Biomedical Engineering Reference
In-Depth Information
night in a bed that would perfectly fit each visitor. As the visitors discovered
to their cost, however, it was the guest who was altered to fit the bed, rather
than the bed to fit the guest. Short visitors were stretched to fit, and tall visitors
had suitable parts of their bodies cut off so they would fit. The result, it seems,
was invariably fatal. The hero Theseus put a stop to this unpleasant practice
by subjecting Procrustes to his own method. The term “Procrustes” became a
criticism for the practice of unjustifiably forcing data to look like they fit
another set. More recently, Procrustes statistics has lost its negative associa-
tions and is used in shape analysis.
The Procrustes problem is an optimal fitting problem of least square type:
given two configurations of
N
points in
D
dimensions
P
and
Q
{}
p
i
2
{
.
The notation is
P
,
Q
are the
N
-by-
D
matrices whose rows are the coordinates
of the points
p
i
,
q
i
, and
T
(
P
) is the corresponding matrix of transformed
points. The standard case is when
T
is a rigid-body transformation.
4,5
One
can additionally consider scaling, i.e., look for the minimum of similarity
transformations.
4
If
T
is affine, we are faced with a standard least square.
6
q
i
, one seeks the transformation
T
which minimizes
G
(
T
)
T
P
()
Q
}
rigid-body transfor-
mations has known solutions. A matrix representation of the rotational part
can be computed using Singular Value Decomposition (SVD).
4,6-9
First replace
P
and
Q
by their demeaned versions
Solutions:
The classical Procrustes problem, i.e.,
T
{
}
p
i
→
p
i
p
q
i
→
q
i
q
This reduces the problem to the orthogonal Procrustes problem in which we
wish to determine the orthogonal rotation
R
. Central to the problem is the
D
-by-
D
correlation matrix
K
P
T
Q
,
as this matrix quantifies how much
the points in
Q
are “predicted” by points in
P
. If
P
T
,
p
T
p
T
,
[
…
]
is a
p
i
q
i
T
i
K
i
where
K
i
matrix of row vectors (and the same for
Q
),
K
, then:
UDV
t
U
T
diag 1, 1, det
VU
T
K
⇒
R
V
(
(
)
)
UDV
T
where
K
is the SVD of
K
.
It is essential for most medical registration applications that
R
does not
include any reflections. This can be detected from the determinant of
VU
T
,
which should be
1 for a rotation with no reflection, and will be
1 if there
is a reflection. In the above equation,
qRp
.
takes this into account.
This approach has been widely used in medical image registration, first for
intermodality registration,
10,11
and more recently in image-guided surgery.
12
The theory of errors has been advanced in the medical application domain
through the work of Fitzpatrick and colleagues.
5,13
Finally the translation
t
