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# os_meg

## (Toolbox/os_meg.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    G = os_meg(L,Channel,Param,Order,imegsens,irefsens,Verbose);

### Help Text

    OS_MEG - Calculate the (overlapping) sphere models for MEG
     function G = os_meg(L,Channel,Param,Order,imegsens,irefsens,Verbose);
     function G = os_meg(L,Channel,Param,Order,imegsens,irefsens);
     function G = os_meg(L,Channel,Param,Order);
     Modified for CME, not MME, as Order = 1.
     Calculate the magnetic field, spherical head, arbitrary orientation

     INPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     L : a 3 x nL array, each column a source location (x y z coordinates); nL sources
     Channel is a BrainStorm channel structure
     Param an array of structures:
     For every channel i:
     Param(i).Center = a vector of the x, y, z locations for the sphere model
     (assume the same center for every sphere for the classical spherical head model);

     Param(i).Radii = a vector containing the radius in meters of the concentric spheres  ;
     Can be a scalar for the single-sphere head model

     Param(i).EEGType = [] % Leave it empty for MEG;

     Order: Defines the source order for which to compute the forward problem:
      -1 current dipole
       0 focal(magnetic) dipole
       1 1st order current multipole

     imegsens is the index to the MEG sensors in the Channel information
     irefsens is the index to the MEG reference sensors in the Channel
     if imegsens (irefsens) is not given, then routine (expensively)
     searches the Channel structure for 'MEG' ('MEG REF') values

     Verbose : toggle verbose mode (1 is default)

     OUTPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     G is the gain matrix: each column is the forward field of each source

### Cross-Reference Information

This function calls

    blk_diag                           C:\BrainStorm_2001\Toolbox\blk_diag.m
    blk_lex                            C:\BrainStorm_2001\Toolbox\blk_lex.m
    bst_message_window                 C:\BrainStorm_2001\Toolbox\bst_message_window.m
    colnorm                            C:\BrainStorm_2001\Toolbox\colnorm.m
    cross_mat                          C:\BrainStorm_2001\Toolbox\cross_mat.m
    good_channel                       C:\BrainStorm_2001\Toolbox\good_channel.m
    inorcol                            C:\BrainStorm_2001\Toolbox\inorcol.m
    mby3check                          C:\BrainStorm_2001\Toolbox\mby3check.m
    norcol                             C:\BrainStorm_2001\Toolbox\norcol.m
    vec                                C:\BrainStorm_2001\Toolbox\vec.m

This function is called by

    bst_headmodeler                    C:\BrainStorm_2001\Toolbox\bst_headmodeler.m
    montreal_data_create               C:\BrainStorm_2001\Toolbox\montreal_data_create.m
    simulator                          C:\BrainStorm_2001\Toolbox\simulator.m

### Listing of function C:\BrainStorm_2001\Toolbox\os_meg.m

    function G = os_meg(L,Channel,Param,Order,imegsens,irefsens,Verbose);
    %OS_MEG - Calculate the (overlapping) sphere models for MEG
    % function G = os_meg(L,Channel,Param,Order,imegsens,irefsens,Verbose);
    % function G = os_meg(L,Channel,Param,Order,imegsens,irefsens);
    % function G = os_meg(L,Channel,Param,Order);
    % Modified for CME, not MME, as Order = 1.
    % Calculate the magnetic field, spherical head, arbitrary orientation
    %
    % INPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %
    % L : a 3 x nL array, each column a source location (x y z coordinates); nL sources
    % Channel is a BrainStorm channel structure
    % Param an array of structures:
    % For every channel i:
    % Param(i).Center = a vector of the x, y, z locations for the sphere model
    % (assume the same center for every sphere for the classical spherical head model);
    %
    % Param(i).Radii = a vector containing the radius in meters of the concentric spheres  ;
    % Can be a scalar for the single-sphere head model
    %
    % Param(i).EEGType = [] % Leave it empty for MEG;
    %
    % Order: Defines the source order for which to compute the forward problem:
    %  -1 current dipole
    %   0 focal(magnetic) dipole
    %   1 1st order current multipole
    %
    % imegsens is the index to the MEG sensors in the Channel information
    % irefsens is the index to the MEG reference sensors in the Channel
    % if imegsens (irefsens) is not given, then routine (expensively)
    % searches the Channel structure for 'MEG' ('MEG REF') values
    %
    % Verbose : toggle verbose mode (1 is default)
    %
    % OUTPUTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %
    % G is the gain matrix: each column is the forward field of each source

    %<autobegin> ---------------------- 09-Jul-2004 22:17:14 -----------------------
    % --------- Automatically Generated Comments Block Using AUTO_COMMENTS ---------
    %
    % CATEGORY: Forward Modeling
    %
    % Alphabetical list of external functions (non-Matlab):
    %   toolbox\blk_diag.m
    %   toolbox\blk_lex.m
    %   toolbox\bst_message_window.m
    %   toolbox\colnorm.m
    %   toolbox\cross_mat.m
    %   toolbox\good_channel.m
    %   toolbox\inorcol.m
    %   toolbox\mby3check.m
    %   toolbox\norcol.m
    %   toolbox\vec.m
    %
    % Subfunctions in this file, in order of occurrence in file:
    %   c = cross(a,b);
    %   k = kronmat(a,b);
    %   G = sarvas(L,P,Order);
    %   G = sarvas_dipole(L,P,Order);
    %   D = sarvas_partial(L,P);
    %
    % At Check-in: $Author: Mosher $  $Revision: 28 $  $Date: 7/09/04 8:42p $
    %
    % This software is part of BrainStorm Toolbox Version 2.0 (Alpha) 09-Jul-2004
    %
    % Principal Investigators and Developers:
    % ** Richard M. Leahy, PhD, Signal & Image Processing Institute,
    %    University of Southern California, Los Angeles, CA
    % ** John C. Mosher, PhD, Biophysics Group,
    %    Los Alamos National Laboratory, Los Alamos, NM
    % ** Sylvain Baillet, PhD, Cognitive Neuroscience & Brain Imaging Laboratory,
    %    CNRS, Hopital de la Salpetriere, Paris, France
    %
    % See BrainStorm website at http://neuroimage.usc.edu for further information.
    %
    % Copyright (c) 2004 BrainStorm by the University of Southern California
    % This software distributed  under the terms of the GNU General Public License
    % as published by the Free Software Foundation. Further details on the GPL
    % license can be found at http://www.gnu.org/copyleft/gpl.html .
    %
    % FOR RESEARCH PURPOSES ONLY. THE SOFTWARE IS PROVIDED "AS IS," AND THE
    % UNIVERSITY OF SOUTHERN CALIFORNIA AND ITS COLLABORATORS DO NOT MAKE ANY
    % WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF
    % MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, NOR DO THEY ASSUME ANY
    % LIABILITY OR RESPONSIBILITY FOR THE USE OF THIS SOFTWARE.
    %<autoend> ------------------------ 09-Jul-2004 22:17:14 -----------------------

    % NOT OPTIMIZED FOR BRAINSTORM STRUCTURE.
    %  But it's a whole lot better now (October 99 version).

    % John C. Mosher, Ph.D.

    % ------------------------ HISTORY -------------------------------------------
    % 29-Oct-1999 JCM tried to speed up cross products, which are a substantial portion of
    %                 the total calculation time, for a lot of dipoles.
    %                 Also added the waitbar feature for big calculations.
    %
    % 28-Nov-2000 SB  Modified for updated 3rd-order gradient correction of CTF data
    % 24-Jan-2002 JCM in Paris, modified code to run old partial gradient calculation
    %                 in this code
    % 20-Feb-2002 JCM fixed missing translocation of the partial gradient location
    %                 information. Put center shifting into the partial_gradient
    %                 code itself. Major change to gradient interface to handle
    %                 head center in different location.
    % 20-Feb-2002 SB  Fixed different version to handle case of m x 3 L matrix,
    %                 force L to be 3 x N or return error.
    % 12-Mar-2002 JCM Force L to be 3 x N or return error.
    % 19-Jun-2002 JCM Realized version differences between toolbox and developers
    %                 toolbox, carefully merged two version together into toolbox,
    %                 updated comments, history, switched waitbar to message window
    % 02-Jul-2002  SB Fixed display using the 'overwrite' option of bst_message_window
    % ...........     Added progression messages during computation of large CME gain matrices
    % 03-Sep-2002  SB Fixed bug when Channel contains magnetometers and when head center is at origin ([0 0 0])
    % 20-Nov-2002  SB Updated computation with CTF 3rd order gradient correction
    %                 Now must pass the entire Channel structure to OS_MEG
    %                 MEG and MEG REF channels are extracted using GOOD_CHANNEL
    %                 Lightly altered display
    % 21-Oct-2003 JCM added optional imegsens and irefsens indexed inputs
    % 09-Mar-2004 SB  added Verbose argument
    % -----------------------------------------------------------------------------

    % Which verbose mode ?
    if 0  % deprecated code - get a warning in Matlab 6.5.0 because 'Verbose' is an argument and has to be declared as GLOBAL before first use
        global Verbose % Pass Verbose to subfunctions
    end

    if nargin == 5
        Verbose = imegsens;
        clear imegsens
    elseif nargin < 7 ^ ~exist('Verbose','var')
        Verbose = 1;% Default
    end

    % Indices of MEG and MEG REF channels in Channel structure array:
    if ~exist('imegsens','var') | isempty(imegsens),
       imegsens = good_channel(Channel,[],'MEG');
    end

    if isempty(imegsens)
        error('No MEG channels available')
    end

    if ~exist('irefsens','var'),
       irefsens = good_channel(Channel,[],'MEG REF');
    end

    if ~isempty(irefsens)
        ChannelRef = Channel(irefsens); % Fill out a specific channel structure with MEG REF channels only
        [ChannelRef(:).Type] = deal('MEG'); % Ref channels will be treated as regular MEG sensors when Sarvas is called to compute their forward fields
        refFlag = 1;
    else
        refFlag = 0;
    end
    Channel = Channel(imegsens); % Keep Channel as a MEG-channel only channel set.
    Param = Param(imegsens);

    % ----------------------------

    NumCoils = size(Channel(1).Loc,2); % number of coils, assumed same for all channels
    NumSensors = length(Channel); % how many channels

    if(size([Channel.Loc],2) ~= NumSensors*NumCoils),
       errordlg({'Sorry, OS_MEG not equipped to handle different number of coils'});
       G = [];
       return
    end

    % load up the old parameter array
    % P.sensor is 3 x nR,each column a sensor location
    % P.orient is 3 x nR, the sensor orientation
    % P.center is 3 x nR, the sphere center for each sensor

    [P(1:NumCoils)] = deal(struct('sensor',zeros(3,NumSensors),...
       'orient',zeros(3,NumSensors),...
       'center',zeros(3,NumSensors),'weight',[]));
    AllLocs = [Channel.Loc]; % remap all Locations
    AllLocs = reshape(AllLocs,NumCoils*3,size(AllLocs,2)/NumCoils);
    AllOrient = [Channel.Orient];
    AllOrient = AllOrient*inorcol(AllOrient);
    AllOrient= reshape(AllOrient,NumCoils*3,size(AllOrient,2)/NumCoils);
    AllWeight = [Channel.Weight];
    AllWeight = reshape(AllWeight(:),NumCoils,length(AllWeight(:))/NumCoils);

    for j = 1:NumCoils,
       P(j).sensor = AllLocs([-2:0]+j*3,:);
       P(j).orient = AllOrient([-2:0]+j*3,:);
       P(j).weight = AllWeight(j,:);
       P(j).center = [Param.Center]; %one center for both coils
    end

    [m,n] = size(L);
    if(m~=3), % should be 3 x m
       % Old Mosher convention was to give L as m x 3
       % Newer Mathworks convention is for sets of vectors to
       %  be 3 x m (except paradoxically the "patch" command).
       % Error to user, force correction in calling code.
       if Verbose
           bst_message_window('wrap',{'LOCATION GIVEN AS M X 3.',...
                   'Please adjust calling code to handle new convention'});
       end

       error('Matrix not given as 3 x n. Correct calling code');
    end

    G = 0;
    for i = 1:length(P),
       G = G + sarvas(L,P(i),Order); % local call below
    end

    % is there special reference channel considerations?
    %  See Channel.mat structure description in the ParameterDescriptions document.
    if (refFlag) % Gradient correction is available as well

        % read the CTF reference channel information
        meanCenter = mean([Param.Center],2); % mean head center of all of the channels
        % create a temporary parameters file with the same center for all reference channels.
        [RefParam(1:length(irefsens))] = ...
            deal(struct('Center',meanCenter));
        % recursively call

        % Forward model on all reference sensors
        % JCM 19-Jun-2002 switched to feval of mfilename
        Gr = feval(mfilename,L,ChannelRef,RefParam,Order,[1:length(irefsens)],[]); % refs now called as MEG

        % Apply nth-order gradient correction on good channels only

        global ChannelFlag
        if isempty(ChannelFlag)
            ChannelFlag = ones(size(G,1),1); % Take all channels
        end

        %Weight by the current nth-order correction coefficients
        G = G - Channel(1).Comment*Gr; %CHEAT Need to fix the badchannel issue
    end

    clear global Verbose

    % ------------- SUBFUNCTIONS, First the simple utilities, then more complicated -----
    % -----------------------------------------------------------------------------------
    function c = cross(a,b);
    % fast and simple, and row major should be faster
    % 10/29/99 conversion to row major was slightly faster
    %  in the calculation, but overall slower in the transposes needed.
    %  retained the column major multiplies below
    c = zeros(size(a));
    c(1,:) = a(2,:).*b(3,:) - a(3,:).*b(2,:);
    c(2,:) = a(3,:).*b(1,:) - a(1,:).*b(3,:);
    c(3,:) = a(1,:).*b(2,:) - a(2,:).*b(1,:);

    % ----------------------------------------------------------------------------
    function k = kronmat(a,b);
    % column by column, not element by matrix
    k = [a([1 1 1],:) .* b; ...
          a([2 2 2],:) .* b; ...
          a([3 3 3],:) .* b];

    % ----------------------------------------------------------------------
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Local Sarvas functions %%%%%%%%%%%%%%%%%%%%%%%%
    % First, the standard interface to above
    % If the Order is -1 or 0, then same as before. Order 1 is handled separately

    function G = sarvas(L,P,Order);

    global Verbose

    % Bronzan Sarvas forward model, spherical head
    % Order = -1 is current dipole
    % Order = 0 is magnetic dipole of BST 2000
    % Order = 1 is the new 1st order current multipole
    % L is 3 x nL
    %
    % P.sensor is 3 x nR,each column a sensor location
    % P.orient is 3 x nR, the sensor orientation
    % P.center is 3 x nR, the sphere center for each sensor

    % January 18, 2002 from sarvas_partial function of 1995

    % Used old parameter convention to continue to handle Sylvain's exceptions
    %  for the CTF weighting coils

    %  if P.center in nonexistant or null, then assumed to be
    %  all zeros.

    %%%%%%%%%%%%%%%%%%%% which multipolar model to run %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    switch Order
    case {-1,0}  %%%%%%%%%%   CURRENT or MAGNETIC DIPOLE %%%%%%%%%%%%%%%%%%%

       G = sarvas_dipole(L,P,Order); % BST 2000 function still used here

    case {1} %%%%%%%%%%%%%%%%%%%%%%%%%  CURRENT MULTIPOLE MODEL %%%%%%%%%%%%%%%%%%555
       % new CME multipole, not MME multipole

       % each set of three columns corresponds to a dipole
       % first call the dipolar term
       % SB 02-Jul-2002
       if(size(L,2) > 5*size(P.sensor,2)) & size(P.sensor,2) > 1 & Verbose, %arbitrary definition of a lot
           MESSAGE = 1; % let's show messages
       else
           MESSAGE = 0; % let's keep quiet
       end

       if MESSAGE
           bst_message_window('wrap','Initiating the computation of CME modeling . . .')
       end

       Gdip = sarvas_dipole(L,P,-1);

       % now call the quadrupole component
       if MESSAGE
           bst_message_window('Refining model . . .') % Not very explicit but gives hint about the progression here - John any better message ?
       end

       Gquad = sarvas_partial(L,P); % form the gradient in a local function
       % weights are not applied in this routine, applied separately in the
       %  below forloop

       % For example:
       % Gquad(4:6,2) is the gradient of the second coil wrt to the second dipole
       % So we must Vec for each coil to get it into the proper format

       nR = size(P.sensor,2); % number of sensors
       nL = size(L,2);  % number of source points

       G = zeros(nR,12*nL); % the full gain matrix

       for i = 1:nL, % for each dipole
          ndx = [-11:0]+i*12; % indexer for this source
          G(:,ndx(1:3)) = Gdip(:,[-2:0]+i*3); % map next dipole to the first columns
          temp = Gquad(:,[-2:0]+i*3); % next block of partial gradients
          for j = 1:nR, % for each sensor position, also apply appropriate weight
             G(j,ndx(4:12)) = vec(temp([-2:0]+j*3,:))' * P.weight(j);
          end
       end

    end

    G = G*1e-7; % mu_o over 4 pi

    %--------------------------------------------------------------------------
    function G = sarvas_dipole(L,P,Order);

    global Verbose

    % SARVAS MEG Forward Model, spherical head

    %  if P.center in nonexistant or null, then assumed to be
    %  all zeros.

    if(~isfield(P,'center')), % user did not provide
       P.center = []; % initialize to null
    end
    if(isempty(P.center)), % user gave as null
       P.center = zeros(size(P.sensor));  % set to coordinate origin
    end

    P.sensor = P.sensor - P.center; % shift sensor coordinates

    % SB 03-Sep-2002
    % Now there is the issue of having a sensor array being a mixture of magnetometers and gradiometers.
    % Magnetometers are referred as pseudo-gradiometers: the corresponding .Loc field of the Channel array of structures
    % is still 3x2. For magnetometers though, the 2nd colum is filled with zeros (i.e.: [0 0 0]').
    % Same story holds for the .Orient field.
    % The full gain matrices are computed for these channels located at [0 0 0] but when the orientation vector is applied
    % the net field is set to 0. Therefore, the second call to sarvas_dipole produces a null field which is substracted frim the field
    % from the magnetometer (i.e the fisrt coil of the pseudo-gradiometer).
    % Calculations fail (divide by zero) when head center is also at [0 0 0].
    % I'm therefore testing this out here and fix things by virtually translating the virtual coils located at [0 0 0]
    % to [1 1 1] (arbitrary). The net field will still be null when the orientation is applied anyway.
    % Any more elegant fix is welcome at this point.
    iMag = find(norcol(P.sensor)==0); % Indices of channels located at P.center.
    if ~isempty(iMag)
        P.sensor(:,iMag) = repmat([1 1 1]',1,length(iMag)); % Move them away (arbitrary location).
    end

    nR = size(P.sensor,2); % number of sensors
    nL = size(L,2);  % number of source points

    Rn2 = sum(P.sensor.^2,1); % distance to sensor squared
    Rn = sqrt(Rn2); % distance

    if (nR >= nL), % more sensors than dipoles
       if(Order == 1),
          G = zeros(nR,12*nL); % gain matrix
       else
          G = zeros(nR,3*nL);  % gain matrix
       end

       for Li = 1:nL,
          Lmat = L(:,Li+zeros(1,nR)); % matrix of location repeated
          Lmat = Lmat - P.center; % each center shifted relative to its center
          D = P.sensor - Lmat;  % distance from souce to sensors
          Dn2 = sum(D.^2,1); % distance squared
          Dn = sqrt(Dn2);  % distance
          R_dot_D = sum(P.sensor .* D);  % dot product of sensor and distance
          R_dot_Dhat = R_dot_D ./ Dn;  % dot product of sensor and distance

          F = Dn2 .* Rn + Dn .* R_dot_D;  % Sarvas' function F

          GF_dot_o = Dn2 .* sum(P.sensor.*P.orient) ./ Rn + ...
             (2 * Rn + R_dot_Dhat) .* sum(D.*P.orient) + ...
             Dn .* sum((D+P.sensor).*P.orient);

          tempF = GF_dot_o ./ F.^2;

          if(Order == -1), % current dipole model
             temp = cross(Lmat,P.orient) ./ F([1 1 1],:) - ...
                cross(Lmat,P.sensor) .* tempF([1 1 1],:);
             G(:,Li*3+[-2 -1 0]) = temp';
          elseif(Order == 0) % magnetic dipole model
             temp = P.sensor .* tempF([1 1 1],:) - P.orient ./ F([1 1 1],:);
             G(:,Li*3+[-2 -1 0]) = temp';
          elseif(Order == 1),  % 1st order multipole
             % first the dipole
             temp_m = P.sensor .* tempF([1 1 1],:) - P.orient ./ F([1 1 1],:);
             % then the quadrupole
             temp1 = -(2*Rn + R_dot_Dhat + Dn);
             temp2 = -sum(D.*P.orient) ./ Dn;
             temp3 = -(2*sum(P.sensor.*P.orient)./Rn + sum((D+P.sensor).*P.orient)./Dn - ...
                sum(D.*P.orient).*R_dot_D./(Dn2.*Dn));

             GGpF_dot_o = temp1([1 1 1],:) .* P.orient + ...
                temp2([1 1 1],:).*P.sensor + temp3([1 1 1],:) .* D;
             temp1 = -(2*Rn + R_dot_Dhat);
             GpF = temp1([1 1 1],:) .* D - Dn([1 1 1],:) .* P.sensor;
             temp1 = 1 ./ F.^2;
             temp2 = 2*GF_dot_o./F;
             temp_q = temp1(ones(1,9),:) .* (kronmat(GGpF_dot_o,P.sensor) + ...
                kronmat(GpF,P.orient - temp2([1 1 1],:).*P.sensor));
             G(:,Li*12+[-11:0]) = [temp_m;temp_q]';
          end

       end

    else  % more dipoles than sensors nL > nR
       if(Order == 1)
          G = zeros(12*nL,nR);  % 1st order multipole gain matrix transposed
       else
          G = zeros(3*nL,nR);  % gain matrix transposed
       end

       % if there are a lot of dipoles, let's watch on the screen
       % JCM 18-Jun-2002 no more waitbar, use message window
       if (nL > 5*nR) & nR > 1 & Verbose, %arbitrary definition of a lot
          MESSAGE = 1; % let's show messages
          %       bst_message_window('wrap',sprintf(...
          %          'Making order %.0f matrix of %.0f sensors x %.0f sources',Order,nR,nL));
          %       bst_message_window('append','Making first sensor . . .'); % prepare for overwrite
          bst_message_window('overwrite',sprintf(...
              'Making Current Dipole matrix of %.0f sensors x %.0f sources',nR,nL));
          bst_message_window('append','Making first sensor . . .'); % prepare for overwrite   else
      else % SB 03-Sep-2002 : else was probably missing
          MESSAGE = Verbose; % let's be quiet
      end

       for Ri = 1:nR,
          if(MESSAGE), % want to show the user progress?
             if(~rem(Ri,30)), % every tenth sensor
                bst_message_window('overwrite',sprintf('Progress report:....... %.0f of %.0f . . .',Ri,nR));
             end
          end
          Rmat = P.sensor(:,Ri+zeros(1,nL)); % matrix of sensor repeated
          Omat = P.orient(:,Ri+zeros(1,nL)); % orientations
          Lmat = L - P.center(:,Ri+zeros(1,nL)); % shift centers to this coordinate

          D = Rmat - Lmat;
          Dn2 = sum(D.^2,1); % distance squared
          Dn = sqrt(Dn2);  % distance
          R_dot_D = sum(Rmat .* D);  % dot product of sensor and distance
          R_dot_Dhat = R_dot_D ./ Dn;  % dot product of sensor and distance

          F = Dn2 * Rn(Ri) + Dn .* R_dot_D;  % Sarvas' function F

          GF_dot_o = Dn2 * sum(P.sensor(:,Ri).*P.orient(:,Ri)) / Rn(Ri) + ...
             (2 * Rn(Ri) + R_dot_D ./ Dn) .* sum(D.*Omat) + ...
             Dn .* sum((D+Rmat).*Omat);

          tempF = GF_dot_o ./ F.^2;

          if(Order == -1), % current dipole model
             temp = cross(Lmat,Omat) ./ F([1 1 1],:) - ...
                cross(Lmat,Rmat) .* tempF([1 1 1],:);
          elseif(Order == 0) % magnetic dipole model
             temp = Rmat .* tempF([1 1 1],:) - Omat ./ F([1 1 1],:);
          elseif(Order == 1),  % 1st order multipole
             % first the dipole
             temp_m = Rmat .* tempF([1 1 1],:) - Omat ./ F([1 1 1],:);
             % then the quadrupole
             temp1 = -(2*Rn(Ri) + R_dot_Dhat + Dn);
             temp2 = -sum(D.*Omat) ./ Dn;
             temp3 = -(2*sum(P.sensor(:,Ri).*P.orient(:,Ri))./Rn(Ri) + sum((D+Rmat).*Omat)./Dn - ...
                sum(D.*Omat).*R_dot_D./(Dn2.*Dn));

             GGpF_dot_o = temp1([1 1 1],:) .* Omat + ...
                temp2([1 1 1],:).*Rmat + temp3([1 1 1],:) .* D;
             temp1 = -(2*Rn(Ri) + R_dot_Dhat);
             GpF = temp1([1 1 1],:) .* D - Dn([1 1 1],:) .* Rmat;
             temp1 = 1 ./ F.^2;
             temp2 = 2*GF_dot_o./F;
             temp_q = temp1(ones(1,9),:) .* (kronmat(GGpF_dot_o,Rmat) + ...
                kronmat(GpF,Omat - temp2([1 1 1],:).*Rmat));
             temp = [temp_m;temp_q];
          else % unimplemented order
             disp('SARVAS: Unimplemented source order');
             G = [];
             return
          end

          G(:,Ri) = temp(:);
       end

       G = G';

    end

    if(isfield(P,'weight')),
       Weights = P.weight(:); %make sure column
       % scale each row by its appropriate weight
       G = Weights(:,ones(1,size(G,2))) .* G;
    end

    % ----------------------------------------------------------------------------
    %%%%%%%%%%%%%%%%%%%%%%   Partial calculation for the sarvas %%%%%%%%%%%%%%%%%%%

    function D = sarvas_partial(L,P);
    %SARVAS_PARTIAL Calculate the partial of the Sarvas Formula w.r.t L
    % function D = sarvas_partial(L,P);
    % For dipole locations in L and sensor information in R, calculate the
    % partial of the Sarvas formula (dipole in a sphere, arbitrary sensor
    % orientation) with respect to the dipole location.  The result is a matrix
    % D of partials information.
    % If there are M sensors and P dipoles, then D
    % is 3*M by 3*P.  Let the corresponding moments Q be 3 by M.
    % Then partial = D * blk_diag(Q,1) is 3*M by P.  Each column of partial
    % corresponds to a different dipole in L.  Each set of three rows
    % corresponds to a different sensor location.  Thus partial(4:6,2) is the
    % partial of the Sarvas formula at the second sensor location, with respect
    % to the second dipole.
    %
    % Used to calculate the Cramer-Rao Lower Bounds
    % structure P has fields .weight, length of the number of sensors, which
    %  is applied to each row of the gain matrix. Weight is usually 1 or -1
    %  Field .center is 3 x number of sensors, head center for each coil.
    %  P.sensor and P.orient are the location and orientation information, resp.
    %  P.weight is not applied, must be applied separately, see above calling code.

    % Author: John C. Mosher, Ph.D.
    % Los Alamos National Laboratory
    % Los Alamos, NM 87545
    % email: mosher@LANL.Gov

    % August 15, 1995 author
    % 20 Feb 2002 JCM extensive I/O changes to adapt into os_meg, handle arbitrary
    %                 head center

    L = mby3check(L,0);        % matrices are now 3 x <>, but don't warn if 3 x 3

    numSens = size(P.sensor,2); % number of sensors
    numDip = size(L,2);  % number of source points

    % use some older notation for programming reuse below
    R = [P.sensor]; % sensor locations
    Rs = [P.orient]; % sensor orientations

    if(~isfield(P,'center')), % user did not provide
       P.center = []; % initialize to null
    end
    if(isempty(P.center)), % user gave as null
       P.center = zeros(size(R));  % set to coordinate origin
    end

    % each channel coil is shifted to its relative location
    R = R - P.center; % shift sensor coordinates

    % SB 03-Sep-2002
    % Now there is the issue of having a sensor array being a mixture of magnetometers and gradiometers.
    % Magnetometers are referred as pseudo-gradiometers: the corresponding .Loc field of the Channel array of structures
    % is still 3x2. For magnetometers though, the 2nd colum is filled with zeros (i.e.: [0 0 0]').
    % Same story holds for the .Orient field.
    % The full gain matrices are computed for these channels located at [0 0 0] but when the orientation vector is applied
    % the net field is set to 0. Therefore, the second call to sarvas_dipole produces a null field which is substracted frim the field
    % from the magnetometer (i.e the fisrt coil of the pseudo-gradiometer).
    % Calculations fail (divide by zero) when head center is also at [0 0 0].
    % I'm therefore testing this out here and fix things by virtually translating the virtual coils located at [0 0 0]
    % to [1 1 1] (arbitrary). The net field will still be null when the orientation is applied anyway.
    % Any more elegant fix is welcome at this point.
    iMag = find(norcol(R)==0); % Indices of channels located at P.center.
    if ~isempty(iMag)
        R(:,iMag) = repmat([1 1 1]',1,length(iMag)); % Move them away (arbitrary location).
    end

    Rn = colnorm(R);         % distance to sensor
    iRn = 1../Rn;            % inverse distance
    o3 = ones(3,1);            % col vector of three ones

    % three by three matrices per sensor and dipole
    D = zeros(numSens*3,3*numDip);

    if(size(L,2) > 5*size(R,2)) & size(R,2) > 1 & Verbose, %arbitrary definition of a lot
        MESSAGE = Verbose; % let's show messages
        bst_message_window({...
                'Computing quadrupolar moments',...
                sprintf('for sources %.0f of %.0f . . .',100,numDip)...
            });

    else
        MESSAGE = Verbose; % let's be quiet
    end

    for Dip = 1:numDip        % foreach dipole,

        %################ main loop ########################

       % if there are a lot of dipoles, let's watch on the screen
       % SB 02-Jul-2002 no more waitbar, use message window

       if(MESSAGE), % want to show the user progress?
           if(~rem(Dip,100)), % every tenth source
               bst_message_window('overwrite',sprintf('for sources %.0f of %.0f . . .',Dip,numDip));
           end
       end

       ThisDipole = L(:,Dip);        % next dipole
       Lmat = repmat(ThisDipole,1,numSens); % repeat this dipole for all sensors
       Lmat = Lmat - P.center; % shift each location relative to the sensor's center

       % let "a" be the same as Sarvas' "a", a = sensor - dipole.
       a = R - Lmat;
       an = colnorm(a);        % norm of each a
       ian=1../an;            % inverse of norm a
       ian3 = ian.^3;            % inverse cubed of norm a

       aDotR = sum(a.*R,1);

       % Form F

       F = (an .* Rn + aDotR) .* an;
       iF = 1../F;

       % From gradient F

       tmp1 = an.^2 .* iRn + aDotR .* ian + 2*an + 2*Rn;
       tmp1 = tmp1(o3,:).* R;
       tmp2 = an + 2*Rn + aDotR .* ian;
       tmp2 = tmp2(o3,:) .* Lmat;

       gradF = tmp1 - tmp2;

       % take partials of (Rs dot grad(F)) wrt r_q.  Result is 3 by 1 per sensor

       tmp1 = aDotR.*ian3 - 2*(iRn+ian);
       tmp1 = (tmp1(o3,:) .* a) - (R .* ian(o3,:));
       tmp1 = tmp1 .* (o3*sum(Rs.*R,1));

       tmp2 = an + 2*Rn + ian.*aDotR;
       tmp2 = tmp2(o3,:) .* Rs;

       tmp3 = aDotR.*ian3 - ian;
       tmp3 = tmp3(o3,:).*a - (R .* ian(o3,:));
       tmp3 = tmp3 .* (o3*sum(Rs.*Lmat,1));

       partRsGradF = tmp1 - tmp2 - tmp3;

       % take partial of F wrt r_q

       tmp1 = aDotR .* ian;
       tmp1 = tmp1(o3,:).*a;

       partF = -2*Rn(o3,:).*a - an(o3,:).*R - tmp1;

       % now take partial of ((gradF dot s) / F^2).

       tmp1 = F(o3,:) .* partRsGradF;

       tmp2 = 2*sum(gradF.*Rs,1);
       tmp2 = tmp2(o3,:) .* partF;

       partGradFdotRsOverF2 = (tmp1 - tmp2) .* iF(o3,:).^3;

       % now take partial of inverse of F

       partInvF = -partF .* iF(o3,:).^2;

       %  Now we are ready to generate the partial of the Sarvas' formula wrt dipole
       %  location.  The result is a 3 by 1 per sensor location; however, we want
       %  to separate out the dipole moment q.  So our "partials matrix" D is 3 x 3
       %  per sensor location.  We will concatenate into a 3*M by 3 matrix for each
       %  dipole.

       % Rs cross q divide by F
       % Want the cross product matrix tensor

       tmp1 = blk_diag(cross_mat(Rs),3); % each set of three columns is a tensor
       tmp1 = tmp1 .* kron(iF,ones(3)); % divide each tensor by F
       tmp1 = blk_lex(tmp1,3);     % now each set of three rows is a tensor

       % L cross Rs dot q time partInvF
       % Want the direct (outer) product of partInvF and (L cross Rs)

       tmp2 = cross(Lmat,Rs); % cross products
       % each set of three columns is the outer product
       tmp2 = partInvF * blk_diag(tmp2,1)';
       tmp2 = blk_lex(tmp2,3);        % now each set of three rows is a tensor

       % (R cross q)*(gradF dot Rs)/F^2
       % Want scalar times the cross product tensor

       tmp3 = sum(gradF .* Rs,1) .* iF.^2;
       tmp3 = kron(tmp3,ones(3));    % repeat scalar for each submatrix
       tmp3 = tmp3 .* blk_diag(cross_mat(R),3);
       tmp3 = blk_lex(tmp3,3);        % vertically stacked now

       % ((L cross R) dot q) * partial(gradF dot Rs over F^2)
       % Want direct (outer) product of partGradFdotRsOverF2 and (L cross R)

       tmp4 = cross(Lmat,R);
       tmp4 = partGradFdotRsOverF2 * blk_diag(tmp4,1)';
       tmp4 = blk_lex(tmp4,3);

       % Now combine into the appropriate columns of D.

       D(:,(Dip-1)*3 + [1:3]) = tmp1 + tmp2 - tmp3 - tmp4;

       %################ end main loop ########################
    end                 % next dipole

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