• transformation matrix
  • affine parameter matrix
• representation in .air files
• initialization files
• default initialization
• integration with other registration and display packages
• Programs that incorporate the affine transformation model:
  • alignlinear

Transformation Matrix

The affine model requires that lines that are parallel before transformation remain parallel after transformation. In the AIR package, the affine model is parameterized in terms of twelve parameters defined below. These parameters do not involve explicit definition of rotations, etc.

The affine transformation for converting from an internal coordinate in the standard file to the corresponding internal coordinate in the reslice file is best expressed as a homogenous transformation matrices:

(reslice file internal coordinates)=A*(standard file internal coordinates)

where

• A specifies the parameters of the affine transformation

Affine parameter matrix

A=



where:

a,b,c,d,e,f,g,h,i,j,k and m are independent parameters

Representation in .air files

Since the .air file format requires that the matrix stored there represents a transformation that will convert coordinates from a version of the standard file that has been interpolated to cubic voxels, the matrix A must be modified before storage in a .air file.

This is accomplished by:

• dividing a, e and i by sxoom=(standard file voxel x size) / (smallest standard file voxel size)
• dividing b, f and j by syoom=(standard file voxel y size) / (smallest standard file voxel size)
• dividing c, g and k by szoom=(standard file voxel z size) / (smallest standard file voxel size)

Representation in initialization files

Initialization files for the affine model consist of the following in ASCII format:

• a
• b
• c
• d
• e
• f
• g
• h
• i
• j
• k
• m

Default initialization

The default initialization for this model was modified in AIR 5.1 to make it identical to the default for other models even when voxel sizes in the two files are different

If no initialization file is specified, the default initialization for programs using an affine model is:

• a =sx_size / rx_size
• b=0
• c =0
• d =((rx_dim-1) - (sx_dim-1)*(sx_size / rx_size)) / 2
• e=0
• f =sy_size / ry_size
• g =0
• h =((ry_dim-1) - (sy_dim-1)*(sy_size / ry_size)) / 2
• i=0
• j =0
• k =sz_size / rz_size
• m =((rz_dim-1) - (sz_dim-1)*(sz_size / rz_size)) / 2

where:

• sx_size is the voxel x size of the standard file
• sy_size is the voxel y size of the standard file
• sz_size is the voxel z size of the standard file
• rx_size is the voxel x size of the reslice file
• ry_size is the voxel y size of the reslice file
• rz_size is the voxel z size of the reslice file
• sx_dim is the x dimension of the standard file
• sy_dim is the y dimension of the standard file
• sz_dim is the z dimension of the standard file
• rx_dim is the x dimension of the reslice file
• ry_dim is the y dimension of the reslice file
• rz_dim is the z dimension of the reslice file

This results in the exact centers of the two files being aligned to one another.

Other registration and display packages

There are many different ways to define an affine transformation. Without explicit equations such as those provided above, the terms a,b,c,d,e,f,g,h,i,j,k and m are ambiguous. Aside from d, h and m which specify x-, y- and z-axis shifts respectively in a version of the reslice file that has been interpolated to cubic voxels and which are expressed in units of cubified voxels, the physical meanings of the other parameters are difficult to interpret in isolation.

If your alternative package generates a linear algebraic transformation matrix of its own, don't forget that transformation matrices are dependent upon the coordinate system used and that the AIR internal coordinate system used to define transformation matrices may differ from that of your alternative package.