# C.20.2.1 Registration Module Attribute Descriptions

## C.20.2.1.1 Frame of Reference Transformation Matrix

The Frame of Reference Transformation Matrix (3006,00C6)
^{
A
}
M
_{
B
}
describes how to transform a point (
^{
B
}
x,
^{
B
}
y,
^{
B
}
z) with respect to RCS
_{
B
}
into (
^{
A
}
x,
^{
A
}
y,
^{
A
}
z) with respect to RCS
_{
A
}
according to the equation below.

[pic]

The Matrix Registration is expressible as multiple matrices, each in a separate item of the Matrix Sequence (0070,030A). The equation below specifies the order of the matrix multiplication where
*
M
*
*
*_{
1
}
,
*
M
*
*
*_{
2
}
and
*
M
*
*
*_{
3
}
are the first, second and third items in the sequence.

[pic]

where [pic] = [pic]

Registration often involves two or more RCS, each with a corresponding Frame of Reference Transformation Matrix. For example, another Frame of Reference Transformation Matrix
^{
A
}
M
_{
C
}
can describe how to transform a point (
*
*^{
C
}
*
x,
*
*
*^{
C
}
*
y,
*
*
*^{
C
}
*
z
*
) with respect to RCS
*
*_{
C
}
into (
*
*^{
A
}
*
x,
*
*
*^{
A
}
*
y,
*
*
*^{
A
}
*
z
*
) with respect to RCS
_{
A
}
. It is straightforward to find the Frame of Reference Transformation Matrix
*
*^{
B
}
M
*
*_{
C
}
that describes how to transform the point (
*
*^{
C
}
*
x,
*
*
*^{
C
}
*
y,
*
*
*^{
C
}
*
z
*
) with respect to RCS
_{
C
}
into the point (
*
*^{
B
}
*
x,
*
*
*^{
B
}
*
y,
*
*
*^{
B
}
*
z
*
) with respect to RCS
_{
B.
}
The solution is to invert
^{
A
}
M
_{
B
}
and multiply by
^{
A
}
M
_{
C
}
, as shown below:

[pic]

## C.20.2.1.2 Frame of Reference Transformation Matrix Type

There are three types of Registration Matrices:

RIGID: This is a registration involving only translations and rotations. Mathematically, the matrix is constrained to be orthonormal and describes six degrees of freedom: three translations, and three rotations.

RIGID_SCALE: This is a registration involving only translations, rotations and scaling. Mathematically, the matrix is constrained to be orthogonal and describes nine degrees of freedom: three translations, three rotations and three scales. This type of transformation is sometimes used in atlas mapping.

AFFINE: This is a registration involving translations, rotations, scaling and shearing. Mathematically, there are no constraints on the elements of the Frame of Reference Transformation Matrix, so it conveys twelve degrees of freedom. This type of transformation is sometimes used in atlas mapping.

See the PS 3.17 Annex on Transforms and Mappings for more detail.