Biomedical Engineering Reference
In-Depth Information
The Procrustes algorithm is used for determining rigid-body or affine trans-
formations. Point landmarks can also be used to determine nonaffine transfor-
mations using algorithms such as the thin-plate spline described in Chapter 13.
3.4.1.2 Errors in Rigid-Body Point Registration
It is clear that the localization of the fiducial points is never perfect. This is
what Fitzpatrick et al.
5
call the fiducial localization error (FLE). The least
square residual itself is called the fiducial registration error (FRE), and this
has a distribution described by Sibson.
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Fitzpatrick et al. stress that what
really matters is not the value of FRE, as the fiducials are not the points of
interest, but what they defined as the target registration error (TRE), i.e., the
error induced by FLE at a given target. Explicitly, if the FLE is
, the rotation
and translations which solve the Procrustes problem are going to be depen-
dent on
:
T
(
R
,
t
). The TRE at the target
x
is then
T
(
x
)
T
(
x
)
, and to
first order in
, this decreases as in where
N
is the number of fiducial
points.
5
This result can be summarized as follows. The squared expectation
value of TRE at position
x
, (coordinates (
x
1
,…,
x
D
) in
D
dimensions) is going
to be (to first order)
1/
N
,
D
D
x
i
2
1
----
1
----
2
2
--------------------
〈
TRE
x
()
〉
FLE
〈
〉
i
2
2
i
j
i
where
are the singular values of the marker locations, and are related to
the distribution of markers with respect to the principal axes of the point
distribution.
3.4.2
Surface Matching
Boundaries, or surfaces, in medical images are frequently more distinct
than landmarks, and various segmentation algorithms can successfully
locate high contrast surfaces. This is especially true of the skin surface—the
boundary between tissue and air—which is high contrast in most imaging
modalities, with the important exception of certain tracers in nuclear med-
icine emission tomography. If equivalent surfaces can be automatically seg-
mented from two images to be combined, then rigid-body registration can
be achieved by fitting the surfaces together. The surface matching algo-
rithms described below are normally only used for rigid-body registration.
An alternative approach that does not require automatic segmentation but
which can be thought of as an interactive version of surface matching, is to
provide the user with an interactive image transformation package that
allows the user to translate and rotate one image with respect to the other,
while displaying the edge-map from the image being transformed on top of
the intensity values of the reference image.
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