<s>
In	O
statistics	O
,	O
Procrustes	B-Algorithm
analysis	I-Algorithm
is	O
a	O
form	O
of	O
statistical	O
shape	O
analysis	O
used	O
to	O
analyse	O
the	O
distribution	O
of	O
a	O
set	O
of	O
shapes	O
.	O
</s>
<s>
The	O
name	O
Procrustes	O
(	O
)	O
refers	O
to	O
a	O
bandit	O
from	O
Greek	O
mythology	O
who	O
made	O
his	O
victims	O
fit	O
his	O
bed	O
either	O
by	O
stretching	B-Algorithm
their	O
limbs	O
or	O
cutting	O
them	O
off	O
.	O
</s>
<s>
an	O
orthogonal	O
Procrustes	O
problem	O
is	O
a	O
method	O
which	O
can	O
be	O
used	O
to	O
find	O
out	O
the	O
optimal	O
rotation	O
and/or	O
reflection	B-Algorithm
(	O
i.e.	O
,	O
the	O
optimal	O
orthogonal	O
linear	O
transformation	O
)	O
for	O
the	O
Procrustes	B-Algorithm
Superimposition	I-Algorithm
(	O
PS	O
)	O
of	O
an	O
object	O
with	O
respect	O
to	O
another	O
.	O
</s>
<s>
a	O
constrained	O
orthogonal	O
Procrustes	O
problem	O
,	O
subject	O
to	O
det(R )	O
=	O
1	O
(	O
where	O
R	O
is	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
)	O
,	O
is	O
a	O
method	O
which	O
can	O
be	O
used	O
to	O
determine	O
the	O
optimal	O
rotation	O
for	O
the	O
PS	O
of	O
an	O
object	O
with	O
respect	O
to	O
another	O
(	O
reflection	B-Algorithm
is	O
not	O
allowed	O
)	O
.	O
</s>
<s>
In	O
some	O
contexts	O
,	O
this	O
method	O
is	O
called	O
the	O
Kabsch	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
When	O
a	O
shape	O
is	O
compared	O
to	O
another	O
,	O
or	O
a	O
set	O
of	O
shapes	O
is	O
compared	O
to	O
an	O
arbitrarily	O
selected	O
reference	O
shape	O
,	O
Procrustes	B-Algorithm
analysis	I-Algorithm
is	O
sometimes	O
further	O
qualified	O
as	O
classical	O
or	O
ordinary	O
,	O
as	O
opposed	O
to	O
Generalized	B-Algorithm
Procrustes	I-Algorithm
analysis	I-Algorithm
(	O
GPA	O
)	O
,	O
which	O
compares	O
three	O
or	O
more	O
shapes	O
to	O
an	O
optimally	O
determined	O
"	O
mean	O
shape	O
"	O
.	O
</s>
<s>
Procrustes	B-Algorithm
superimposition	I-Algorithm
(	O
PS	O
)	O
is	O
performed	O
by	O
optimally	O
translating	B-Algorithm
,	O
rotating	B-General_Concept
and	O
uniformly	B-Algorithm
scaling	I-Algorithm
the	O
objects	O
.	O
</s>
<s>
The	O
aim	O
is	O
to	O
obtain	O
a	O
similar	O
placement	O
and	O
size	O
,	O
by	O
minimizing	O
a	O
measure	O
of	O
shape	O
difference	O
called	O
the	O
Procrustes	B-Algorithm
distance	I-Algorithm
between	O
the	O
objects	O
.	O
</s>
<s>
This	O
is	O
sometimes	O
called	O
full	O
,	O
as	O
opposed	O
to	O
partial	O
PS	O
,	O
in	O
which	O
scaling	B-Algorithm
is	O
not	O
performed	O
(	O
i.e.	O
</s>
<s>
In	O
some	O
cases	O
,	O
both	O
full	O
and	O
partial	O
PS	O
may	O
also	O
include	O
reflection	B-Algorithm
.	O
</s>
<s>
Reflection	B-Algorithm
allows	O
,	O
for	O
instance	O
,	O
a	O
successful	O
(	O
possibly	O
perfect	O
)	O
superimposition	O
of	O
a	O
right	O
hand	O
to	O
a	O
left	O
hand	O
.	O
</s>
<s>
Thus	O
,	O
partial	O
PS	O
with	O
reflection	B-Algorithm
enabled	O
preserves	O
size	O
but	O
allows	O
translation	O
,	O
rotation	O
and	O
reflection	B-Algorithm
,	O
while	O
full	O
PS	O
with	O
reflection	B-Algorithm
enabled	O
allows	O
translation	O
,	O
rotation	O
,	O
scaling	B-Algorithm
and	O
reflection	B-Algorithm
.	O
</s>
<s>
Optimal	O
translation	O
and	O
scaling	B-Algorithm
are	O
determined	O
with	O
much	O
simpler	O
operations	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
Often	O
,	O
these	O
points	O
are	O
selected	O
on	O
the	O
continuous	O
surface	O
of	O
complex	O
objects	O
,	O
such	O
as	O
a	O
human	O
bone	O
,	O
and	O
in	O
this	O
case	O
they	O
are	O
called	O
landmark	B-General_Concept
points	I-General_Concept
.	O
</s>
<s>
The	O
shape	O
of	O
an	O
object	O
can	O
be	O
considered	O
as	O
a	O
member	O
of	O
an	O
equivalence	O
class	O
formed	O
by	O
removing	O
the	O
translational	B-Algorithm
,	O
rotational	B-General_Concept
and	O
uniform	B-Algorithm
scaling	I-Algorithm
components	O
.	O
</s>
<s>
For	O
example	O
,	O
translational	B-Algorithm
components	O
can	O
be	O
removed	O
from	O
an	O
object	O
by	O
translating	B-Algorithm
the	O
object	O
so	O
that	O
the	O
mean	O
of	O
all	O
the	O
object	O
's	O
points	O
(	O
i.e.	O
</s>
<s>
Now	O
translate	B-Algorithm
these	O
points	O
so	O
that	O
their	O
mean	O
is	O
translated	O
to	O
the	O
origin	O
,	O
giving	O
the	O
point	O
.	O
</s>
<s>
Likewise	O
,	O
the	O
scale	O
component	O
can	O
be	O
removed	O
by	O
scaling	B-Algorithm
the	O
object	O
so	O
that	O
the	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
distance	O
(	O
RMSD	O
)	O
from	O
the	O
points	O
to	O
the	O
translated	O
origin	O
is	O
1	O
.	O
</s>
<s>
Removing	O
the	O
rotational	B-General_Concept
component	O
is	O
more	O
complex	O
,	O
as	O
a	O
standard	O
reference	O
orientation	O
is	O
not	O
always	O
available	O
.	O
</s>
<s>
Fix	O
the	O
reference	O
object	O
and	O
rotate	O
the	O
other	O
around	O
the	O
origin	O
,	O
until	O
you	O
find	O
an	O
optimum	O
angle	O
of	O
rotation	O
such	O
that	O
the	O
sum	O
of	O
the	O
squared	O
distances	O
(	O
SSD	O
)	O
between	O
the	O
corresponding	O
points	O
is	O
minimised	O
(	O
an	O
example	O
of	O
least	B-Algorithm
squares	I-Algorithm
technique	O
)	O
.	O
</s>
<s>
When	O
the	O
object	O
is	O
three-dimensional	O
,	O
the	O
optimum	O
rotation	O
is	O
represented	O
by	O
a	O
3-by-3	O
rotation	B-Algorithm
matrix	I-Algorithm
R	O
,	O
rather	O
than	O
a	O
simple	O
angle	O
,	O
and	O
in	O
this	O
case	O
singular	O
value	O
decomposition	O
can	O
be	O
used	O
to	O
find	O
the	O
optimum	O
value	O
for	O
R	O
(	O
see	O
the	O
solution	O
for	O
the	O
constrained	O
orthogonal	O
Procrustes	O
problem	O
,	O
subject	O
to	O
det(R )	O
=	O
1	O
)	O
.	O
</s>
<s>
The	O
difference	O
between	O
the	O
shape	O
of	O
two	O
objects	O
can	O
be	O
evaluated	O
only	O
after	O
"	O
superimposing	O
"	O
the	O
two	O
objects	O
by	O
translating	B-Algorithm
,	O
scaling	B-Algorithm
and	O
optimally	O
rotating	B-General_Concept
them	O
as	O
explained	O
above	O
.	O
</s>
<s>
This	O
measure	O
is	O
often	O
called	O
Procrustes	B-Algorithm
distance	I-Algorithm
.	O
</s>
<s>
Notice	O
that	O
other	O
more	O
complex	O
definitions	O
of	O
Procrustes	B-Algorithm
distance	I-Algorithm
,	O
and	O
other	O
measures	O
of	O
"	O
shape	O
difference	O
"	O
are	O
sometimes	O
used	O
in	O
the	O
literature	O
.	O
</s>
<s>
However	O
,	O
Generalized	B-Algorithm
Procrustes	I-Algorithm
analysis	I-Algorithm
provides	O
a	O
better	O
method	O
to	O
achieve	O
this	O
goal	O
.	O
</s>
<s>
GPA	O
applies	O
the	O
Procrustes	B-Algorithm
analysis	I-Algorithm
method	O
to	O
optimally	O
superimpose	O
a	O
set	O
of	O
objects	O
,	O
instead	O
of	O
superimposing	O
them	O
to	O
an	O
arbitrarily	O
selected	O
shape	O
.	O
</s>
<s>
Generalized	O
and	O
ordinary	O
Procrustes	B-Algorithm
analysis	I-Algorithm
differ	O
only	O
in	O
their	O
determination	O
of	O
a	O
reference	O
orientation	O
for	O
the	O
objects	O
,	O
which	O
in	O
the	O
former	O
technique	O
is	O
optimally	O
determined	O
,	O
and	O
in	O
the	O
latter	O
one	O
is	O
arbitrarily	O
selected	O
.	O
</s>
<s>
Scaling	B-Algorithm
and	O
translation	O
are	O
performed	O
the	O
same	O
way	O
by	O
both	O
techniques	O
.	O
</s>
<s>
When	O
only	O
two	O
shapes	O
are	O
compared	O
,	O
GPA	O
is	O
equivalent	O
to	O
ordinary	O
Procrustes	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
if	O
the	O
Procrustes	B-Algorithm
distance	I-Algorithm
between	O
mean	O
and	O
reference	O
shape	O
is	O
above	O
a	O
threshold	O
,	O
set	O
reference	O
to	O
mean	O
shape	O
and	O
continue	O
to	O
step	O
2	O
.	O
</s>
<s>
The	O
shape	O
of	O
an	O
object	O
can	O
be	O
considered	O
as	O
a	O
member	O
of	O
an	O
equivalence	O
class	O
formed	O
by	O
taking	O
the	O
set	O
of	O
all	O
sets	O
of	O
k	O
points	O
in	O
n	O
dimensions	O
,	O
that	O
is	O
Rkn	O
and	O
factoring	O
out	O
the	O
set	O
of	O
all	O
translations	O
,	O
rotations	O
and	O
scalings	B-Algorithm
.	O
</s>
<s>
This	O
will	O
give	O
a	O
manifold	B-Architecture
of	O
dimension	O
kn-4	O
.	O
</s>
<s>
the	O
remaining	O
points	O
form	O
the	O
Bookstein	B-Algorithm
coordinates	I-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
common	O
to	O
consider	O
shape	O
and	O
scale	O
that	O
is	O
with	O
translational	B-Algorithm
and	O
rotational	B-General_Concept
components	O
removed	O
.	O
</s>
