<s>
Discrete	B-Algorithm
tomography	I-Algorithm
focuses	O
on	O
the	O
problem	O
of	O
reconstruction	O
of	O
binary	B-Algorithm
images	I-Algorithm
(	O
or	O
finite	O
subsets	O
of	O
the	O
integer	O
lattice	O
)	O
from	O
a	O
small	O
number	O
of	O
their	O
projections	O
.	O
</s>
<s>
In	O
general	O
,	O
tomography	B-Algorithm
deals	O
with	O
the	O
problem	O
of	O
determining	O
shape	O
and	O
dimensional	O
information	O
of	O
an	O
object	O
from	O
a	O
set	O
of	O
projections	O
.	O
</s>
<s>
From	O
the	O
mathematical	O
point	O
of	O
view	O
,	O
the	O
object	O
corresponds	O
to	O
a	O
function	O
and	O
the	O
problem	O
posed	O
is	O
to	O
reconstruct	O
this	O
function	O
from	O
its	O
integrals	O
or	O
sums	O
over	O
subsets	O
of	O
its	O
domain	B-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
the	O
tomographic	B-Algorithm
inversion	O
problem	O
may	O
be	O
continuous	O
or	O
discrete	O
.	O
</s>
<s>
domain	B-Algorithm
and	O
the	O
range	O
of	O
the	O
function	O
are	O
continuous	O
and	O
line	O
integrals	O
are	O
used	O
.	O
</s>
<s>
In	O
discrete	B-Algorithm
tomography	I-Algorithm
the	O
domain	B-Algorithm
of	O
the	O
function	O
may	O
be	O
either	O
discrete	O
or	O
continuous	O
,	O
and	O
the	O
range	O
of	O
the	O
function	O
is	O
a	O
finite	O
set	O
of	O
real	O
,	O
usually	O
nonnegative	O
numbers	O
.	O
</s>
<s>
In	O
continuous	O
tomography	B-Algorithm
when	O
a	O
large	O
number	O
of	O
projections	O
is	O
available	O
,	O
accurate	O
reconstructions	O
can	O
be	O
made	O
by	O
many	O
different	O
algorithms	O
.	O
</s>
<s>
It	O
is	O
typical	O
for	O
discrete	B-Algorithm
tomography	I-Algorithm
that	O
only	O
a	O
few	O
projections	O
(	O
line	O
sums	O
)	O
are	O
used	O
.	O
</s>
<s>
a	O
binary	B-Algorithm
image	I-Algorithm
from	O
a	O
small	O
number	O
of	O
projections	O
.	O
</s>
<s>
The	O
name	O
discrete	B-Algorithm
tomography	I-Algorithm
is	O
due	O
to	O
Larry	O
Shepp	O
,	O
who	O
organized	O
the	O
first	O
meeting	O
devoted	O
to	O
this	O
topic	O
(	O
DIMACS	O
Mini-Symposium	O
on	O
Discrete	B-Algorithm
Tomography	I-Algorithm
,	O
September	O
19	O
,	O
1994	O
,	O
Rutgers	O
University	O
)	O
.	O
</s>
<s>
Discrete	B-Algorithm
tomography	I-Algorithm
has	O
strong	O
connections	O
with	O
other	O
mathematical	O
fields	O
,	O
such	O
as	O
number	O
theory	O
,	O
discrete	O
mathematics	O
,	O
computational	O
complexity	O
theory	O
and	O
combinatorics	O
.	O
</s>
<s>
In	O
fact	O
,	O
a	O
number	O
of	O
discrete	B-Algorithm
tomography	I-Algorithm
problems	O
were	O
first	O
discussed	O
as	O
combinatorial	O
problems	O
.	O
</s>
<s>
In	O
the	O
same	O
year	O
,	O
David	O
Gale	O
found	O
the	O
same	O
consistency	O
conditions	O
,	O
but	O
in	O
connection	O
with	O
the	O
network	B-Algorithm
flow	I-Algorithm
problem	O
.	O
</s>
<s>
The	O
problem	O
of	O
reconstructing	O
a	O
binary	B-Algorithm
image	I-Algorithm
from	O
a	O
small	O
number	O
of	O
projections	O
generally	O
leads	O
to	O
a	O
large	O
number	O
of	O
solutions	O
.	O
</s>
<s>
Coloring	O
a	O
grid	O
using	O
colors	O
with	O
the	O
restriction	O
that	O
each	O
row	O
and	O
each	O
column	O
has	O
a	O
specific	O
number	O
of	O
cells	O
of	O
each	O
color	O
is	O
known	O
as	O
the	O
−atom	O
problem	O
in	O
the	O
discrete	B-Algorithm
tomography	I-Algorithm
community	O
.	O
</s>
<s>
or	O
)	O
,	O
greedy	B-Algorithm
algorithms	I-Algorithm
(	O
see	O
for	O
approximation	O
guarantees	O
)	O
,	O
and	O
Monte	O
Carlo	O
algorithms	O
.	O
</s>
<s>
A	O
form	O
of	O
discrete	B-Algorithm
tomography	I-Algorithm
also	O
forms	O
the	O
basis	O
of	O
nonograms	O
,	O
a	O
type	O
of	O
logic	O
puzzle	O
in	O
which	O
information	O
about	O
the	O
rows	O
and	O
columns	O
of	O
a	O
digital	O
image	O
is	O
used	O
to	O
reconstruct	O
the	O
image	O
.	O
</s>
