<s>
In	O
the	O
mathematical	O
fields	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
and	O
approximation	B-Algorithm
theory	I-Algorithm
,	O
box	B-Algorithm
splines	I-Algorithm
are	O
piecewise	B-Algorithm
polynomial	I-Algorithm
functions	O
of	O
several	O
variables	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
are	O
considered	O
as	O
a	O
multivariate	O
generalization	O
of	O
basis	O
splines	O
(	O
B-splines	B-Algorithm
)	O
and	O
are	O
generally	O
used	O
for	O
multivariate	O
approximation/interpolation	O
.	O
</s>
<s>
Geometrically	O
,	O
a	O
box	B-Algorithm
spline	I-Algorithm
is	O
the	O
shadow	O
(	O
X-ray	B-Algorithm
)	O
of	O
a	O
hypercube	B-Operating_System
projected	O
down	O
to	O
a	O
lower-dimensional	O
space	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
and	O
simplex	O
splines	O
are	O
well	O
studied	O
special	O
cases	O
of	O
polyhedral	O
splines	O
which	O
are	O
defined	O
as	O
shadows	O
of	O
general	O
polytopes	O
.	O
</s>
<s>
When	O
the	O
number	O
of	O
vectors	O
is	O
the	O
same	O
as	O
the	O
dimension	O
of	O
the	O
domain	O
(	O
i.e.	O
,	O
)	O
then	O
the	O
box	B-Algorithm
spline	I-Algorithm
is	O
simply	O
the	O
(	O
normalized	O
)	O
indicator	O
function	O
of	O
the	O
parallelepiped	O
formed	O
by	O
the	O
vectors	O
in	O
:	O
</s>
<s>
Adding	O
a	O
new	O
direction	O
,	O
to	O
or	O
generally	O
when	O
the	O
box	B-Algorithm
spline	I-Algorithm
is	O
defined	O
recursively	O
:	O
</s>
<s>
Considering	O
tempered	O
distributions	O
a	O
box	B-Algorithm
spline	I-Algorithm
associated	O
with	O
a	O
single	O
direction	O
vector	O
is	O
a	O
Dirac-like	O
generalized	O
function	O
supported	O
on	O
for	O
.	O
</s>
<s>
Then	O
the	O
general	O
box	B-Algorithm
spline	I-Algorithm
is	O
defined	O
as	O
the	O
convolution	O
of	O
distributions	O
associated	O
the	O
single-vector	O
box	B-Algorithm
splines	I-Algorithm
:	O
</s>
<s>
Then	O
the	O
box	B-Algorithm
spline	I-Algorithm
has	O
degrees	O
of	O
continuity	O
:	O
.	O
</s>
<s>
When	O
(	O
and	O
vectors	O
in	O
span	O
)	O
the	O
box	B-Algorithm
spline	I-Algorithm
is	O
a	O
compactly	O
supported	O
function	O
whose	O
support	O
is	O
a	O
zonotope	O
in	O
formed	O
by	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
the	O
direction	O
vectors	O
.	O
</s>
<s>
Since	O
zonotopes	O
are	O
centrally	O
symmetric	O
,	O
the	O
support	O
of	O
the	O
box	B-Algorithm
spline	I-Algorithm
is	O
symmetric	O
with	O
respect	O
to	O
its	O
center	O
:	O
</s>
<s>
For	O
applications	O
,	O
linear	O
combinations	O
of	O
shifts	O
of	O
one	O
or	O
more	O
box	B-Algorithm
splines	I-Algorithm
on	O
a	O
lattice	O
are	O
used	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
have	O
been	O
useful	O
in	O
characterization	O
of	O
hyperplane	O
arrangements	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
multidimensional	O
signal	O
processing	O
,	O
box	B-Algorithm
splines	I-Algorithm
can	O
provide	O
multivariate	O
interpolation	O
kernels	O
(	O
reconstruction	O
filters	O
)	O
tailored	O
to	O
non-Cartesian	O
sampling	O
lattices	O
,	O
and	O
crystallographic	O
lattices	O
(	O
root	O
lattices	O
)	O
that	O
include	O
many	O
information-theoretically	O
optimal	O
sampling	O
lattices	O
.	O
</s>
<s>
In	O
the	O
2-D	O
setting	O
the	O
three-direction	O
box	B-Algorithm
spline	I-Algorithm
is	O
used	O
for	O
interpolation	O
of	O
hexagonally	O
sampled	O
images	O
.	O
</s>
<s>
In	O
the	O
3-D	O
setting	O
,	O
four-direction	O
and	O
six-direction	O
box	B-Algorithm
splines	I-Algorithm
are	O
used	O
for	O
interpolation	O
of	O
data	O
sampled	O
on	O
the	O
(	O
optimal	O
)	O
body-centered	O
cubic	O
and	O
face-centered	O
cubic	O
lattices	O
respectively	O
.	O
</s>
<s>
The	O
seven-direction	O
box	B-Algorithm
spline	I-Algorithm
has	O
been	O
used	O
for	O
modelling	O
surfaces	O
and	O
can	O
be	O
used	O
for	O
interpolation	O
of	O
data	O
on	O
the	O
Cartesian	O
lattice	O
as	O
well	O
as	O
the	O
body	O
centered	O
cubic	O
lattice	O
.	O
</s>
<s>
Generalization	O
of	O
the	O
four	O
-	O
and	O
six-direction	O
box	B-Algorithm
splines	I-Algorithm
to	O
higher	O
dimensions	O
can	O
be	O
used	O
to	O
build	O
splines	O
on	O
root	O
lattices	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
are	O
key	O
ingredients	O
of	O
hex-splines	O
and	O
Voronoi	O
splines	O
that	O
,	O
however	O
,	O
are	O
not	O
refinable	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
have	O
found	O
applications	O
in	O
high-dimensional	O
filtering	O
,	O
specifically	O
for	O
fast	O
bilateral	O
filtering	O
and	O
non-local	O
means	O
algorithms	O
.	O
</s>
<s>
Moreover	O
,	O
box	B-Algorithm
splines	I-Algorithm
are	O
used	O
for	O
designing	O
efficient	O
space-variant	O
(	O
i.e.	O
,	O
non-convolutional	O
)	O
filters	O
.	O
</s>
<s>
Box	B-Algorithm
splines	I-Algorithm
are	O
useful	O
basis	O
functions	O
for	O
image	O
representation	O
in	O
the	O
context	O
of	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
problems	O
as	O
the	O
spline	O
spaces	O
generated	O
by	O
box	B-Algorithm
splines	I-Algorithm
spaces	O
are	O
closed	O
under	O
X-ray	B-Algorithm
and	O
Radon	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
In	O
this	O
application	O
while	O
the	O
signal	O
is	O
represented	O
in	O
shift-invariant	O
spaces	O
,	O
the	O
projections	O
are	O
obtained	O
,	O
in	O
closed-form	O
,	O
by	O
non-uniform	O
translates	O
of	O
box	B-Algorithm
splines	I-Algorithm
.	O
</s>
<s>
In	O
the	O
context	O
of	O
image	O
processing	O
,	O
box	B-Algorithm
spline	I-Algorithm
frames	O
have	O
been	O
shown	O
to	O
be	O
effective	O
in	O
edge	O
detection	O
.	O
</s>
