1. Index

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2. Coordinate systems

Freesurfer uses at least 4 voxel coordinate systems and 4 "RAS" coordinate systems.

0. Stored in volume file     
     original volume        ========>     RAS ("scanner RAS", c_(r,a,s)!=0 in general)
          |                                |
          | 1. calculated                  | identity
          |                                |
          V       2. calculated            V
   conformed volume         ========>     RAS (to have the same c_(r,a,s) as above)
      L^3 with S mm voxel                |
          |                                |
          | identity                       | 3. calculated (translation)
          |                                |
          V       4. fixed(standard)       V
   conformed volume         ========>    SurfaceRAS with c_(r,a,s) = 0
      L^3 with S mm voxel                

The functional analysis stream uses another coordinate system to map from the src volume.

     original volume        ========>    RAS ("scanner RAS", c_(r,a,s) != 0 in general)    
          |                                |
          | identity                       |  calculated
          V       5. fixed("standard")     V
     original volume        ========>   tkregRAS where c_(r,a,s) = 0 
          |                                |
          | 6. calculated                  |  mri2fmri (registration will give this)
          |                                |
          V        7. fixed("standard")    V                        
     overlay volume         ========>    fRAS where c_(r,a,s) = 0

All these coordinate systems make it a rather difficult task to trace to the original source volume voxel index from surface or functional index. If you can follow the arrows, you can get the necessary transforms easily.

The transform 2 (CORONAL coordinates) is calculated so that the following equation holds. That is, the direction cosine part is fixed, but not the translation part. In this way, the conformed volume always in the CORONAL orientation.

              [-1  0  0 s1][S 0 0 0][L/2]   [c_r]       s1 = c_r + S*L/2
              [ 0  0  1 s2][0 S 0 0][L/2] = [c_a]  ==>  s2 = c_a - S*L/2
              [ 0 -1  0 s3][0 0 S 0][L/2]   [c_s]       s3 = c_s + S*L/2
              [ 0  0  0  1][0 0 0 1][ 1 ]   [ 1 ]

where c_(r,a,s) is from the "scanner RAS". This "scanner RAS" has the physical meaning of "Right-Anterior-Superior" directions of the head.

The transform 4 (surfaceRASFromConformedVoxel) is fixed as

              [-1  0  0  S*L/2][S 0 0 0]
              [ 0  0  1 -S*L/2][0 S 0 0]
              [ 0 -1  0  S*L/2][0 0 S 0]
              [ 0  0  0    1  ][0 0 0 1]

The transform 1 (conformedVoxelFromVoxel) is calculated by (in a matrix sense)

     xform1 = inv(xform2) * xform0  = [1/S 0  0  0][-1  0   0  s1] * xform0
                                      [ 0 1/S 0  0][ 0  0  -1  s3] 
                                      [ 0  0 1/S 0][ 0  1   0 -s2]
                                      [ 0  0  0  1][ 0  0   0  1 ]   

The transform 3(SurfaceRASFromRAS) is calculated by (in a matrix sense). Note that it is independent of conformed voxel size S and the length L:

     xform3 = xform4 * inv(xform2) = [ 1 0 0  -c_r]
                                     [ 0 1 0  -c_a]
                                     [ 0 0 1  -c_s]
                                     [ 0 0 0    1 ]

Because of the xform3 (changing only translation part), it is easy to calculate SurfaceRASFromVoxel (xform3*xform0) and is given by

     SurfaceRASFromVoxel = [  3x3 part   (t1 - c_r)]
                           [  same as    (t2 - c_a)]
                           [  xform0     (t3 - c_s)]
                           [    0            1     ]

where t1,t2,t3 are the translation part of xform0.

The transform 5 and the transform 7 are calculated by the requirement

              [-1  0  0 s1][xsize  0     0    0][width/2 ]   [0]
              [ 0  0  1 s2][  0  ysize   0    0][height/2] = [0]
              [ 0 -1  0 s3][  0    0   zsize  0][depth/2 ]   [0]
              [ 0  0  0  1][  0    0     0    1][    1   ]   [1]

Here the name "RAS" lost the meaning completely. This "RAS" is just to be used for alignment purpose only. The width/height/depth are for the appropriate volume. The reason is that the original volume could be sagittal or horizontal.

3. Relationship between lores and hires surfaceRAS

                conformed vol ------------->     surfRAS         
                          |                         |
                          |                         |   surfRASToRAS
                          V                         V
                 high  res    ------------>        RAS
                          |                         |
                          |                         |    Xfm
                          V                         V
                 low  res     ------------>        RAS
                          |                         |
                          |                         |   RASToSurfRAS
                          V                         V
                conformed vol ------------->     surfRAS        

Therefore, the map from the high res surfaceRAS to the low res surfaceRAS is given by

     highresSurfRASTolowresSurfRAS = RASTosurfRAS(lowres) * Xfm * sufRASToRAS(highres)

which is not equal to Xfm. In fact,

       RASTosurfRAS(lowres) = 1 0 0 -c_r(lowres)        surfRASTORAS(highres) =  1 0 0  c_r(highres)
                              0 1 0 -c_a(lowres)                                 0 1 0  c_a(highres)
                              0 0 1 -c_s(lowres)                                 0 0 1  c_s(highres)
                              0 0 0     1                                        0 0 0    1

or write these as

       RASToSurfRAS(lowres) = [ 1  -C(lowres)]      surfRASToRAS(highres) = [ 1 C(highres)]
                              [ 0   1        ]                              [ 0 1         ]

       Xfm                  = [ R   T ]
                              [ 0   1 ]

Then,

       highresSurfRASTolowresSurfRAS = [ 1  -C(lowres)] x [ R   T ]  x [ 1   C(highres)]
                                       [ 0   1        ]   [ 0   1 ]    [ 0   1         ]

                                     = [ R   R*C(highres)-C(lowres) + T ]
                                       [ 0          1                   ]

The conclusion is that the transform is non-trivial. Even when R = 1 and T = 0 (no rotation), we have

      highresSurfRASTolowresSurfRAS =  [ 1  C(highres)-C(lowres)]
                                       [ 0          1           ]

Therefore, when translating highres vertex positions into lowres vertex position, we must make sure that c_(ras) for highres and lowres must match exactly.

4. Direction cosines

Each volume format stores information in each own way. COR volume (COR-001, ...) stores it in COR-.info file. ["DICOM"] stores it in the file. Note that Analyze file (.img)stores it in .mat file, meanwhile bflort (.bfloat or .bshort) stores it in .bhdr file. For these formats it is essential to have these files.

5. MNI Talairach in tkmedit

["tkmedit"] shows a non-identity Talairach even though talairach.xfm is an identity matrix. The reason is described in [http://www.mrc-cbu.cam.ac.uk/Imaging/Common/mnispace.shtml Cambride Imagers-MNI space]. ["tkmedit"] uses Approach 2 described in the web page.

There are two talairach transformed used in ["tksurfer"]. One is MNI talairach and the other is modified talairach. The modified talairach is done in tksurfer.c conv_initialize():

For Z < 0:

 conv_mnital_to_tal_m_ltz = MatrixIdentity (4, NULL);
  stuff_four_by_four (conv_mnital_to_tal_m_ltz,
                      0.99,       0,     0, 0,
                      0.00,  0.9688, 0.042, 0,
                      0.00, -0.0485, 0.839, 0,
                      0.00,       0,     0, 1);

For Z > 0:
  
  conv_mnital_to_tal_m_gtz = MatrixIdentity (4, NULL);
  stuff_four_by_four (conv_mnital_to_tal_m_gtz,
                      0.99,       0,      0, 0,
                      0.00,  0.9688,  0.046, 0,
                      0.00, -0.0485, 0.9189, 0,
                      0.00,       0,      0, 1);