<s>
The	O
Kubelka-Munk	B-Algorithm
theory	I-Algorithm
,	O
devised	O
by	O
Paul	O
Kubelka	O
and	O
Franz	O
Munk	O
,	O
is	O
a	O
fundamental	O
approach	O
to	O
modeling	O
the	O
appearance	O
of	O
paint	O
films	O
.	O
</s>
<s>
In	O
their	O
article	O
,	O
"	O
fundamental	O
differential	O
equations	O
"	O
are	O
developed	O
using	O
a	O
two-stream	B-Algorithm
approximation	I-Algorithm
for	O
light	O
diffusing	O
through	O
a	O
coating	O
whose	O
absorption	O
and	O
remission	B-Algorithm
(	O
back-scattering	B-Algorithm
)	O
coefficients	O
are	O
known	O
.	O
</s>
<s>
The	O
total	O
remission	B-Algorithm
from	O
a	O
coating	O
surface	O
is	O
the	O
summation	O
of	O
:	O
1	O
)	O
the	O
reflectance	O
of	O
the	O
coating	O
surface	O
;	O
2	O
)	O
the	O
remission	B-Algorithm
from	O
the	O
interior	O
of	O
the	O
coating	O
;	O
and	O
3	O
)	O
the	O
remission	B-Algorithm
from	O
the	O
surface	O
of	O
the	O
substrate	O
.	O
</s>
<s>
This	O
case	O
yielded	O
the	O
Kubelka	O
–	O
Munk	O
equation	O
,	O
which	O
describes	O
the	O
remission	B-Algorithm
from	O
a	O
sample	O
composed	O
of	O
an	O
infinite	O
number	O
of	O
infinitesimal	O
layers	O
,	O
each	O
having	O
as	O
an	O
absorption	O
fraction	O
and	O
as	O
a	O
remission	B-Algorithm
fraction	O
.	O
</s>
<s>
The	O
authors	O
noted	O
that	O
the	O
remission	B-Algorithm
from	O
an	O
infinite	O
number	O
of	O
these	O
infinitesimal	O
layers	O
is	O
"	O
solely	O
a	O
function	O
of	O
the	O
ratio	O
of	O
the	O
absorption	O
and	O
back-scatter	B-Algorithm
(	O
remission	B-Algorithm
)	O
constants	O
,	O
but	O
not	O
in	O
any	O
way	O
on	O
the	O
absolute	O
numerical	O
values	O
of	O
these	O
constants	O
"	O
.	O
</s>
<s>
Sometimes	O
these	O
improvements	O
are	O
touted	O
as	O
extensions	O
of	O
Kubelka	B-Algorithm
–	I-Algorithm
Munk	I-Algorithm
theory	I-Algorithm
,	O
sometimes	O
as	O
embracing	O
more	O
general	O
mathematics	O
of	O
which	O
the	O
Kubelka	O
–	O
Munk	O
equation	O
is	O
a	O
special	O
case	O
,	O
and	O
sometimes	O
as	O
an	O
alternate	O
approach	O
.	O
</s>
<s>
In	O
the	O
following	O
,	O
is	O
the	O
fraction	O
of	O
incident	O
light	O
that	O
is	O
remitted	O
(	O
reflected	O
)	O
by	O
a	O
coated	O
substrate	O
under	O
consideration	O
,	O
is	O
the	O
remission	B-Algorithm
fraction	O
from	O
the	O
substrate	O
alone	O
,	O
is	O
the	O
remission	B-Algorithm
fraction	O
from	O
the	O
coating	O
,	O
is	O
remission	B-Algorithm
fraction	O
of	O
an	O
infinitely	O
thick	O
layer	O
,	O
is	O
the	O
fraction	O
of	O
incident	O
light	O
transmitted	O
by	O
the	O
sample	O
under	O
consideration	O
,	O
and	O
X	O
is	O
the	O
coating	O
thickness	O
.	O
</s>
<s>
In	O
the	O
original	O
article	O
,	O
there	O
is	O
a	O
solution	O
for	O
remission	B-Algorithm
from	O
a	O
coating	O
of	O
finite	O
thickness	O
.	O
</s>
<s>
The	O
Kubelka	B-Algorithm
–	I-Algorithm
Munk	I-Algorithm
theory	I-Algorithm
is	O
also	O
used	O
in	O
the	O
paper	O
industry	O
to	O
predict	O
optical	O
properties	O
of	O
paper	O
,	O
avoiding	O
a	O
labor-intensive	O
trial-and-error	O
approach	O
.	O
</s>
<s>
There	O
is	O
a	O
simpler	O
method	O
adapted	O
from	O
the	O
Kubelka	B-Algorithm
–	I-Algorithm
Munk	I-Algorithm
theory	I-Algorithm
,	O
in	O
which	O
the	O
band	O
gap	O
is	O
calculated	O
by	O
plotting	O
,	O
versus	O
,	O
where	O
is	O
the	O
absorption	O
coefficient	O
.	O
</s>
<s>
By	O
resolving	O
the	O
Kubelka	O
–	O
Munk	O
equation	O
for	O
the	O
ratio	O
of	O
absorption	O
to	O
scatter	O
,	O
one	O
can	O
obtain	O
a	O
"	O
remission	B-Algorithm
function	O
"	O
:	O
</s>
<s>
We	O
may	O
define	O
and	O
as	O
absorption	O
and	O
back-scattering	B-Algorithm
coefficients	O
,	O
which	O
replace	O
the	O
absorption	O
and	O
remission	B-Algorithm
fractions	O
and	O
in	O
the	O
Kubelka	O
–	O
Munk	O
equation	O
above	O
.	O
</s>
<s>
One	O
special	O
case	O
has	O
received	O
much	O
attention	O
in	O
diffuse	B-Algorithm
reflectance	I-Algorithm
spectroscopy	I-Algorithm
,	O
that	O
of	O
an	O
opaque	O
(	O
infinitely	O
thick	O
)	O
coating	O
,	O
which	O
can	O
be	O
applied	O
to	O
a	O
sample	O
modeled	O
as	O
an	O
infinite	O
number	O
of	O
infinitesimal	O
layers	O
.	O
</s>
<s>
The	O
two-stream	B-Algorithm
approximation	I-Algorithm
was	O
embraced	O
by	O
the	O
early	O
practitioners	O
.	O
</s>
<s>
The	O
remission	B-Algorithm
function	O
(	O
also	O
called	O
the	O
Kubelka	O
–	O
Munk	O
function	O
)	O
was	O
almost	O
abandoned	O
in	O
favor	O
of	O
"	O
log( 	O
1/R	O
)	O
"	O
.	O
</s>
<s>
In	O
the	O
equation	O
,	O
and	O
are	O
the	O
measured	O
remission	B-Algorithm
by	O
and	O
transmission	O
through	O
a	O
sample	O
of	O
layers	O
,	O
each	O
layer	O
having	O
absorption	O
and	O
remission	B-Algorithm
fractions	O
of	O
and	O
.	O
</s>
<s>
The	O
term	O
"	O
failure	O
of	O
the	O
Kubelka	B-Algorithm
–	I-Algorithm
Munk	I-Algorithm
theory	I-Algorithm
"	O
has	O
been	O
applied	O
because	O
it	O
does	O
not	O
"	O
remain	O
valid	O
in	O
strongly	O
absorbing	O
materials	O
"	O
.	O
</s>
<s>
where	O
is	O
the	O
linear	O
absorption	O
coefficient	O
,	O
is	O
the	O
back-scatter	B-Algorithm
coefficient	O
,	O
and	O
is	O
their	O
sum	O
,	O
often	O
called	O
the	O
extinction	O
coefficient	O
.	O
</s>
<s>
The	O
total	O
remission	B-Algorithm
is	O
2/3	O
≈	O
0.667	O
.	O
</s>
<s>
For	O
two	O
plane	O
parallel	O
layers	O
,	O
x	O
and	O
y	O
,	O
having	O
different	O
properties	O
,	O
the	O
transmission	O
remission	B-Algorithm
,	O
and	O
absorption	O
fractions	O
for	O
the	O
two	O
layers	O
can	O
be	O
calculated	O
from	O
the	O
properties	O
of	O
the	O
individual	O
layers	O
(	O
)	O
from	O
the	O
following	O
equations	O
:	O
</s>
<s>
Consequently	O
,	O
for	O
non-homogeneous	O
layers	O
,	O
the	O
remission	B-Algorithm
from	O
the	O
first	O
layer	O
that	O
occurs	O
in	O
the	O
denominator	O
is	O
the	O
remission	B-Algorithm
when	O
illuminated	O
from	O
the	O
reverse	O
(	O
not	O
the	O
forward	O
)	O
direction	O
,	O
so	O
we	O
will	O
need	O
to	O
know	O
the	O
value	O
for	O
for	O
the	O
next	O
step	O
,	O
that	O
is	O
when	O
the	O
layerx	O
is	O
layer2	O
,	O
and	O
layery	O
is	O
layer1	O
:	O
</s>
<s>
Their	O
"	O
Mixbox	O
"	O
approach	O
works	O
by	O
converting	O
the	O
inputs	O
into	O
a	O
version	O
of	O
CMYK	O
(	O
phthalo	O
blue	O
,	O
quinacridone	O
magenta	O
,	O
Hansa	O
yellow	O
,	O
and	O
titanium	O
white	O
)	O
plus	O
a	O
residue	O
(	O
to	O
account	O
for	O
the	O
gamut	O
difference	O
)	O
,	O
performing	O
the	O
K	O
–	O
M	O
mixing	O
in	O
that	O
latent	B-Algorithm
space	I-Algorithm
,	O
and	O
then	O
producing	O
the	O
output	O
in	O
RGB	O
.	O
</s>
