Biomedical Engineering Reference
In-Depth Information
(Dis)similarity measures for monomodal registration tasks are best suited for
motion correction of gated PET data. We thus restrict the discussion in the rest of this
section to these monomodal measures. For a more detailed discussion of similarity
measures for multimodal studies, such as mutual information (which measures the
amount of shared information of two images) or normalized gradient fields (which
measures deviations in the gradients of the input images), we refer to [
95
].
A common dissimilarity measure for monomodal image registration is SSD,
which is introduced in the following Definition. Differences of the reference and
the template image are measured quadratically.
SSD
Definition 2.1 (
D
- Sum of squared differences).
The
sum of squared
differences (
SSD
)
of two images
T
:
Ω
→
R
and
R
:
Ω
→
R
on a domain
3
Ω
⊂
R
is defined as
1
2
SSD
2
dx
D
(
T ,R
)
:
=
Ω
(
T
(
x
)
−R
(
x
))
.
(2.3)
SSD measures the point-wise distances of image intensities. For images with
locally similar intensities the measure gets low. SSD has to be minimized and has
its optimal value at 0.
Large differences in the input images are often induced by noise which conse-
quently leads to a high energy in the data term. L
1
-like distance measures are often
used in optical flow techniques [
19
] to overcome this problem. The L
1
distance of
two images
T
and
R
is defined as
x
x
dx
L
1
(
T ,R
)
:
=
Ω
|T
(
)
−R
(
)
|
.
(2.4)
Using this functional as a data term in Eq. (
2.1
) raises problems as we require
differentiability of all components during optimization. As the absolute value
function is not differentiable at zero, the idea is thus to create a differentiable version
of the L
1
distance by adding a differentiable outer function
ψ
to the difference in
Eq. (
2.4
) with a similar behavior to the absolute value function.
SAD
Definition 2.2 (
D
- Sum of absolute differences).
The (approximated)
sum of absolute differences (SAD)
of two images
T
:
Ω
→
R
and
R
:
Ω
→
R
3
on a domain
Ω
⊂
R
is defined as
SAD
D
(
T ,R
)
=
Ω
ψ
(
T
(
x
)
−R
(
x
))
dx
,
:
(2.5)
(continued)
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