DICOM PS3.17 2023e - Explanatory Information |
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The Affine Transform Matrix is of the following form.
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.
This transform requires the matrix obey orthonormal transformation properties:
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:
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}.
The matrix above consists of two parts: a rotation and translation as shown below;
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.
The following constraint applies:
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:
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:
where M_{RBWS} is a matrix of type RIGID_SCALE and M_{RB} is a matrix of type RIGID.
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):
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.
DICOM PS3.17 2023e - Explanatory Information |
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