y if (and only if)
(x, y) ~ A. The axioms A must satisfy are:
(i) If x and y are 6- (4-)adjacent then x and y are A-adjacent.
(ii) If x and y are A-adjacent then x and y ate 26- (8-)adjacent.
If (A, S) is a binary digital picture then we refer to S as the set of object points of
the picture, we refer to S c as the set of background points of the picture, and we refer
to A as the
adjacency relation
of the picture.
The set S of object points is usually derived from pictorial data, but the adjacency
relation A is chosen by the user. In the above definition the adjacency relation is
explicitly included as part of a binary digital picture; we have found it helpful to
think in this way. However, we are not the first to incorporate adjacency relations
A THEORY OF BINARY DIGITAL PICTURES 225
into the mathematical structure of a digital picture. Previous authors did so
implicitly when using terms like "connectedness in the sense of the background."
Nevertheless, the definition just given represents a slight but significant departure
from the usual conceptual framework of digital geometry, because the adjacency
relation has been freed of all dependence on the set of object points: there is no
longer any notion of "adjacency in the S (or S c) sense." In the rest of this paper
(A, S) will be a 2- or 3-dimensional binary digital picture.
MORE ELEMENTARY TERMINOLOGY
In this section and the next we adapt the terminology of [15] to our new definition
of digital pictures.
If two points are A-adjacent then each is called an A-neighbor of the other. A
point is A-adjacent to a set if it is A-adjacent to some member of that set. Two
disjoint sets of lattice-points will be said to be A-adjacent if there is a point in one
subset that is A-adjacent to some point in the other. An A-path is a sequence of
distinct lattice-points such that any two consecutive points in the sequence are
A-adjacent. If each point in an A-path belongs to some set T then we shall call the
path an A-path in T. An A-path whose first point is x and whose last point is y will
be called an A-path
from x to y,
or alternatively an A-path that
links x
to y. We
shall say S is A-connected if every pair of points in S is linked by an A-path. An
A-connected subset of S that is not A-adjacent to any other point in S will be called
an A-component of S. Thus S is A-connected iff S contains just one A-component.
If S and T are disjoint sets of lattice-points then the (A, S)-border of T (or,
alternatively, the A-border of T with respect to S) is defined to be the set of all
points in T that are A-adjacent to a point in S.
If A is an adjacency relation on 12 or Z 3 and X is a window then
A(X)
denotes
the adjacency relation on the set of lattice-points in X such that x is A(X)-adjacent
to y ill' there is a unit cell in X which contains both x and y, and x is A-adjacent
to y.
We shall frequently want to use these definitions in the special cases where A is
the a-adjacency relation for some a; it will then be very convenient to use the prefix
"a" in place of "A." Thus we might refer to an 18-component (= an A-component
where A is the 18-adjacency relation), or "the (6, S)-border of T" (= the (A, S)-
border of T where A is the 6-adjacency relation. This use of the numbers 6, 18, 26, 4,
and 8 as prefixes is fully consistent with the usage established by previous authors
[15,10,13,12], etc.).
If K is any unit cell in R 2 then we define K* to be the union of K with the four
other unit cells that have an edge in common with K. Similarly, if K is any unit cell
in R 3 then we define K* to be the union of K with the six other unit cells that have
a face in common with K.
Finally, we define
A(a, ~, "/, S)
to be the adjacency relation on Z 2 or Z 3 in which
two points x, y are adjacent iff either x and y are a-neighbors in S or x and y are
~8-neighbors in S c or x and y are ,/-neighbors, and exactly one of x and y belongs
to S.
ADJACENCY GRAPHS
Let X be any window. The X-adjacency graph of (A, S), denoted by adj(A, X, S),
is a (possibly infinite) bipartite graph each of whose vertices is a whole A(X)-com-