Algorithms for labeling, counting, and computing connected objects in binary three dimensional arra

Park, C. M.

Rosenfeld, A.

English

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TJNIVERSI'TY OF MARYLAND
COMPUTER SCIENCE CENTER
COLLEGE PARK, MARYLAND
N	 ILT`	 _
°	 (ACCESSION	 Rai ) J	 H )
oe
LA. 	 jE^1 G E 5 	 ^./ (CODE
a (NASA CR OR TMX OR AD NUMBER)
	 (CATEGbRY)
rTechnical Report TR-156
	 May, 1971
AT-(40-1) -3662
CONNE(T IVITY AND GENUS
IN THREE DIMENSIONS
Chan Mo Park
and
Azriel Rosenfeld
/jle t a/ - v v ^ 1 -- & o^
Algorithms are presented for labeling connected ob-
jects in a binary three-dimensional array, for counting
such objects, and for computing the genus of the array.
The support of the U.S. Atomic Energy Commission, under
Contract AT-(40-1)-.3662, is gratefully acknowledged. The
help of Mrs. E eanor	 aters in preparing the manuscript
is also deeply appreciated.
s6
E
4.
1. Introduction
A three dimensional volume of "digital spejce " may be
represented by a stack of planes each of which contains a
cross-section of the volume. Objects in such a volume
can be specified by assigning the value I to their points,
while points in the space between or within the objects
have value 0.
This paper describes an algorithm for assign.Lng laiaels
to connected objects, so that each point of a given object
receives the same label, while points belonging to different
objects receive different labels. In particular, this al-
gorithm can be used to count the objects (number of objects =
number of distinct labels used) and to compute their volumes
(volume of an object = number of times its label occurs).
The principle employed in the construction of the al-
gorithm is to scan the volume element by element in a cer-
tain sequence and to examine each point found to have the
value 1. For each such point which is not connected to
any object found previously, a new label is assigned, the
number of objects is incremented by 1 and the volume for
the new object is initiated to be 1. If the point is con-
nected to just one object found earlier, the volume of the
object to which this element is connected is incremented
r
t
%I
ri
by 1. However, the point may be connected to several ob-
jects found earlier but not previously known to be con-
nected. In this case, the volumes of these objects must
be combined and the number of objects must be reduced.
Also included in this paper is an algorithm to compute
the three-dimensional genus.
R
	
	 A detailed discussion of the topological properties
of three-dimensional arrays may be found in S. B. Gray,
Local properties of binary images in two and three di-
mensions". Informa.ion International, Inc., January 21,
1970.
f
I.
k
N
^r
t'
6
2. Definitions and notation
An explanation of the definitions and notation used
in the following sections is given here.
a) The volume to be analyzed is an r by s by t hexahedron
(rectangular parallelepiped). An arbitrary point (or
element) in the volume will be denoted by P.
	
, where
^, 7 .k
l s i s r, 1 s j s s, l s k s t.
i
(1,1 1) —^ r,l,l)
7
lk(r,
s,l)	 s
(l,s,t	 (r,s,t)
The volume can be regarded as a stack of t planes each
of which consists of r by s points.
b) In defining connectivity, two elements are regarded as
adjacent if the distance between them is one unit.
Thus, any element is adjacent to six other elements:
one element on the plane above, four elements on the
same plane, and one element on the plane below.
c) The volume is processed in the following sequential
order:
i increases most rapidly
increases next most rapidly7	 P Y
k increases least rapidly
iaa
Thus the order of scan is:
p 1 1 1,11 p 2, 1,1 . . . . pr, 1, 1
p l, 2, 1 p 2, 2, 1 .... pr, 2, 1
p l, s, 1 p 2, s, 1 pr, s, 1
p l, 1, 2 p2, 1,2 .... pr, 1, 2
pl, 2, 2 p 2, 2, 2 .... pr, 2, 2
d) The following are he major variables in the algorithm:
count = the number of objects in the volume
label, = the label of the Xth object found
volume, = the volume of the Ath object
P..= an arbitrary element
Li jk	 = the 'label of the element Pi,j,k
neighborh = the index of the label of an element which
is connected to the element being examined.
oldlabel = the original label which was assigned when
the object was found
newlabel = the new label assigned to the object be-
01
cause it is found to be connected to an
object labeled earlier.
X = index for the arrays of labels and volumnes
a = index for the arrays of oldlabels and newlabels
I
r3. The labeling algorithm
a) Initialization
The number of objects, various arrays and the
indexes are cleared:
count «- 0
label.	 0i
volume.	 0
1
oldlabel. - 0
1
newl abe 1. «- 01
«- 0
•- 0
b) Pick the next element, Pi,j,k'
If it is 0 go to (g).
If it is 1 go to (c).
c) If k > 1, examine the label L.i,j,k-1 of the neighbor
of Pi,j,k on the plane above.
d) Examine the labels of the elements in the same plane
which could be connected to Pig k and which havej•
already been processed. In general these are
Li,j-1,k and L+-1,j ^ k (unless Pi,j,k is on a face
of the volume) .
) If at least one of these labels is not 0 go to (f).
Otherwise, a new object has been found; set
count «- count + 1
%	 % + 1
%I
1ti
4
r
i
label «- f (k) where f is a label generation
function*
volume, 1
Li " j o k «-
 label  X
and go to (g) .
f) If Pifj,k has neighbors that have already been
labeled,
i)	 Form an array "neighbor" of the indices (%'s)
of the labels of these neighbors. Assume
that there are v such indices. Find the mini-
mum element in the array, i.e.. Xmin - min
(neighbor h)
	
lshsv
Adjust "count"
count •- count - G-1)
iii) Determine the label of the current point, and
save it
Li 	«- label ( X	 )j• k
iv) Update the volumes
Volume	 = ^ volume neighbor + 1
min h=1	 h
Volume neighbor  = 0 except for neighbor  = Xmin
v) Change the L's which are connected to Pi, kj
into Li ^ j,k (= label (% in )), because they
may be connected to the next element to be
scanned.
*E.g., for a UNIVAC 1108
f M = "blank" + X/26 * 64 + mod (X, 26)
assuming that labels are single letters or pairs of letters,t	 i.e., A,B, • • • Z, AA,AB, ...ZZ,
^t
1I
i
vi)	 Enter the old and new labels into the arrays
oldlabel and newlabel, respectively. This
information will be used later to assign final
correct labels to the points of objects.
a «- a+1
oldlabel « = label (neighbor h)
newlabel
a
= label (A min)
Repeat (vi) for h = 1, 2, ... , v except for
neighbor h
=X min '
g) If all the elements have been considered, go to
(h); otherwise go to (b).
h) Update the array newlabel:
If
newlabeli = oldlabelj,
then
newlabel. «- newlabel,1
Repeat this step until there is no further change.
i) Update the labels of objects
^f1
Li^j^k = oldlabelm0
then
Li^j.k - newlabelm.
j) Stop
An example is given in Section 4 which shows successive
stages in the application of the algorithm.
If
If the original volume is of large size, it may be
necessary to use tapes to supply the input data and to
record the labels assigned to the objects. If the size
of the volume is such that the labels of all the points
in one plane as well as those of the points in two rows
in the next plane (i.e., rs + 2r points) can be stored in
the core memory at the same time, only one tape (besides
the input tape) is needed to record the labels of the c".1
jects. Otherwise, it is necessary to use two tapes. How-
ever, if the original data on the input tape can be des-
troyed, the input tape can be used to record the labels.
In such cases no additional tape is necessary if the core
memory can hold the labels of the points in two planes (i.e.,
2rs points) at the same time.
4. An example of the labeling al
The figure at the end of thi
ume and the final labels assigned
a point changes from one label to
the old label is indicated inside
aorithm
s section shows an input vol-
to the objects. 4.nen
another during the process
parentheses beside the
new label assigned to that point. The table below indi-
cates the status after each plane is processed.
Plane Count X Label Volume Oldlabel Newlabel Remarks
k=1 3 1 A 20 0 0 3 different ob-
(top 2 B 2 jects, A,	 B and
plane) 3 C 2 C are fouad.
k=2 5 1 A 40 0 0 2 more objects,
2 B 5 D and E are
3 C 2 found.
4 D 4
IC
5 E 2
k=3 6 1 A 60 0 0 A new object,
2 B 10 and
-' 3 C 2 F	 (P10, 3, 3
4 D 8 P10, 4, 3 )	 is
5 E 4 found.
e 6 F 2
k=4 6 1 A 80 F E It is found that(6-► 5-•6) 2 B 10 PiO, 4, 3 (F)	 is con-
^^
4 D 8 nected to F O1	 '4'4L} 5 E 9 which is connected
6 F 0 to P	 (E) .7 G 3 10, 6, 3
Thus L
	
is10,4, 3
changed to E and
V6
 is added to V5
before it is cleared.
The count is reduced
by 1.
	 A new object
(G) is found, which
'
increases the count
back to 6.
Plane Count X Label Volume Oldlabel Newlabel Remarks
k=5	 6	 1 A 120	 F E	 When P	 is
(6-•7-•6) 2 B 10	 H A	 1,.4, 5
3 C
C
2 processed, it is
4 8 not connected to any
points previously
5
5 E 9
F
F 0 labeled.	 So H is
7
6
6 assigned and the
8 H count is increased
to 7.
	 But H is
found later to be
connected to the
object A via P2,4,5'
which reduces the
count back to 6
and changes the
label to A.
I
i
E
E
E (F)
E (F)
E
E
E
E
E
t	 I^.
t
A A A A A A
A A
B	 A A
B	 A A
A A
A A A A A A
c
c
A A A A A A
B	 A A
B	 A D D A
B	 A D D A
B	 A A
B	 A A A A A A
A A A A A A
A A
A A
A A
A A
A A A A A A
G G G
A A A A A A
A A A A A A
A A A A A A
(H)A A A A A A A A
A A A A A A A A
A A A A A A
G G G
t
i
Final LabelInput Picture
1 2 3 4 5 6 7 8 9 10
Top plane 1 0 0 1 1 1 1 1 1 0 0k=1 2 0 0 1 0 0 0 0 1 0 0
3 1 0 1 0 0 0 0 1 0 1
4 1 0 1 0 0 0 0 1 0 1
5 0 0 1 0 0 0 0 1 0 0
6 0 0 1 1 1 1 1 1 0 0
7 0 0 0 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0 0 0 .0
k=2 0 0 1 1 1 1 1 1 0 0
0 0 1 0 0 0 0 1 0 0
1 0 1 0 1 1 0 1 0 0
1 0 1 0 1 1 0 1 0 0
1 0 1 0 0 0 0 1 0 0
0011111101
0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0
k=3 0 0 1 1 1 1 1 1 0 0
1 0 1 0 0 0 0 1 0 0
1 0 1 0 1 1 0 1 0 1
1 0 1 0 1 1 0 1 0 1
1 0 1 0 0 0 0 1 0 0
1 0 1 1 1 1 1 1 0 1
0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0
k=4 0 0 1 1 1 1 1 1 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0 1
0 0 1 0 0 0 0 1 0 1
0 0 1 1 1 1 1 1 0 1
0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 0 0 0
k=5 0 0 1 1 1 1 1 1 0 0
0 0 1 1 1 1 1 1 0 0
0 0 1 1 1 1 1 1 0 0
1 1 1 1 1 1 1 1 0 0
1 1 1 1 1 1 1 1 0 0
0 0 1 1 1 1 1 1 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 0 0 0
A A A A A A
A A
B	 A D D A
B	 A D D A
B	 A 2%
A A A A A A
1
,•
S. Connectivity, volumes and genus
When the labels are not dasired but only the number
of objects (i.e., the connectivity) as well as their vol-
umes, the labeling algorithm described above can be used
with some modification. Although the labels are not re-
quired, it is necessary to assign temporary identifiers to
the points in order to distinguish connected objects from
one another. However, it is no longer necessary to keep
the identifiers of all the points which have already been
processed, but only those of the points on the plane above
and the current plane. The other stages which are omitted
in this case are those steps concerned with the arrays old-
label and newlabel and with updating the labels of objects,
i.e., steps (f vi), (h) and (i).
An even simpler algorithm can be used if one wants to
compute the three-dimensional genus or Euler number. This
is defined as 0-H+C, where 0 is the number of connected
objects, H is the number of holes in objects (topologically
'	 similar to the hole in a doughnut), and C is the number of
a
cavities in
	 components of 0'sobjectsconnected com^	 (i.e.,(	
completely surrounded by 1's). It should be pointed out
that in one and-two dimensions the genus is defined as
(2)E (1) = o and E = 0-H, respectively.
rs
I
The genus can be computed by performing a local count-
ing operation over the given volume. Let
Cl be the number of 19s
Y 
C2 be the number of 1 by 1 by 2 patterns of 1's (in
all orientations)
C4 be the number of 1 by 2 by 2 patterns of 1's (in
all orientations)
C8 be the number of 2 by 2 by 2 patterns of l's
Then E (1) = C1-C2
E(2) = C1-C2+C4
E(3) = C1-C2+C4-C8
A proof of this assertion, for E (3) , is given in Section 6.
In Section 7 an algorithm for computing E 4
	is given.
r
7
r
6	 Proof that E (3) = C1-C2+C4-C8
The proof is by induction on the number of 1's in
the given volume. Specifically, it is shown that if the
rightmost, uppermost 1 in the topmost plane is deleted,
both E (3) = 0-H+C and C1-C2+C4-C8 change by the same
amount. (For the case where there are no 1's it is
certainly true that E (3) = O-H+C = 0 and C1-C2+C4-C8 = 0.)
Let P be this 1; then the neighbors of P whose values af-
fect the Euler number are
a P on the top plane
c d
e f on the next plane below
g h
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7. Algorithm for computing the genus
a) Initialization
Cl 0
C2 «- 0
C4 «- 0
C8 •- 0
b) Pick the next element, P. If it is 0, go
"o (d) . Otherwise go to (c) .
C) i)	 C1	 C1+1
ii) Examine the elements, in the (k-1)st and kth
planes which are neighbors of P ig .	 or
neighbors of those of its neighbors that have
already been considered. In general, there
are four such elements in the (k-1)st plane
and three in the k-th plane (provided P,
is not on a face):
(k-1)st plane: P,i-1, j-l,k-l' Pi, j -1,k-1
	
i-1,j,k-1'	 Pi,j,k-1
kth plane:
	
Pi-l,j-1,k'	 Pi,j-1,k
Pi-1,j,k
^I
Increment C2, C4 and C8 as follows:
C2	 C2 + [Pi-1,j,k + Pi,j -1,k + Pi,j,k-1^
C4	 C4 + [P	 i.
	
A P,	 iA Pi	 '-1,j,k	 -l,j-1,k	 ,j-1,k-'
+ [Pi-1,j,k n Pi-l,j,k-1 A Pi,j,k-11
+ [P i, j -1,k A P i, j,k--1 ^ P i, j-1,k-1 1
C8 «- C8+1 provided ( new C4)-(old C4) = 3
and Pi-1 -1,k-1	 1d) If all the elements have'^een considered go to (e).
Otherwise go to (b).
e) E (3) = C1-C2+C4-C8; stop.
M


Connectivity and genus in three dimensions

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