Biomedical Engineering Reference
In-Depth Information
registration where one image is acquired from a single patient, and the other
image is somehow constructed from an image information database obtained
using imaging of many subjects.
1.3
General Registration Theory
In general, registration is the process by which two or more data sets are brought
into alignment. Registration can be defined as
“the process of finding a set
of transformation operations between two or more data sets such that the
overlap between these sets in a certain common space minimizes a certain
optimization criterion”
.
The registration problem can be mathematically represented as follows:
A parametric shape
S
, either a curve segment or a surface, is a vector function,
3
x
:[
a
,
b
]
→
(1.1)
for curves where
a
and
b
are scalars and
2
3
x
:
(1.2)
→
for surfaces. Both curve and surface data sets are usually in the form of, or can
be easily converted to, a set of 2D or 3D points, which represent the most general
form of 2D or 3D curves and surfaces including free-from curves and surfaces.
Let the points in the first, or model data set,
S
, be denoted by
{
x
i
|
i
=
1
,...,
m
}
,
and those in the second, or experimental data set,
S
, be denoted by
{
y
j
|
j
=
1
,...,
n
}
. We want to find a transformation matrix
T
such that when applied to
S
, the distance from each point on the resulting surface and its corresponding
point on the model surface
S
is zero in the noise free case.
For the case of rigid registration (without considering the scaling factor),
the transformation matrix
T
consists of two components: a rotation matrix
R
,
and a translation vector
t
. The objective of registration is to determine
R
and
t
such that the following criterion is minimized
n
d
2
(
Ry
i
+
t
,
S
)
.
F
(
R
,
t
)
=
(1.3)
i
=
1
where
d
(
y
i
,
S
) denotes the distance of point
y
i
to shape
S
.
If we add the scaling factor as a third component, then the matrix
T
rep-
resents a matrix called the
similarity transformation
matrix. In this case the
