Biomedical Engineering Reference
In-Depth Information
produced by the UL-TPS (left) and the CL-TPS (right) applied to a rectangular
grid for this experiment. Notice that there is a considerable amount of inverse
consistency error in the UL-TPS algorithm while there is no visually detectable
inverse consistency error produced by the CL-TPS algorithm. The blurring of
the grid is due to bilinear interpolation used to deform the grid images with the
error displacements. Both images are created with the same technique, but the
inverse consistent image needs very little interpolation since there is nearly zero
displacement error.
The minimum and maximum Jacobian values of the forward (reverse)
transformation specify the maximum expansion and contraction of the trans-
formation, respectively. The Jacobian error, calculated as
1
2
|
min
{
Jac
(
h
)
}−
1
1
/
max
{
Jac
(
g
)
}|+
2
|
min
{
Jac
(
g
)
}−
1
/
max
{
Jac
(
h
)
}|
, provides an indirect mea-
sure of the inconsistency between the forward and reverse transformations. The
Jacobian error is zero if the forward and reverse transformations are inverses
of one another, but the converse is not true. Table 6.1 shows that the Jacobian
error was 1000 times smaller for the CL-TPS algorithm compared to the UL-TPS
algorithm.
6.5.2
Landmark and Intensity Registration
The five 2D transverse MRI data sets shown in Fig. 6.7 were used to compare
the performance of the unidirectional landmark (UL-TPS); consistent landmark
(CL-TPS); consistent intensity (CI-TPS); and consistent landmark and intensity
Table 6.1: Comparison between the unidirectional (UL-TPS) and consistent
(CL-TPS) thin-plate spline image registration algorithms. The table columns are
the Experiment, (ICC), transformation Direction (TD), average landmark error
(ALE) in pixels, maximum landmark error (MLE), maximum inverse error
(MIE) in pixels, average inverse error (AIE) in pixels, minimum Jacobian value
(MJ), inverse of the maximum Jacobian value (IJ), and the Jacobian error (JE)
Experiment
ICC
TD
ALE
MLE
AIE
MIE
MJ
IJ
JE
UL-TPS
No
Forward
0
.
010
0
.
016
2
.
2
4
.
1
2
.
4
.
8
.
4
Reverse
0
.
0056
0
.
010
2
.
0
4
.
9
2
.
93
.
2
CL-TPS
Yes
Forward
0
00055
0
0011
0
0028
0
0078
0
28
0
48
0
0012
.
.
.
.
.
.
.
(100 iter.)
Reverse
0
00046
0
00094
0
0024
0
0088
0
48
0
28
.
.
.
.
.
.
