The 2D fixed determinant model requires that the real world transformation have a determinant of one (i.e., areas are unchanged by transformation).Since the AIR package allows anisotropic voxels sizes within a given file as well as different voxel sizes between files, these factors must be taken into account when applying a 2D fixed determinant transformation. In the AIR package, the 2D fixed determinant model is parameterized in terms of five parameters.
The 2D fixed determinant transformation for converting from an internal coordinate in the standard file to the corresponding internal coordinate in the reslice file is best expressed as the product of a series of homogenous transformation matrices:
(reslice file internal coordinates)=Zr*D*P*Zs*(standard file internal coordinates)
Zs=
where:
sxoom=(standard file voxel x size) / (smallest standard file voxel size)
syoom=(standard file voxel y size) / (smallest standard file voxel size)
smallest standard file voxel size=min(standard file voxel x, y and z sizes)
This homogenous transformation matrix remaps coordinate locations in the standard file to new coordinates with cubic voxels. The origin remains at (0,0,0).
P=
where:
ssize=min(standard file voxel x, y and z sizes)
rsize=min(reslice file voxel x, y and z sizes)
Coordinate units are modified by this homogenous coordinate transformation matrix to be equivalent to those used in an interpolated version of the reslice file.
D=
where:
a, b, c, d, and f are related to the formal parameters as specified below
E=(1+b*d) / a
Since a*E - b*d=1, this homogenous coordinate transformation matrix performs affine distortions subject to the restriction that areas are not changed.
Zr=
where:
rxoom=(reslice file voxel x size) / (smallest reslice file voxel size)
ryoom=(reslice file voxel y size) / (smallest reslice file voxel size)
smallest reslice file voxel size=min(reslice file voxel x, y and z sizes)
Cubic voxel coordinate locations are remapped to the actual voxel locations in the reslice file by this homogenous coordinate transformation matrix. The origin (0,0,0) remains unchanged.
Since the standard file interpolation matrix Zs is already implicit in the definition of the.air file transformation matrix, it is omitted from the matrix defined above when creating the .air file matrix: Zr*D*P.
Initialization files for the 2D fixed determinant model consist of the following formal parameters in ASCII format:
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 a 2D fixed determinant model is:
This results in the exact centers of the two files being aligned to one another.
There are many different ways to define a 2D fixed determinant transformation. Without explicit equations such as those provided above, the terms a,b,c,d and f are ambiguous. Aside from
c and f which specify x- and y-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.
Modified: July 21, 2002