P Transforms and Mappings (Informative) (Current)
Prev		Next

P Transforms and Mappings (Informative)

The Affine Transform Matrix is of the following form.

Equation P-1.

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

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:

Equation P-2.

for all combinations of j = 1,2,3 and k = 1,2,3 where δ = 1 for i = j and zero otherwise.

The expansion into non-matrix equations is:
- M₁₁ M₁₁ + M₂₁ M₂₁ + M₃₁ M₃₁ = 1 where j = 1, k = 1
- M₁₁ M₁₂ + M₂₁ M₂₂ + M₃₁ M₃₂ = 0 where j = 1, k = 2
- M₁₁ M₁₃ + M₂₁ M₂₃ + M₃₁ M₃₃ = 0 where j = 1, k = 3
- M₁₂ M₁₁ + M₂₂ M₂₁ + M₃₂ M₃₁ = 0 where j = 2, k = 1
- M₁₂ M₁₂ + M₂₂ M₂₂ + M₃₂ M₃₂ = 1 where j = 2, k = 2
- M₁₂ M₁₃ + M₂₂ M₂₃ + M₃₂ M₃₃ = 0 where j = 2, k = 3
- M₁₃ M₁₁ + M₂₃ M₂₁ + M₃₃ M₃₁ = 0 where j = 3, k = 1
- M₁₃ M₁₂ + M₂₃ M₂₂ + M₃₃ M₃₂ = 0 where j = 3, k = 2
- M₁₃ M₁₃ + M₂₃ M₂₃ + M₃₃ M₃₃ = 1 where j = 3, k = 3
The Frame of Reference Transformation Matrix ^AM_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.

Equation P-3.

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

Rotation:

Equation P-4.

Translation:

Equation P-5.

The first column [M₁₁,M₂₁,M₃₁ ] are the direction cosines (projection) of the X-axis of RCS_B with respect to RCS_A . The second column [M₁₂,M₂₂,M₃₂] are the direction cosines (projection) of the Y-axis of RCS_B with respect to RCS_A. The third column [M₁₃,M₂₃,M₃₃] are the direction cosines (projection) of the Z-axis of RCS_B with respect to RCS_A. The fourth column [T₁,T₂,T₃] 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:

Equation P-6.

for all combinations of j = 1,2,3 and k = 1,2,3 where δ = 1 for i=j and zero otherwise.

The expansion into non-matrix equations is:
- M₁₁ M₁₁ + M₂₁ M₂₁ + M₃₁ M₃₁ = S₁ ² where j = 1, k = 1
- M₁₁ M₁₂ + M₂₁ M₂₂ + M₃₁ M₃₂ = 0 where j = 1, k = 2
- M₁₁ M₁₃ + M₂₁ M₂₃ + M₃₁ M₃₃ = 0 where j = 1, k = 3
- M₁₂ M₁₁ + M₂₂ M₂₁ + M₃₂ M₃₁ = 0 where j = 2, k = 1
- M₁₂ M₁₂ + M₂₂ M₂₂ + M₃₂ M₃₂ = S₂ ² where j = 2, k = 2
- M₁₂ M₁₃ + M₂₂ M₂₃ + M₃₂ M₃₃ = 0 where j = 2, k = 3
- M₁₃ M₁₁ + M₂₃ M₂₁ + M₃₃ M₃₁ = 0 where j = 3, k = 1
- M₁₃ M₁₂ + M₂₃ M₂₂ + M₃₃ M₃₂ = 0 where j = 3, k = 2
- M₁₃ M₁₃ + M₂₃ M₂₃ + M₃₃ M₃₃ = S₃ 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:

Equation P-7.

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):

Equation P-8.

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.

Prev		Next
O.6 Spatial Fiducials	Home	Q Breast Imaging Report (Informative)