<s>
The	O
Hexagonal	B-Algorithm
Efficient	I-Algorithm
Coordinate	I-Algorithm
System	I-Algorithm
(	O
HECS	O
)	O
,	O
formerly	O
known	O
as	O
Array	B-Data_Structure
Set	O
Addressing	O
(	O
ASA	O
)	O
,	O
is	O
a	O
coordinate	O
system	O
for	O
hexagonal	O
grids	O
that	O
allows	O
hexagonally	O
sampled	O
images	O
to	O
be	O
efficiently	O
stored	O
and	O
processed	O
on	O
digital	O
systems	O
.	O
</s>
<s>
The	O
HECS	O
system	O
enables	O
the	O
use	O
of	O
hexagonal	O
sampling	O
for	O
digital	B-Algorithm
imaging	I-Algorithm
applications	O
without	O
requiring	O
significant	O
additional	O
processing	O
to	O
address	O
the	O
hexagonal	O
array	B-Data_Structure
.	O
</s>
<s>
The	O
advantages	O
of	O
sampling	O
on	O
a	O
hexagonal	O
grid	O
instead	O
of	O
the	O
standard	O
rectangular	O
grid	O
for	O
digital	B-Algorithm
imaging	I-Algorithm
applications	O
include	O
:	O
more	O
efficient	O
sampling	O
,	O
consistent	O
connectivity	O
,	O
equidistant	O
neighboring	O
pixels	B-Algorithm
,	O
greater	O
angular	O
resolution	O
,	O
and	O
higher	O
circular	O
symmetry	O
.	O
</s>
<s>
The	O
Hexagonal	B-Algorithm
Efficient	I-Algorithm
Coordinate	I-Algorithm
System	I-Algorithm
(	O
HECS	O
)	O
is	O
based	O
on	O
the	O
idea	O
of	O
representing	O
the	O
hexagonal	O
grid	O
as	O
a	O
set	O
of	O
two	O
rectangular	O
arrays	O
which	O
can	O
be	O
individually	O
indexed	O
using	O
familiar	O
integer-valued	O
row	O
and	O
column	O
indices	O
.	O
</s>
<s>
where	O
the	O
coordinates	O
represent	O
the	O
array	B-Data_Structure
,	O
row	O
,	O
and	O
column	O
,	O
respectively	O
.	O
</s>
<s>
The	O
hexagonal	O
grid	O
is	O
separated	O
into	O
rectangular	O
arrays	O
by	O
taking	O
every	O
other	O
row	O
as	O
one	O
array	B-Data_Structure
and	O
the	O
remaining	O
rows	O
as	O
the	O
other	O
array	B-Data_Structure
,	O
as	O
shown	O
in	O
the	O
figure	O
.	O
</s>
<s>
The	O
addresses	O
of	O
the	O
nearest	B-General_Concept
neighbors	I-General_Concept
of	O
a	O
pixel	B-Algorithm
(	O
or	O
grid	O
point	O
)	O
are	O
easily	O
determined	O
by	O
simple	O
expressions	O
which	O
are	O
functions	O
of	O
the	O
pixel	B-Algorithm
's	O
coordinates	O
,	O
as	O
shown	O
.	O
</s>
<s>
where	O
is	O
the	O
logical	O
XOR	B-Application
operator	O
and	O
is	O
the	O
logical	O
AND	O
operator	O
.	O
</s>
<s>
The	O
hexagonal	O
discrete	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
HDFT	O
)	O
was	O
developed	O
by	O
Mersereau	O
and	O
was	O
converted	O
to	O
an	O
HECS	O
representation	O
by	O
Rummelt	O
.	O
</s>
<s>
Let	O
be	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
ofx	O
.	O
</s>
<s>
Notice	O
that	O
the	O
HECS	O
representation	O
of	O
the	O
HDFT	O
overcomes	O
Mersereau	O
's	O
"	O
insurmountable	O
difficulty	O
"	O
since	O
it	O
is	O
a	O
separable	O
kernel	O
,	O
which	O
led	O
to	O
the	O
development	O
of	O
the	O
hexagonal	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
General	O
balanced	B-Algorithm
ternary	I-Algorithm
(	O
GBT	O
)	O
is	O
based	O
on	O
a	O
hierarchy	O
of	O
cells	O
,	O
where	O
at	O
every	O
level	O
the	O
cells	O
are	O
each	O
aggregates	O
of	O
cells	O
from	O
the	O
previous	O
level	O
.	O
</s>
