<s>
In	O
computer	B-Application
vision	I-Application
,	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
a	O
matrix	B-Architecture
,	O
that	O
relates	O
corresponding	B-Algorithm
points	I-Algorithm
in	O
stereo	B-Algorithm
images	I-Algorithm
assuming	O
that	O
the	O
cameras	O
satisfy	O
the	O
pinhole	B-Algorithm
camera	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
above	O
relation	O
which	O
defines	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
was	O
published	O
in	O
1981	O
by	O
H	O
.	O
Christopher	O
Longuet-Higgins	O
,	O
introducing	O
the	O
concept	O
to	O
the	O
computer	B-Application
vision	I-Application
community	O
.	O
</s>
<s>
Richard	O
Hartley	O
and	O
Andrew	O
Zisserman	O
's	O
book	O
reports	O
that	O
an	O
analogous	O
matrix	B-Architecture
appeared	O
in	O
photogrammetry	B-Application
long	O
before	O
that	O
.	O
</s>
<s>
Finally	O
,	O
it	O
shows	O
how	O
the	O
3D	O
coordinates	O
of	O
the	O
image	O
points	O
can	O
be	O
determined	O
with	O
the	O
aid	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
The	O
essential	B-General_Concept
matrix	I-General_Concept
can	O
be	O
seen	O
as	O
a	O
precursor	O
to	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
,	O
.	O
</s>
<s>
Both	O
matrices	O
can	O
be	O
used	O
for	O
establishing	O
constraints	O
between	O
matching	O
image	O
points	O
,	O
but	O
the	O
fundamental	B-General_Concept
matrix	I-General_Concept
can	O
only	O
be	O
used	O
in	O
relation	O
to	O
calibrated	O
cameras	O
since	O
the	O
inner	O
camera	O
parameters	O
(	O
matrices	O
and	O
)	O
must	O
be	O
known	O
in	O
order	O
to	O
achieve	O
the	O
normalization	O
.	O
</s>
<s>
If	O
,	O
however	O
,	O
the	O
cameras	O
are	O
calibrated	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
can	O
be	O
useful	O
for	O
determining	O
both	O
the	O
relative	O
position	O
and	O
orientation	O
between	O
the	O
cameras	O
and	O
the	O
3D	O
position	O
of	O
corresponding	O
image	O
points	O
.	O
</s>
<s>
where	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
and	O
is	O
a	O
3-dimensional	O
translation	O
vector	O
.	O
</s>
<s>
The	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
then	O
defined	O
as	O
:	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
representation	O
of	O
the	O
cross	O
product	O
with	O
.	O
</s>
<s>
since	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Properties	O
of	O
the	O
matrix	B-Architecture
representation	O
of	O
the	O
cross	O
product	O
.	O
</s>
<s>
which	O
is	O
the	O
constraint	O
that	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
defines	O
between	O
corresponding	O
image	O
points	O
.	O
</s>
<s>
Not	O
every	O
arbitrary	O
matrix	B-Architecture
can	O
be	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
for	O
some	O
stereo	O
cameras	O
.	O
</s>
<s>
To	O
see	O
this	O
notice	O
that	O
it	O
is	O
defined	O
as	O
the	O
matrix	B-Architecture
product	O
of	O
one	O
rotation	B-Algorithm
matrix	I-Algorithm
and	O
one	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
,	O
both	O
.	O
</s>
<s>
The	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
must	O
have	O
two	O
singular	O
values	O
which	O
are	O
equal	O
and	O
another	O
which	O
is	O
zero	O
.	O
</s>
<s>
The	O
multiplication	O
of	O
the	O
rotation	B-Algorithm
matrix	I-Algorithm
does	O
not	O
change	O
the	O
singular	O
values	O
which	O
means	O
that	O
also	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
has	O
two	O
singular	O
values	O
which	O
are	O
equal	O
and	O
one	O
which	O
is	O
zero	O
.	O
</s>
<s>
The	O
properties	O
described	O
here	O
are	O
sometimes	O
referred	O
to	O
as	O
internal	O
constraints	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
If	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
multiplied	O
by	O
a	O
non-zero	O
scalar	O
,	O
the	O
result	O
is	O
again	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
which	O
defines	O
exactly	O
the	O
same	O
constraint	O
as	O
does	O
.	O
</s>
<s>
Here	O
,	O
the	O
last	O
equation	O
is	O
a	O
matrix	B-Architecture
constraint	O
,	O
which	O
can	O
be	O
seen	O
as	O
9	O
constraints	O
,	O
one	O
for	O
each	O
matrix	B-Architecture
element	O
.	O
</s>
<s>
These	O
constraints	O
are	O
often	O
used	O
for	O
determining	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
from	O
five	O
corresponding	B-Algorithm
point	I-Algorithm
pairs	O
.	O
</s>
<s>
The	O
essential	B-General_Concept
matrix	I-General_Concept
has	O
five	O
or	O
six	O
degrees	O
of	O
freedom	O
,	O
depending	O
on	O
whether	O
or	O
not	O
it	O
is	O
seen	O
as	O
a	O
projective	O
element	O
.	O
</s>
<s>
The	O
rotation	B-Algorithm
matrix	I-Algorithm
and	O
the	O
translation	O
vector	O
have	O
three	O
degrees	O
of	O
freedom	O
each	O
,	O
in	O
total	O
six	O
.	O
</s>
<s>
If	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
considered	O
as	O
a	O
projective	O
element	O
,	O
however	O
,	O
one	O
degree	O
of	O
freedom	O
related	O
to	O
scalar	O
multiplication	O
must	O
be	O
subtracted	O
leaving	O
five	O
degrees	O
of	O
freedom	O
in	O
total	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
corresponding	O
image	O
points	O
it	O
is	O
possible	O
to	O
estimate	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
which	O
satisfies	O
the	O
defining	O
epipolar	B-Algorithm
constraint	I-Algorithm
for	O
all	O
the	O
points	O
in	O
the	O
set	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
image	O
points	O
are	O
subject	O
to	O
noise	O
,	O
which	O
is	O
the	O
common	O
case	O
in	O
any	O
practical	O
situation	O
,	O
it	O
is	O
not	O
possible	O
to	O
find	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
which	O
satisfies	O
all	O
constraints	O
exactly	O
.	O
</s>
<s>
Depending	O
on	O
how	O
the	O
error	O
related	O
to	O
each	O
constraint	O
is	O
measured	O
,	O
it	O
is	O
possible	O
to	O
determine	O
or	O
estimate	O
an	O
essential	B-General_Concept
matrix	I-General_Concept
which	O
optimally	O
satisfies	O
the	O
constraints	O
for	O
a	O
given	O
set	O
of	O
corresponding	O
image	O
points	O
.	O
</s>
<s>
The	O
most	O
straightforward	O
approach	O
is	O
to	O
set	O
up	O
a	O
total	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
,	O
commonly	O
known	O
as	O
the	O
eight-point	B-General_Concept
algorithm	I-General_Concept
.	O
</s>
<s>
Given	O
that	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
has	O
been	O
determined	O
for	O
a	O
stereo	O
camera	O
pair	O
--	O
for	O
example	O
,	O
using	O
the	O
estimation	O
method	O
above	O
--	O
this	O
information	O
can	O
be	O
used	O
for	O
determining	O
also	O
the	O
rotation	O
and	O
translation	O
(	O
up	O
to	O
a	O
scaling	O
)	O
between	O
the	O
two	O
camera	O
's	O
coordinate	O
systems	O
.	O
</s>
<s>
The	O
diagonal	O
entries	O
of	O
are	O
the	O
singular	O
values	O
of	O
which	O
,	O
according	O
to	O
the	O
internal	O
constraints	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
,	O
must	O
consist	O
of	O
two	O
identical	O
and	O
one	O
zero	O
value	O
.	O
</s>
<s>
Second	O
,	O
it	O
must	O
be	O
shown	O
that	O
this	O
is	O
a	O
matrix	B-Architecture
representation	O
of	O
the	O
cross	O
product	O
for	O
some	O
.	O
</s>
<s>
According	O
to	O
the	O
general	O
properties	O
of	O
the	O
matrix	B-Architecture
representation	O
of	O
the	O
cross	O
product	O
it	O
then	O
follows	O
that	O
must	O
be	O
the	O
cross	O
product	O
operator	O
of	O
exactly	O
one	O
vector	O
.	O
</s>
<s>
Third	O
,	O
it	O
must	O
also	O
need	O
to	O
be	O
shown	O
that	O
the	O
above	O
expression	O
for	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
To	O
be	O
a	O
proper	O
rotation	B-Algorithm
matrix	I-Algorithm
it	O
must	O
also	O
satisfy	O
.	O
</s>
<s>
Let	O
be	O
normalized	O
vector	O
in	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
.	O
</s>
<s>
To	O
summarize	O
,	O
given	O
there	O
are	O
two	O
opposite	O
directions	O
which	O
are	O
possible	O
for	O
and	O
two	O
different	O
rotations	O
which	O
are	O
compatible	O
with	O
this	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
The	O
above	O
determination	O
of	O
and	O
assumes	O
that	O
satisfy	O
the	O
internal	O
constraints	O
of	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
.	O
</s>
<s>
Many	O
methods	O
exist	O
for	O
computing	O
given	O
corresponding	O
normalized	O
image	O
coordinates	O
and	O
,	O
if	O
the	O
essential	B-General_Concept
matrix	I-General_Concept
is	O
known	O
and	O
the	O
corresponding	O
rotation	O
and	O
translation	O
transformations	O
have	O
been	O
determined	O
.	O
</s>
