<s>
An	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
is	O
a	O
k-d	B-Data_Structure
tree	I-Data_Structure
defined	O
implicitly	O
above	O
a	O
rectilinear	O
grid	O
.	O
</s>
<s>
Each	O
inner	O
node	B-Data_Structure
's	O
split	O
plane	O
is	O
positioned	O
on	O
a	O
grid	O
plane	O
of	O
the	O
underlying	O
grid	O
,	O
partitioning	O
the	O
node	B-Data_Structure
's	O
grid	O
into	O
two	O
subgrids	O
.	O
</s>
<s>
The	O
terms	O
"	O
min/max	O
k-d	B-Data_Structure
tree	I-Data_Structure
"	O
and	O
"	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
"	O
are	O
sometimes	O
mixed	O
up	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
first	O
publication	O
using	O
the	O
term	O
"	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
"	O
did	O
actually	O
use	O
explicit	O
min/max	O
k-d	B-Data_Structure
trees	I-Data_Structure
but	O
referred	O
to	O
them	O
as	O
"	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
"	O
to	O
indicate	O
that	O
they	O
may	O
be	O
used	O
to	O
ray	O
trace	O
implicitly	O
given	O
iso	O
surfaces	O
.	O
</s>
<s>
Nevertheless	O
,	O
this	O
publication	O
used	O
also	O
slim	O
k-d	B-Data_Structure
trees	I-Data_Structure
which	O
are	O
a	O
subset	O
of	O
the	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
with	O
the	O
restriction	O
that	O
they	O
can	O
only	O
be	O
built	O
over	O
integer	O
hyperrectangles	O
with	O
sidelengths	O
that	O
are	O
powers	O
of	O
two	O
.	O
</s>
<s>
Implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
as	O
defined	O
here	O
have	O
recently	O
been	O
introduced	O
,	O
with	O
applications	O
in	O
computer	O
graphics	O
.	O
</s>
<s>
As	O
it	O
is	O
possible	O
to	O
assign	O
attributes	O
to	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
nodes	O
,	O
one	O
may	O
refer	O
to	O
an	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
which	O
has	O
min/max	O
values	O
assigned	O
to	O
its	O
nodes	O
as	O
an	O
"	O
implicit	O
min/max	O
k-d	B-Data_Structure
tree	I-Data_Structure
"	O
.	O
</s>
<s>
Implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
are	O
generally	O
not	O
constructed	O
explicitly	O
.	O
</s>
<s>
When	O
accessing	O
a	O
node	B-Data_Structure
,	O
its	O
split	O
plane	O
orientation	O
and	O
position	O
are	O
evaluated	O
using	O
the	O
specific	O
splitting-function	O
defining	O
the	O
tree	O
.	O
</s>
<s>
Their	O
corresponding	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
are	O
full	O
binary	O
trees	O
,	O
which	O
have	O
for	O
n	O
leaf	O
nodes	O
n	O
-	O
1	O
inner	O
nodes	O
.	O
</s>
<s>
Their	O
corresponding	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
are	O
non-degenerated	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
.	O
</s>
<s>
complete	O
splitting-functions	O
are	O
non-degenerated	O
splitting-functions	O
whose	O
corresponding	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
's	O
leaf	O
nodes	O
are	O
single	O
grid	O
cells	O
such	O
that	O
they	O
have	O
one	O
inner	O
node	B-Data_Structure
less	O
than	O
the	O
amount	O
of	O
gridcells	O
given	O
in	O
the	O
grid	O
.	O
</s>
<s>
The	O
corresponding	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
are	O
complete	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
.	O
</s>
<s>
It	O
creates	O
fairly	O
balanced	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
by	O
using	O
k-dimensional	O
integer	O
hyperrectangles	O
hyprec[2][k]	O
belonging	O
to	O
each	O
node	B-Data_Structure
of	O
the	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
.	O
</s>
<s>
The	O
hyperrectangles	O
define	O
which	O
gridcells	O
of	O
the	O
rectilinear	O
grid	O
belong	O
to	O
their	O
corresponding	O
node	B-Data_Structure
.	O
</s>
<s>
If	O
the	O
volume	O
of	O
this	O
hyperrectangle	O
equals	O
one	O
,	O
the	O
corresponding	O
node	B-Data_Structure
is	O
a	O
single	O
grid	O
cell	O
and	O
is	O
therefore	O
not	O
further	O
subdivided	O
and	O
marked	O
as	O
leaf	O
node	B-Data_Structure
.	O
</s>
<s>
An	O
advantage	O
of	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
is	O
that	O
their	O
split	O
plane	O
's	O
orientations	O
and	O
positions	O
need	O
not	O
to	O
be	O
stored	O
explicitly	O
.	O
</s>
<s>
For	O
complete	O
implicit	B-Data_Structure
k-d	I-Data_Structure
trees	I-Data_Structure
it	O
is	O
possible	O
to	O
pre-allocate	O
a	O
correctly	O
sized	O
array	O
of	O
attributes	O
and	O
to	O
assign	O
each	O
inner	O
node	B-Data_Structure
of	O
the	O
tree	O
to	O
a	O
unique	O
element	O
in	O
that	O
allocated	O
array	O
.	O
</s>
<s>
As	O
a	O
complete	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
has	O
one	O
inner	O
node	B-Data_Structure
less	O
than	O
grid	O
cells	O
,	O
it	O
is	O
known	O
in	O
advance	O
how	O
many	O
attributes	O
need	O
to	O
be	O
stored	O
.	O
</s>
<s>
Implicit	O
max-k-d	O
trees	O
are	O
used	O
for	O
ray	B-Algorithm
casting	I-Algorithm
isosurfaces/MIP	O
(	O
maximum	B-Application
intensity	I-Application
projection	I-Application
)	O
.	O
</s>
<s>
The	O
attribute	O
assigned	O
to	O
each	O
inner	O
node	B-Data_Structure
is	O
the	O
maximal	O
scalar	O
value	O
given	O
in	O
the	O
subgrid	O
belonging	O
to	O
the	O
node	B-Data_Structure
.	O
</s>
<s>
The	O
low	O
storage	O
requirements	O
of	O
the	O
implicit	O
max	B-Data_Structure
kd-tree	I-Data_Structure
and	O
the	O
favorable	O
visualization	O
complexity	O
of	O
ray	B-Algorithm
casting	I-Algorithm
allow	O
to	O
ray	B-Algorithm
cast	I-Algorithm
(	O
and	O
even	O
change	O
the	O
isosurface	O
for	O
)	O
very	O
large	O
scalar	O
fields	O
at	O
interactive	O
framerates	O
on	O
commodity	O
PCs	O
.	O
</s>
<s>
Similarly	O
an	O
implicit	O
min/max	B-Data_Structure
kd-tree	I-Data_Structure
may	O
be	O
used	O
to	O
efficiently	O
evaluate	O
queries	O
such	O
as	O
terrain	O
line	O
of	O
sight	O
.	O
</s>
<s>
Given	O
an	O
implicit	B-Data_Structure
k-d	I-Data_Structure
tree	I-Data_Structure
spanned	O
over	O
an	O
k-dimensional	O
grid	O
with	O
n	O
gridcells	O
.	O
</s>
<s>
Ray	B-Algorithm
casting	I-Algorithm
iso-surfaces/MIP	O
an	O
underlying	O
scalar	O
field	O
using	O
the	O
corresponding	O
implicit	O
max	O
k''-d	O
tree	O
takes	O
roughly	O
time	O
.	O
</s>
