Biomedical Engineering Reference
In-Depth Information
3. Speed/Complexity/Resource requirements
4. Clinical use
5. Assumption verification
In some cases the ground-truth information about the transformation is
given, e.g., in phantom studies (see Section 7.7.2). Based on this information
a magnitude and orientation error can be computed to quantify the accuracy.
Definition 9 (E
MAG
) For a transformation t the (averaged) magnitude error
with regard to the ground-truth transformation t
gt
is defined as
log
kt(x)k
kt
gt
(x)k
dx ;
Z
1
jj
E
MAG
(t;t
gt
) :=
(7.21)
where jj denotes
R
1 dx.
The magnitude error measures the (average) error in length. The optimal
value for E
MAG
is 0. E
MAG
is not bounded above.
Definition 10 (E
O
) For a transformation t the (averaged) orientation error
with regard to the ground-truth transformation t
gt
is defined as
Z
1
2jj
ht(x);t
gt
(x)i + 1
E
O
(t;t
gt
) :=
1
q
dx ;
(7.22)
(kt(x)k
2
+ 1)
(kt
gt
(x)k
2
+ 1)
·
where jj denotes
R
1 dx.
The orientation error E
O
indicates deviations regarding the angle and is scaled
between 0 and 1. Again, the optimal value is 0.
An alternative definition of the orientation error in degrees is:
Definition 11 (E
deg
) For a transformation t the (averaged) orientation error
in degrees with regard to the ground-truth transformation t
gt
is defined as
0
1
A
dx : (7.23)
Z
ht(x);t
gt
(x)i + 1
1
jj
E
deg
(t;t
gt
) :=
@
q
arccos
(kt(x)k
2
+ 1)
(kt
gt
(x)k
2
+ 1)
·
E
deg
is scaled between 0
and 180
and the optimal value is 0
.
Since for real life data, usually no ground-truth information about the
transformation is known, the precision and accuracy of the registration result
has to be measured using image intensities. The lack of ground-truth infor-
mation for real data asks for software and hardware phantoms with known
transformations. See Section 7.7 and [70] for a more detailed discussion about
validation.
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