Annex P Transforms and Mappings (Informative)

The Homogenous Transform Matrix is of the following form.

[pic]

This matrix requires the bottom row to be [0 0 0 1].

The matrix can be of type: RIGID, RIGID_SCALE and AFFINE. These different types represent different conditions on the allowable values for the matrix elements.

RIGID:

This transform requires the matrix obey orthonormal transformation properties:

[pic] for all combinations of j = 1,2,3 and k = 1,2,3 where delta = 1 for i = j and zero otherwise.

The expansion into non-matrix equations is:

M 11 M 11 + M 21 M 21 + M 31 M 31 = 1 where j = 1, k = 1

M 11 M 12 + M 21 M 22 + M 31 M 32 = 0 where j = 1, k = 2

M 11 M 13 + M 21 M 23 + M 31 M 33 = 0 where j = 1, k = 3

M 12 M 11 + M 22 M 21 + M 32 M 31 = 0 where j = 2, k = 1

M 12 M 12 + M 22 M 22 + M 32 M 32 = 1 where j = 2, k = 2

M 12 M 13 + M 22 M 23 + M 32 M 33 = 0 where j = 2, k = 3

M 13 M 11 + M 23 M 21 + M 33 M 31 = 0 where j = 3, k = 1

M 13 M 12 + M 23 M 22 + M 33 M 32 = 0 where j = 3, k = 2

M 13 M 13 + M 23 M 23 + M 33 M 33 = 1 where j = 3, k = 3

The Frame of Reference Transformation Matrix 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 .

[pic]

The matrix above consists of two parts: a rotation and translation as shown below;

Rotation: [pic] Translation: [pic]

The first column [M 11 ,M 21 ,M 31 ] are the direction cosines (projection) of the X-axis of RCS B with respect to RCS A . The second column [M 12 ,M 22 ,M 32 ] are the direction cosines (projection) of the Y-axis of RCS B with respect to RCS A. The third column [M 13 ,M 23 ,M 33 ] are the direction cosines (projection) of the Z-axis of RCS B with respect to RCS A. The fourth column [T 1 ,T 2 ,T 3 ] is the origin of RCS B with respect to RCS A .

There are three degrees of freedom representing rotation, and three degrees of freedom representing translation, giving a total of six degrees of freedom.

RIGID_SCALE

The following constraint applies:

[pic] for all combinations of j = 1,2,3 and k = 1,2,3 where delta = 1 for i=j and zero otherwise.

The expansion into non-matrix equations is:

M 11 M 11 + M 21 M 21 + M 31 M 31 = S 1 2 where j = 1, k = 1

M 11 M 12 + M 21 M 22 + M 31 M 32 = 0 where j = 1, k = 2

M 11 M 13 + M 21 M 23 + M 31 M 33 = 0 where j = 1, k = 3

M 12 M 11 + M 22 M 21 + M 32 M 31 = 0 where j = 2, k = 1

M 12 M 12 + M 22 M 22 + M 32 M 32 = S 2 2 where j = 2, k = 2

M 12 M 13 + M 22 M 23 + M 32 M 33 = 0 where j = 2, k = 3

M 13 M 11 + M 23 M 21 + M 33 M 31 = 0 where j = 3, k = 1

M 13 M 12 + M 23 M 22 + M 33 M 32 = 0 where j = 3, k = 2

M 13 M 13 + M 23 M 23 + M 33 M 33 = S 3 2 where j = 3, k = 3

The above equations show a simple way of extracting the spatial scaling parameters Sj from a given matrix. The units of S j 2 is the RCS unit dimension of one millimeter.

This type can be considered a simple extension of the type RIGID. The RIGID_SCALE is easily created by pre-multiplying a RIGID matrix by a diagonal scaling matrix as follows:

[pic]

where M RBWS is a matrix of type RIGID_SCALE and M RB is a matrix of type RIGID.

AFFINE:

No constraints apply to this matrix, so it contains twelve degrees of freedom. This type of Frame of Reference Transformation Matrix allows shearing in addition to rotation, translation and scaling.

For a RIGID type of Frame of Reference Transformation Matrix, the inverse is easily computed using the following formula (inverse of an orthonormal matrix):

annex. [pic]

For RIGID_SCALE and AFFINE types of Registration Matrices, the inverse cannot be calculated using the above equation, and must be calculated using a conventional matrix inverse operation.