Biomedical Engineering Reference
In-Depth Information
been stated as: Find the transformation
h
:
→
that maps the template im-
age volume
T
into correspondence with the target image volume
S
. Alterna-
tively, the problem can be stated as: Find the transformation
g
:
→
that
transforms
S
into correspondence with
T
. For inverse consistent registration,
the previous two statements are combined into a single problem and restated
as:
Problem Statement:
Jointly estimate the transformations
h
and
g
such that
h
maps
T
to
S
and
g
maps
S
to
T
such that the inverse
consistency constraint
||
h
−
g
−
1
||+||
g
−
h
−
1
||
is minimized.
The image volumes T and S can be of any dimension such as 1D, 2D, 3D, 4D, or
higher dimensional and in general can be multivalued. Image data sets may rep-
resent information such as anatomical structures like the brain, heart, lungs,
etc., or could represent symbolic information such as structure names, ob-
ject features, curvature, brain function, etc., or could represent image frames
in movies that need to be matched for morphing, interpolating transitional
frames, etc., or images of a battlefield with tanks, artillery, etc., or images col-
lected from satellites or robots that need to be fused into a composite image,
etc.
The transformations are vector-valued functions that map the image coor-
dinate system
to itself, i.e.,
h
:
→
and
g
:
→
. Regularization con-
straints are placed on
h
and
g
so that they preserve topology. Throughout
it is assumed that
h
(
x
)
=
x
+
u
(
x
),
h
−
1
(
x
)
=
x
+
u
(
x
),
g
(
x
)
=
x
+
w
(
x
) and
g
−
1
(
x
)
=
x
+
w
(
x
) where
h
(
h
−
1
(
x
))
=
x
and
g
(
g
−
1
(
x
))
=
x
. The vector-valued
functions
u
,
w
,
u
, and
w
are called displacement fields since they define the
transformation in terms of a displacement from a location
x
. All of the func-
tions
h
,
g
,
h
−
1
,
g
−
1
,
u
,
u
,
w
, and
w
are (3
×
1) vector-valued functions defined on
the
.
Registration is defined using a symmetric similarity cost function that de-
scribes the distance between the transformed template
T
◦
h
and target
S
, and
the distance between the transformed target
S
◦
g
and template
T
. To ensure the
desired properties, the transformations
h
and
g
are jointly estimated by minimiz-
ing the similarity cost function while satisfying regularization constraints and
inverse transformation consistency constraints. The regularization constraints
can be enforced on the transformations by constraining them to satisfy the laws
of continuum mechanics [25].
