eeg_interp_sph_spline - Spherical Spline Interpolation of Potential
Useage: [FV,C] = eeg_interp_sph_spline(Zi,Ei)
where: Zi is Nelec x 1, an EEG/ERP measurement at time t
Ei is Nelec x 3, [X Y Z] electrode positions.
The origin of Ei is assumed (0,0,0).
FV => interpolated spherical surface (see sphere_tri)
FV.faces => triangulation of FV.vertices
FV.vertices => cartesian coordinates (Nx3)
FV.Cdata => spherical spline potential at FV.vertices
C => interpolation coefficients of Ei (includes co = C(1))
Notes: This function calculates the spherical spline of
Perrin et al (1989). Electroenceph. & Clin.
Neurophysiology, 72: 184-187. Corrigenda (1990),
Electroenceph. & Clin. Neurophysiology, 76: 565.
(see comments in the .m file for details).0001 function [FV,C] = eeg_interp_sph_spline(Zi,Ei) 0002 0003 % eeg_interp_sph_spline - Spherical Spline Interpolation of Potential 0004 % 0005 % Useage: [FV,C] = eeg_interp_sph_spline(Zi,Ei) 0006 % 0007 % where: Zi is Nelec x 1, an EEG/ERP measurement at time t 0008 % Ei is Nelec x 3, [X Y Z] electrode positions. 0009 % The origin of Ei is assumed (0,0,0). 0010 % 0011 % FV => interpolated spherical surface (see sphere_tri) 0012 % 0013 % FV.faces => triangulation of FV.vertices 0014 % FV.vertices => cartesian coordinates (Nx3) 0015 % FV.Cdata => spherical spline potential at FV.vertices 0016 % 0017 % C => interpolation coefficients of Ei (includes co = C(1)) 0018 % 0019 % Notes: This function calculates the spherical spline of 0020 % Perrin et al (1989). Electroenceph. & Clin. 0021 % Neurophysiology, 72: 184-187. Corrigenda (1990), 0022 % Electroenceph. & Clin. Neurophysiology, 76: 565. 0023 % (see comments in the .m file for details). 0024 0025 % $Revision: 1.2 $ $Date: 2005/07/13 06:03:50 $ 0026 0027 % Licence: GNU GPL, no implied or express warranties 0028 % History: 08/2001 Darren.Weber_at_radiology.ucsf.edu, with 0029 % mathematical advice from 0030 % Dr. Murk Bottema (Flinders University of SA) 0031 % 10/2003 Darren.Weber_at_radiology.ucsf.edu, with 0032 % mathematical advice and LegendreP function from 0033 % Dr. Tom.Ferree_at_radiology.ucsf.edu 0034 % revised, tested & initial verification complete 0035 % 0036 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0037 0038 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0039 % Check for correct size & orientation of Ei & Zi 0040 [e1,e2] = size(Ei); 0041 [v1,v2] = size(Zi); 0042 if e1 < e2, Ei = Ei'; [e1,e2] = size(Ei); end 0043 if v1 < v2, Zi = Zi'; [v1,v2] = size(Zi); end 0044 if ~and(isequal(e1,v1),and(isequal(e2,3),isequal(v2,1))) 0045 error('...Ei must be Nx3 & Zi must be Nx1'); 0046 end 0047 nElectrodes = e1; % The number of electrodes 0048 clear e1 e2 v1 v2; 0049 0050 0051 % ------------------------------------------------------------------------- 0052 % estimate spherical radius of the electrodes and 0053 % obtain spherical projections of Ei 0054 [r,x,y,z] = elec_sphere_project(Ei(:,1),Ei(:,2),Ei(:,3)); 0055 %Ei = [ x y z ]; clear x y z; 0056 0057 % create spherical interpolation surface 0058 FV = sphere_tri('ico',4,r); 0059 0060 % ------------------------------------------------------------------------- 0061 % Calculate the cosines, if Ei is Nx3, COS is NxN matrix 0062 % We use (Ei,Ei) here because it gives the cosines between 0063 % each electrode and every other electrode. This is required 0064 % here because we solve a linear system of equations below 0065 % that will find the interpolated value at a given electrode 0066 % location, which must be equal to the measured 0067 % potential at that location. 0068 EiCOS = elec_cosines(Ei,Ei); 0069 0070 % create zeros on the diagonal elements 0071 % [not sure why this works, but it does.] 0072 for i = 1:length(EiCOS), EiCOS(i,i) = 0; end 0073 0074 % ------------------------------------------------------------------------- 0075 % Calculate g(x), nElectrodes x nElectrodes 0076 Gx = eeg_interp_sph_spline_g(EiCOS); 0077 0078 % ------------------------------------------------------------------------- 0079 % calculate the spherical interpolation coefficients 0080 C = eeg_interp_sph_spline_c(Zi,Gx); clear Gx 0081 0082 % ------------------------------------------------------------------------- 0083 % cosines between electrodes and interpolation points 0084 FvCOS = elec_cosines(Ei,FV.vertices); 0085 0086 % Calculate g(x), nElectrodes x NinterpolationPoints 0087 Gx = eeg_interp_sph_spline_g(FvCOS); 0088 0089 eegversion = '$Revision: 1.2 $'; 0090 fprintf('EEG_INTERP_SPH_SPLINE [v %s]\n',eegversion(11:15)); tic 0091 0092 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0093 % Now that Ci is solved, we can obtain interpolated potentials at Ej (Eq. 1) 0094 % U(Ej) = c(0) + ( for i=1:n, sum = (sum + (c(i) * g(cos(Ei,Ej)))) ) 0095 % U(Ej) = c(0) + sum( Ci * g(x) ) 0096 0097 % Solve Eq 1. (where FV.Cdata = U) 0098 Co = C(1); 0099 Ci = C(2:end); 0100 FV.Cdata = Co + ( Ci' * Gx ); 0101 0102 t=toc; fprintf('...done (%6.2f sec)\n',t); 0103 0104 return