<s>
that	O
correspond	O
to	O
topological	B-Architecture
properties	O
(	O
e.g.	O
,	O
connectedness	O
)	O
or	O
topological	B-Architecture
features	O
(	O
e.g.	O
,	O
boundaries	O
)	O
of	O
objects	O
.	O
</s>
<s>
Concepts	O
and	O
results	O
of	O
digital	B-Algorithm
topology	I-Algorithm
are	O
used	O
to	O
specify	O
and	O
justify	O
important	O
(	O
low-level	O
)	O
image	B-General_Concept
analysis	I-General_Concept
algorithms	O
,	O
</s>
<s>
including	O
algorithms	O
for	O
thinning	B-Algorithm
,	O
border	O
or	O
surface	O
tracing	O
,	O
counting	O
of	O
components	O
or	O
tunnels	O
,	O
or	O
region-filling	O
.	O
</s>
<s>
Digital	B-Algorithm
topology	I-Algorithm
was	O
first	O
studied	O
in	O
the	O
late	O
1960s	O
by	O
the	O
computer	B-General_Concept
image	I-General_Concept
analysis	I-General_Concept
researcher	O
Azriel	O
Rosenfeld	O
(	O
19312004	O
)	O
,	O
whose	O
publications	O
on	O
the	O
subject	O
played	O
a	O
major	O
role	O
in	O
establishing	O
and	O
developing	O
the	O
field	O
.	O
</s>
<s>
The	O
term	O
"	O
digital	B-Algorithm
topology	I-Algorithm
"	O
was	O
itself	O
invented	O
by	O
Rosenfeld	O
,	O
who	O
used	O
it	O
in	O
a	O
1973	O
publication	O
for	O
the	O
first	O
time	O
.	O
</s>
<s>
A	O
related	O
work	O
called	O
the	O
grid	B-Algorithm
cell	I-Algorithm
topology	I-Algorithm
,	O
which	O
could	O
be	O
considered	O
as	O
a	O
link	O
to	O
classic	O
combinatorial	O
topology	B-Architecture
,	O
appeared	O
in	O
the	O
book	O
of	O
Pavel	O
Alexandrov	O
and	O
Heinz	O
Hopf	O
,	O
Topologie	O
I	O
(	O
1935	O
)	O
.	O
</s>
<s>
proposed	O
digital	O
connectivity	O
such	O
as	O
4-connectivity	O
and	O
8-connectivity	B-Algorithm
in	O
two	O
dimensions	O
as	O
well	O
as	O
6-connectivity	O
and	O
26-connectivity	O
in	O
three	O
dimensions	O
.	O
</s>
<s>
Theodosios	O
Pavlidis	O
(	O
1982	O
)	O
suggested	O
the	O
use	O
of	O
graph-theoretic	O
algorithms	O
such	O
as	O
the	O
depth-first	B-Algorithm
search	I-Algorithm
method	O
for	O
finding	O
connected	O
components	O
.	O
</s>
<s>
Vladimir	O
A	O
.	O
Kovalevsky	O
(	O
1989	O
)	O
extended	O
the	O
Alexandrov	O
–	O
Hopf	O
2D	O
grid	B-Algorithm
cell	I-Algorithm
topology	I-Algorithm
to	O
three	O
and	O
higher	O
dimensions	O
.	O
</s>
<s>
He	O
also	O
proposed	O
(	O
2008	O
)	O
a	O
more	O
general	O
axiomatic	O
theory	O
of	O
locally	O
finite	O
topological	B-Architecture
spaces	O
and	O
abstract	O
cell	O
complexes	O
formerly	O
suggested	O
by	O
Ernst	O
Steinitz	O
(	O
1908	O
)	O
.	O
</s>
<s>
It	O
is	O
the	O
Alexandrov	O
topology	B-Architecture
.	O
</s>
<s>
The	O
book	O
from	O
2008	O
contains	O
new	O
definitions	O
of	O
topological	B-Architecture
balls	O
and	O
spheres	O
independent	O
of	O
a	O
metric	O
and	O
numerous	O
applications	O
to	O
digital	B-General_Concept
image	I-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
In	O
the	O
early	O
1980s	O
,	O
digital	B-Algorithm
surfaces	I-Algorithm
were	O
studied	O
.	O
</s>
<s>
This	O
definition	O
contains	O
a	O
total	O
of	O
nine	O
types	O
of	O
digital	B-Algorithm
surfaces	I-Algorithm
.	O
</s>
<s>
The	O
digital	B-Algorithm
manifold	I-Algorithm
was	O
studied	O
in	O
the	O
1990s	O
.	O
</s>
<s>
A	O
recursive	O
definition	O
of	O
the	O
digital	O
k-manifold	O
was	O
proposed	O
intuitively	O
by	O
Chen	O
and	O
Zhang	O
in	O
1993	O
.	O
</s>
<s>
pixels	B-Algorithm
)	O
to	O
ensure	O
the	O
basic	O
topological	B-Architecture
duality	O
of	O
separation	O
and	O
connectedness	O
.	O
</s>
<s>
to	O
open	O
or	O
closed	O
sets	O
in	O
the	O
3D	O
grid	B-Algorithm
cell	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
Grid	B-Algorithm
cell	I-Algorithm
topology	I-Algorithm
also	O
applies	O
to	O
multilevel	O
(	O
e.g.	O
,	O
color	O
)	O
2D	O
or	O
3D	O
images	O
,	O
</s>
<s>
Digital	B-Algorithm
topology	I-Algorithm
is	O
highly	O
related	O
to	O
combinatorial	O
topology	B-Architecture
.	O
</s>
<s>
The	O
main	O
differences	O
between	O
them	O
are	O
:	O
(	O
1	O
)	O
digital	B-Algorithm
topology	I-Algorithm
mainly	O
studies	O
digital	O
objects	O
that	O
are	O
formed	O
by	O
grid	O
cells	O
,	O
and	O
(	O
2	O
)	O
digital	B-Algorithm
topology	I-Algorithm
also	O
deals	O
with	O
non-Jordan	O
manifolds	B-Architecture
.	O
</s>
<s>
A	O
combinatorial	B-Algorithm
manifold	I-Algorithm
is	O
a	O
kind	O
of	O
manifold	B-Architecture
which	O
is	O
a	O
discretization	O
of	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
It	O
usually	O
means	O
a	O
piecewise	O
linear	O
manifold	B-Architecture
made	O
by	O
simplicial	O
complexes	O
.	O
</s>
<s>
A	O
digital	B-Algorithm
manifold	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
combinatorial	B-Algorithm
manifold	I-Algorithm
which	O
is	O
defined	O
in	O
digital	O
space	O
i.e.	O
</s>
<s>
A	O
digital	O
form	O
of	O
the	O
Gauss	O
–	O
Bonnet	O
theorem	O
is	O
:	O
Let	O
M	O
be	O
a	O
closed	O
digital	O
2D	O
manifold	B-Architecture
in	O
direct	O
adjacency	O
(	O
i.e.	O
,	O
a	O
(	O
6	O
,	O
26	O
)	O
-surface	O
in	O
3D	O
)	O
.	O
</s>
