[Eeglablist] Coordinate systems question

[Tineke Grent - 't Jong]

Dear Lazlo,

I've been struggling with this question myself, so maybe I can help you out 
a little bit.

 > I am trying to write code to digest electrode positions data for a 
Polhemus Patriot digitizer. I would like to
 > match up results with some standard coordinates, however I am puzzled 
with the various location files available.
 > The problem is with the origins/axes. Neuroscan sample files seem to 
have axes thru preauriculars and nasion.

The Neuroscan 3DSpace-DX program, that can be used with a Polhemus device 
to localize cap-electrodes, indeed uses a coordinate-system based on the 
preauricular points and the nasion (a so-called head-based coordinate 
system). If you open the program (if you have access to it of course) and 
read in one of the available sample.3DD files, you will see a 3D-plot 
(including X-Y-Z- axes) showing all the digitized electrodes. Channels as 
Fp1-Fp2-O1-O2 are indeed not on the equator in digitized data, but some 1-2 
inches above the equator. However, if you save this .3DD file as a BESA 
elp-file (with locations defined in angles from Z-axis and XY-plane), all 
electrode-locations are not only converted into theta and phi (angle) 
coordinates, but also the whole coordinate-system changes its centroid from 
a head-based to a spherical head-model (using an equator-based centroid; 
with the equator going through Fp1-Fp2-O1-O2, at 90 degrees from Cz (0,0) 
at the pole of the sphere). Standard locations are often described using 
this system, because of its value in dipole-fitting. For clearity about 
BESA assumptions and routines, here is how it was described in an old BESA 
(V2.2) manual. It must be said, however, that more recent BESA versions use 
a more ellipsoid (more realistic) than spherical headmodel, and also 
calculate the sphere differently. This, however, does not change the 
placement of the origin of the sphere, that is, it is still higher than the 
origin of the head-based system.

===============================================================================================

"In order to understand the use of the angular coordinates we will refer to 
Cartesian (x, y, z) coordinates with the origin at the center of the head, 
in which the z-axis points to the vertex (through Cz), the x-axis points to 
the right (through T4) and the y-axis points forward (through Fpz).  The 
center of the equivalent spherical head model is approximately at the 
middle of the posterior commissure.  Theta is the azimuth angle with the 
vertical z-axis and phi is the latitude angle in the horizontal x-y-plane 
(counterclockwise). The easiest way to understand the coordinate system is 
to Read Electrode file 1020.elp and to enter the ELECTRODE menu.  You can 
see that  positive values of theta indicate right, negative values left 
hemispheric location.  Within each hemisphere, negative phi indicates 
clockwise, positive phi counterclockwise displacement from the horizontal 
x-axis.

Defining electrodes from digitized locations
Digitized electrode locations form a 3-D cloud that is approximately 
ellipsoid and follows the contours of the subject's head. To convert these 
to a sphere, BESA performs two steps:
a) finds the best fit sphere to match the electrode cloud, and
b) rotates the sphere to match as well as possible the locations of named 
electrodes with its own location table.
In step a, normally all digitized locations are used. However, locations 
labelled with an 'X' are excluded from the fit, allowing the user to 
generate the best fit sphere to selected locations (e.g. the back of the 
head). In step b, only those locations are used for which BESA recognizes 
the label. By selecting or omitting labels, this allows the user to choose 
which coordinates should be used for rotating the sphere to fit the user 
coordinates.
Transformation parameters
Once the electrode coordinates have been converted into internal spherical 
coordinates, three types of information are available:
a) the location of the sphere center and the radius of the sphere (from 
step a above),
b) the angles through which x, y, and z axes were rotated to get from USER 
to BESA coordinates (from step b above), and
c) a generalized transformation matrix that can be used to convert between 
USER and BESA coordinates for any location in and on the head. This matrix 
is used to generate source locations in USER coordinates, or to input 
hypothesized source locations from functional MRI.
The informations in a and b are obtained directly from the fits described 
above. The generalized transformation matrix, X, is obtained in an 
additional step. Let B be the matrix of electrode locations on the BESA 
sphere (n electrodes x 3 coordinates), and U be the matrix of digitized, 
user coordinates (also n x 3). Then the transformation is defined by
                 U = X B
The 3 x 3 matrix X is defined to give the best possible transformation (in 
the least squares sense) between U and B.
Thus, if applied to any location in the head, the transformation matrix 
does its best to perform the same distortion of the spherical to the 
ellipsoidal volume that it would do to transform the electrode locations."

===============================================================================================

 > If I understand correctly, the system should not influence the 
localisation, until it is  coregistered with te
 > fiducial points (LE,RE, Nasion), since the head model is getting 
transformed to the MRI head using the
 > coordinates of the fiducials .Anyway, I would like to see a bit more 
clearly.

True. The fiducials are the most crucial locations. Those can be used to 
transform (using scaling, translation, and rotation) your electrode-data 
into MRI (MNI or Talairach)-space. In sum, trying to match your results to 
BESA-standard locations is not recommended, because those are calculated 
differently and are based on different head-models.

You also might find the information on Robert Oostenveld's website useful.
http://www.smi.auc.dk/~roberto/index.html

Hope this helps.

Best,

Tineke Grent-'t-Jong
Department of Psychopharmacology
University Utrecht
The Netherlands

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