FreeSurfer 1.0

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Coordinate Systems
class="anchor" id="line-5">
The "official" FreeSurfer Coordinate defintions can be found in the following power-points slides (SEE THESE FIRST):
attachment:fscoordinates.ppt
or PDF:
attachment:fscoordinates.pdf
This also includes a little intro to Affine Transformations.
Coordinate System Overview

class="anchor" id="line-17">

Disclaimer: The stuff below may or may not be correct.

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.

d_hires_surfaceRAS">2.0 Relationship between lores and hires surfaceRAS class="line867">

                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.

FreeSurferTalairach class="line891">FreeSurferTalairach

Direction cosines class="line862">The direction cosines tell you about how the acquired volume x, y, z axes correspond to the R(right), A(anterior), S(superier) axes. You can see this info by running the command, ["mri_info"] <volume>. For example, x_r = -1, x_a = 0, x_s = 0 means that the x axis points to the left (-1) of the R axis (no components along A and S axes). The CORONAL orientation, which is by the way the default freesurfer volume orientation, has x_r = -1, x_a = 0, x_s = 0, y_r = 0, y_a = 0, y_s = -1, z_r = 0, z_a = 1, z_s = 0. This means that the x axis points to the left, the y axis points to the inferior, and the z axis points to the anterior direction. The "c_ras" values specify where the volume center sits in the RAS coordinate system. That is, c_r, c_a. c_s are the RAS coordinate values of a voxel point (width/2, height/2, depth/2). Note that we use the convention of the voxel coordinate such that the center of a voxel corresponds to the integer voxel coordinate position. This is not the usual convention taken by the graphics software like OpenGL. You have to watch out for this 1/2 voxel position shift.

Each volume format stores this direction cosine 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.

MNI Talairach in tkmedit class="line862">["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);

-  ⇤ ← Revision 3 as of 2004-07-30 11:07:54 → 
  Size: 7330
  Editor: DavidTuch
  Comment: Added Index. Added hires->lores surfaceRAS info from TosasFaq.
+   ← Revision 17 as of 2008-02-21 11:15:15 → ⇥
  Size: 9959
  Editor: NickSchmansky
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 1:
-= Index =
+[wiki:Self:FreeSurferWiki top]
 Line 3:
-[[Navigation(children)]]
[[TableOfContents]]
+= FreeSurfer Coordinate Systems =
-Line 6:
+Line 5:
-= Coordinate systems =
+The "official" FreeSurfer Coordinate defintions can be found in the following power-points slides ('''SEE THESE FIRST'''):

 * '''attachment:fscoordinates.ppt'''

or PDF: 

 * '''attachment:fscoordinates.pdf''' 

This also includes a little intro to Affine Transformations.

== 1.0 Coordinate System Overview ==

'''''Disclaimer:''''' The stuff below may or may not be correct.

Freesurfer uses at least 4 voxel coordinate systems and 4 "RAS" coordinate systems.
-Line 8:
+Line 21:
-Freesurfer uses at least 4 voxel coordinate systems and 4 "RAS" coordinate systems. 

0. Stored in volume file
+. Stored in volume file
-Line 17:
+Line 28:
-      L^3 with S mm voxel                |
+        L^3 with S mm voxel                |
-Line 23:
+Line 34:
-      L^3 with S mm voxel
+      L^3 with S mm voxel
}}}
-Line 25:
+Line 37:
+The functional analysis stream uses another coordinate system to map from the src volume.
-Line 26:
+Line 39:
-Functional analysis(fMRI) part uses another coordinate system to map from the src volume. 


     original volume        ========>    RAS ("scanner RAS", c_(r,a,s) != 0 in general)
+{{{
     original volume        ========>    RAS ("scanner RAS", c_(r,a,s) != 0 in general)
-Line 33:
+Line 44:
      original volume        ========>   tkregRAS where c_(r,a,s) = 0
-Line 37:
+Line 48:
           V        7. fixed("standard")    V
-Line 39:
+Line 50:
+}}}
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.
-Line 40:
+Line 53:
-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.
-Line 42:
+Line 55:
-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.
+{{{
-Line 49:
+Line 60:
+}}}
-Line 52:
+Line 63:
-The transform 4 (surfaceRASFromConformedVoxel) is fixed as
+The transform 4 (surface``RAS``From``Conformed``Voxel) is fixed as
-Line 54:
+Line 65:
+{{{
-Line 59:
+Line 70:
+}}}
-Line 60:
+Line 72:
-The transform 1 (conformedVoxelFromVoxel) is calculated by (in a matrix sense)
+The transform 1 (conformed``Voxel``From``Voxel) is calculated by (in a matrix sense)
-Line 62:
+Line 74:
+{{{
     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(Surface``RAS``From``RAS) is calculated by (in a matrix sense). Note that it is independent of conformed voxel size S and the length L:
-Line 63:
+Line 82:
-     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:
+{{{
-Line 75:
+Line 87:
+}}}
Because of the xform3 (changing only translation part), it is easy to calculate Surface``RAS``From``Voxel (xform3*xform0) and is given by
-Line 76:
+Line 90:
-Because of the xform3 (changing only translation part), it is easy to calculate SurfaceRASFromVoxel (xform3*xform0) and is given by
+{{{
-Line 83:
+Line 95:
+}}}
where t1,t2,t3 are the translation part of xform0.
-Line 84:
+Line 98:
-where t1,t2,t3 are the translation part of xform0.
+The transform 5 and the transform 7 are calculated by the requirement
-Line 86:
+Line 100:
-The transform 5 and the transform 7 are calculated by the requirement
+{{{
-Line 93:
+Line 105:
+}}}
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.
-Line 94:
+Line 108:
-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. 

}}}

= Relationship between lores and hires surfaceRAS =
+== 2.0 Relationship between lores and hires surfaceRAS ==
-Line 100:
+Line 110:
                 conformed vol ------------->     surfRAS
-Line 112:
+Line 122:
-                conformed vol ------------->     surfRAS
+                conformed vol ------------->     surfRAS
}}}
Therefore, the map from the high res surfaceRAS to the low res surfaceRAS is given by
-Line 114:
+Line 126:
-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,
-Line 116:
+Line 131:
-     highresSurfRASTolowresSurfRAS = RASTosurfRAS(lowres) * Xfm * sufRASToRAS(highres)

which is not equal to Xfm. In fact,
+{{{
-Line 126:
+Line 136:
+}}}
-Line 127:
+Line 138:
-or write these as
+or write these as
{{{
-Line 134:
+Line 145:
-Then,
+}}}
Then,
{{{
-Line 142:
+Line 153:
+}}}
The conclusion is that the transform is non-trivial. Even when R = 1 and T = 0 (no rotation), we have
-Line 143:
+Line 156:
-The conclusion is that the transform is non-trivial. Even when R = 1 and T = 0 (no rotation), we have
+{{{
-Line 148:
+Line 159:
+}}}
Therefore, when translating highres vertex positions into lowres vertex position, we must make sure that c_(ras) for highres and lowres must match exactly.
-Line 149:
+Line 162:
-Therefore, when translating highres vertex positions into lowres vertex position, we must make sure that c_(ras) for highres and lowres must match exactly.
+== 3.0 FreeSurferTalairach ==
 * FreeSurferTalairach
-Line 151:
+Line 165:
+== 4.0 Direction cosines ==
The direction cosines tell you about how the acquired volume x, y, z axes correspond to the R(right), A(anterior), S(superier) axes.  You can see this info by running the command, ["mri_info"] <volume>.   For example, x_r = -1, x_a = 0, x_s = 0 means that the x axis points to the left (-1) of the R axis (no components along A and S axes).  The '''CORONAL''' orientation, which is by the way the default freesurfer volume orientation, has x_r = -1, x_a = 0, x_s = 0, y_r = 0, y_a = 0, y_s = -1, z_r = 0, z_a = 1, z_s = 0.  This means that the x axis points to the left, the y axis points to the inferior, and the z axis points to the anterior direction.  The "c_ras" values specify where the volume center sits in the RAS coordinate system.  That is, c_r, c_a. c_s are the RAS coordinate values of a voxel point (width/2, height/2, depth/2).  Note that we use the convention of the voxel coordinate such that the center of a voxel corresponds to the
integer voxel coordinate position.   This is not the usual convention taken by the graphics software like OpenGL.  You have to watch out for this 1/2 voxel position shift.

Each volume format stores this direction cosine 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.

== 5.0 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);
-Line 153:
+Line 187:
-= 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.
+{{{
For Z > 0:
-Line 156:
+Line 190:
-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.
+  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);
}}}