vb_eeg_one_shell2

 EEG potential for multiple dipoles at a single EEG sensor
     in spherical brain model
 Calculate EEG potential for each dipole orientation
   eeg = vb_eeg_one_shell2(P, R, sigma)
 INPUT
   Dipole current source
   P : current dipole position   ( NP x 3 )
   EEG sensor
   R : one of sensor position    ( 1 x 3 ) or ( 3 x 1 )

 sigma : Conductivity inside the sphere

 OUTPUT
   eeg : EEG Lead field matrix   ( NP x 3 )

   NP : # of dipole points

 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL.46, 245-259, 1999
 John C. Mosher, Richard M. Leahy and Paul S. Lewis

  2004-12-15 Made by M. Sato

 Copyright (C) 2011, ATR All Rights Reserved.
 License : New BSD License(see VBMEG_LICENSE.txt)

0001 function    eeg = vb_eeg_one_shell2(P, R, sigma)
0002 % EEG potential for multiple dipoles at a single EEG sensor
0003 %     in spherical brain model
0004 % Calculate EEG potential for each dipole orientation
0005 %   eeg = vb_eeg_one_shell2(P, R, sigma)
0006 % INPUT
0007 %   Dipole current source
0008 %   P : current dipole position   ( NP x 3 )
0009 %   EEG sensor
0010 %   R : one of sensor position    ( 1 x 3 ) or ( 3 x 1 )
0011 %
0012 % sigma : Conductivity inside the sphere
0013 %
0014 % OUTPUT
0015 %   eeg : EEG Lead field matrix   ( NP x 3 )
0016 %
0017 %   NP : # of dipole points
0018 %
0019 % IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL.46, 245-259, 1999
0020 % John C. Mosher, Richard M. Leahy and Paul S. Lewis
0021 %
0022 %  2004-12-15 Made by M. Sato
0023 %
0024 % Copyright (C) 2011, ATR All Rights Reserved.
0025 % License : New BSD License(see VBMEG_LICENSE.txt)
0026 
0027 % Dipole position vector
0028 P1 = P(:,1);
0029 P2 = P(:,2);
0030 P3 = P(:,3);
0031 
0032 % Sensor position vector
0033 R  = R(:);
0034 R1 = R(1);
0035 R2 = R(2);
0036 R3 = R(3);
0037 
0038 % Difference from dipole to sensor : (NP x 3)
0039 dP = [(R1-P1), (R2-P2), (R3-P3)];
0040 
0041 % Square norm    :  (NP x 1)
0042 dd  = sum(dP.^2, 2);
0043 d   = sqrt(dd);
0044 ddd = d.*dd;
0045 pp  = sum(P.^2, 2);
0046 
0047 rr  = sum(R.^2);
0048 r   = sqrt(rr);
0049 
0050 % Inner product
0051 dp = sum(dP.*P, 2);
0052 dr = dP*R;
0053 pr = P*R;
0054 
0055 % Coefficient terms
0056 f  = dd .* r + d .* dr ;            % Eq.(11)
0057 c1 = 2*dp./ddd + 1./d - 1./r ;      % Eq.(25)
0058 c2 = 2./ddd + (d + r)./(r.*f) ;     % Eq.(26)
0059 c3 = ( c1 - c2.*pr )./pp;            % Eq.(27)
0060 
0061 % EEG potential
0062 eeg = c3(:,ones(1,3)).*P + c2*R';    % Eq.(27)
0063 
0064 eeg = eeg./(4.0*pi*sigma);
0065 
0066 return