<s>
The	O
Kabsch	B-Algorithm
algorithm	I-Algorithm
,	O
named	O
after	O
Wolfgang	O
Kabsch	O
,	O
is	O
a	O
method	O
for	O
calculating	O
the	O
optimal	O
rotation	B-Algorithm
matrix	I-Algorithm
that	O
minimizes	O
the	O
RMSD	B-General_Concept
(	O
root	B-General_Concept
mean	I-General_Concept
squared	I-General_Concept
deviation	O
)	O
between	O
two	O
paired	O
sets	O
of	O
points	O
.	O
</s>
<s>
It	O
is	O
useful	O
in	O
graphics	O
,	O
cheminformatics	O
to	O
compare	O
molecular	O
structures	O
,	O
and	O
also	O
bioinformatics	O
for	O
comparing	O
protein	O
structures	O
(	O
in	O
particular	O
,	O
see	O
root-mean-square	B-General_Concept
deviation	I-General_Concept
(	O
bioinformatics	O
)	O
)	O
.	O
</s>
<s>
The	O
algorithm	O
only	O
computes	O
the	O
rotation	B-Algorithm
matrix	I-Algorithm
,	O
but	O
it	O
also	O
requires	O
the	O
computation	O
of	O
a	O
translation	O
vector	O
.	O
</s>
<s>
When	O
both	O
the	O
translation	O
and	O
rotation	O
are	O
actually	O
performed	O
,	O
the	O
algorithm	O
is	O
sometimes	O
called	O
partial	O
Procrustes	B-Algorithm
superimposition	I-Algorithm
(	O
see	O
also	O
orthogonal	O
Procrustes	O
problem	O
)	O
.	O
</s>
<s>
Each	O
set	O
of	O
points	O
can	O
be	O
represented	O
as	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
algorithm	O
works	O
in	O
three	O
steps	O
:	O
a	O
translation	O
,	O
the	O
computation	O
of	O
a	O
covariance	O
matrix	B-Architecture
,	O
and	O
the	O
computation	O
of	O
the	O
optimal	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
second	O
step	O
consists	O
of	O
calculating	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
In	O
matrix	B-Architecture
notation	O
,	O
</s>
<s>
which	O
is	O
a	O
cross-covariance	O
matrix	B-Architecture
when	O
and	O
are	O
seen	O
as	O
data	O
matrices	O
.	O
</s>
<s>
First	O
,	O
calculate	O
the	O
SVD	O
of	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
optimal	O
rotation	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
expressed	O
in	O
terms	O
of	O
quaternions	O
.	O
</s>
