<s>
Digital	B-Algorithm
geometry	I-Algorithm
deals	O
with	O
discrete	O
sets	O
(	O
usually	O
discrete	O
point	O
sets	O
)	O
considered	O
to	O
be	O
digitized	O
models	O
or	O
images	O
of	O
objects	O
of	O
the	O
2D	O
or	O
3D	O
Euclidean	O
space	O
.	O
</s>
<s>
The	O
images	O
we	O
see	O
on	O
the	O
TV	O
screen	O
,	O
the	O
raster	B-Algorithm
display	O
of	O
a	O
computer	O
,	O
or	O
in	O
newspapers	O
are	O
in	O
fact	O
digital	B-General_Concept
images	O
.	O
</s>
<s>
Its	O
main	O
application	O
areas	O
are	O
computer	O
graphics	O
and	O
image	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Constructing	O
digitized	O
representations	O
of	O
objects	O
,	O
with	O
the	O
emphasis	O
on	O
precision	O
and	O
efficiency	O
(	O
either	O
by	O
means	O
of	O
synthesis	O
,	O
see	O
,	O
for	O
example	O
,	O
Bresenham	B-Algorithm
's	I-Algorithm
line	I-Algorithm
algorithm	I-Algorithm
or	O
digital	B-General_Concept
disks	O
,	O
or	O
by	O
means	O
of	O
digitization	O
and	O
subsequent	O
processing	O
of	O
digital	B-General_Concept
images	O
)	O
.	O
</s>
<s>
Study	O
of	O
properties	O
of	O
digital	B-General_Concept
sets	O
;	O
see	O
,	O
for	O
example	O
,	O
Pick	O
's	O
theorem	O
,	O
digital	B-General_Concept
convexity	O
,	O
digital	B-General_Concept
straightness	O
,	O
or	O
digital	B-General_Concept
planarity	O
.	O
</s>
<s>
Transforming	O
digitized	O
representations	O
of	O
objects	O
,	O
for	O
example	O
(	O
A	O
)	O
into	O
simplified	O
shapes	O
such	O
as	O
(	O
i	O
)	O
skeletons	O
,	O
by	O
repeated	O
removal	O
of	O
simple	O
points	O
such	O
that	O
the	O
digital	B-Algorithm
topology	I-Algorithm
of	O
an	O
image	O
does	O
not	O
change	O
,	O
or	O
(	O
ii	O
)	O
medial	O
axis	O
,	O
by	O
calculating	O
local	O
maxima	O
in	O
a	O
distance	O
transform	O
of	O
the	O
given	O
digitized	O
object	O
representation	O
,	O
or	O
(	O
B	O
)	O
into	O
modified	O
shapes	O
using	O
mathematical	B-Algorithm
morphology	I-Algorithm
.	O
</s>
<s>
Reconstructing	O
"	O
real	O
"	O
objects	O
or	O
their	O
properties	O
(	O
area	O
,	O
length	O
,	O
curvature	O
,	O
volume	O
,	O
surface	O
area	O
,	O
and	O
so	O
forth	O
)	O
from	O
digital	B-General_Concept
images	O
.	O
</s>
<s>
Study	O
of	O
digital	B-General_Concept
curves	O
,	O
digital	B-Algorithm
surfaces	I-Algorithm
,	O
and	O
digital	B-Algorithm
manifolds	I-Algorithm
.	O
</s>
<s>
Designing	O
tracking	O
algorithms	O
for	O
digital	B-General_Concept
objects	O
.	O
</s>
<s>
Functions	O
on	O
digital	B-General_Concept
space	O
.	O
</s>
<s>
Digital	B-Algorithm
geometry	I-Algorithm
heavily	O
overlaps	O
with	O
discrete	O
geometry	O
and	O
may	O
be	O
considered	O
as	O
a	O
part	O
thereof	O
.	O
</s>
<s>
A	O
2D	O
digital	B-General_Concept
space	O
usually	O
means	O
a	O
2D	O
grid	O
space	O
that	O
only	O
contains	O
integer	O
points	O
in	O
2D	O
Euclidean	O
space	O
.	O
</s>
<s>
A	O
2D	O
image	O
is	O
a	O
function	O
on	O
a	O
2D	O
digital	B-General_Concept
space	O
(	O
See	O
image	B-Algorithm
processing	I-Algorithm
)	O
.	O
</s>
<s>
In	O
Rosenfeld	O
and	O
Kak	O
's	O
book	O
,	O
digital	B-General_Concept
connectivity	O
are	O
defined	O
as	O
the	O
relationship	O
among	O
elements	O
in	O
digital	B-General_Concept
space	O
.	O
</s>
<s>
For	O
example	O
,	O
4-connectivity	O
and	O
8-connectivity	B-Algorithm
in	O
2D	O
.	O
</s>
<s>
Also	O
see	O
pixel	B-Algorithm
connectivity	I-Algorithm
.	O
</s>
<s>
A	O
digital	B-General_Concept
space	O
and	O
its	O
(	O
digital	B-General_Concept
-	O
)	O
connectivity	O
determine	O
a	O
digital	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
In	O
digital	B-General_Concept
space	O
,	O
the	O
digitally	O
continuous	O
function	O
(	O
A	O
.	O
Rosenfeld	O
,	O
1986	O
)	O
and	O
the	O
gradually	B-Algorithm
varied	I-Algorithm
function	I-Algorithm
(	O
L	O
.	O
Chen	O
,	O
1989	O
)	O
were	O
proposed	O
,	O
independently	O
.	O
</s>
<s>
A	O
digitally	O
continuous	O
function	O
means	O
a	O
function	O
in	O
which	O
the	O
value	O
(	O
an	O
integer	O
)	O
at	O
a	O
digital	B-General_Concept
point	O
is	O
the	O
same	O
or	O
off	O
by	O
at	O
most	O
1	O
from	O
its	O
neighbors	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
x	O
and	O
y	O
are	O
two	O
adjacent	O
points	O
in	O
a	O
digital	B-General_Concept
space	O
,	O
|	O
f(x )	O
f(y )	O
|≤1	O
.	O
</s>
<s>
A	O
gradually	B-Algorithm
varied	I-Algorithm
function	I-Algorithm
is	O
a	O
function	O
from	O
a	O
digital	B-General_Concept
space	O
to	O
where	O
and	O
are	O
real	O
numbers	O
.	O
</s>
<s>
So	O
we	O
can	O
see	O
that	O
the	O
gradually	B-Algorithm
varied	I-Algorithm
function	I-Algorithm
is	O
defined	O
to	O
be	O
more	O
general	O
than	O
the	O
digitally	O
continuous	O
function	O
.	O
</s>
<s>
The	O
necessary	O
and	O
sufficient	O
condition	O
for	O
the	O
existence	O
of	O
the	O
gradually	O
varied	O
extension	O
of	O
is	O
:	O
for	O
each	O
pair	O
of	O
points	O
and	O
in	O
,	O
assume	O
and	O
,	O
we	O
have	O
,	O
where	O
is	O
the	O
(	O
digital	B-General_Concept
)	O
distance	O
between	O
and	O
.	O
</s>
