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# zhang_fit

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

### Function Synopsis

    delta = zhang_fit(berg,R,f)

### Help Text

    ZHANG_FIT - Compute 'Berg' parameters for EEG multilayer sphere
     function delta = zhang_fit(berg,R,f)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     MINIMIZATION FUNCTION FOR COMPUTING BERG PARAMETERS FOR EEG MULTILAYER
     SPHERICAL FORWARD MODEL (zhang_fit.m)

     This function serves as a minimization function for determining the Berg
     Eccentricity and Magnitude parameters associated with a series approximation
     of a Single Dipole in a Multilayer Sphere -by- multiple dipoles in a single shell.
     (Zhang: Eq 5i"). This function is intended to be called by the Matlab "fmins"
     function. This version has not been optimized for speed.

     (Ref: Z. Zhang "A fast method to compute surface potentials generated by dipoles
     within multilayer anisotropic spheres" (Phys. Med. Biol. 40, pp335-349,1995)

     INPUTS:
             berg : Contains                                              (2*J-1) x 1
             R    : Concentric Sphere Radii(in meters) from INNERMOST to
                    OUTERMOST                                                  NL x 1
             f: Weighting terms corresponding to sphere/conductivity
                profile of interest                                          nmax x 1

          where: NL = # of sphere layers; nmax = # of series terms; J = # of Berg Dipoles

     OUTPUTS:
              delta: Minimization function output. Best fit of
                     Berg Eccentricity and Magnitude Paramters occurs when
                     this value is minimized                                    1 x 1

     External Functions and Files: None

     By John Ermer 5/5/99
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

### Cross-Reference Information

This function calls

    berg                               C:\BrainStorm_2001\Toolbox\berg.m

This function is called by

    berg                               C:\BrainStorm_2001\Toolbox\berg.m

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

    function delta = zhang_fit(berg,R,f)
    %ZHANG_FIT - Compute 'Berg' parameters for EEG multilayer sphere
    % function delta = zhang_fit(berg,R,f)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %
    % MINIMIZATION FUNCTION FOR COMPUTING BERG PARAMETERS FOR EEG MULTILAYER
    % SPHERICAL FORWARD MODEL (zhang_fit.m)
    %
    % This function serves as a minimization function for determining the Berg
    % Eccentricity and Magnitude parameters associated with a series approximation
    % of a Single Dipole in a Multilayer Sphere -by- multiple dipoles in a single shell.
    % (Zhang: Eq 5i"). This function is intended to be called by the Matlab "fmins"
    % function. This version has not been optimized for speed.
    %
    % (Ref: Z. Zhang "A fast method to compute surface potentials generated by dipoles
    % within multilayer anisotropic spheres" (Phys. Med. Biol. 40, pp335-349,1995)
    %
    % INPUTS:
    %         berg : Contains                                              (2*J-1) x 1
    %         R    : Concentric Sphere Radii(in meters) from INNERMOST to
    %                OUTERMOST                                                  NL x 1
    %         f: Weighting terms corresponding to sphere/conductivity
    %            profile of interest                                          nmax x 1
    %
    %      where: NL = # of sphere layers; nmax = # of series terms; J = # of Berg Dipoles
    %
    % OUTPUTS:
    %          delta: Minimization function output. Best fit of
    %                 Berg Eccentricity and Magnitude Paramters occurs when
    %                 this value is minimized                                    1 x 1
    %
    % External Functions and Files: None
    %
    % By John Ermer 5/5/99
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    %<autobegin> ---------------------- 26-May-2004 11:34:43 -----------------------
    % --------- Automatically Generated Comments Block Using AUTO_COMMENTS ---------
    %
    % CATEGORY: Forward Modeling
    %
    % At Check-in: $Author: Mosher $  $Revision: 13 $  $Date: 5/26/04 10:02a $
    %
    % This software is part of BrainStorm Toolbox Version 2.0 (Alpha) 24-May-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> ------------------------ 26-May-2004 11:34:43 -----------------------

    % History ------------------------------------------------
    % Ermer 5-May-1999 creation
    % JCM 08-Sep-2003 Comments
    % ---------------------------------------------------------

    NL = length(R);           % Number of Sphere Layers
    nmax = length(f);         % Number of Legendre Series Terms
    J = (length(berg)+1)/2;   % Infer Number of Berg Dipoles from Passed data
    mu_berg = berg(1:J);      % Berg Eccentricity Factors 1->J
    lam_berg = [0.0 berg(J+1:2*J-1)]; % Berg Magnitude Factors 2->J
    %
    delta = 0.0;  % Compute Factor to be minimized
    %
    for n = 2:nmax
       term0 = 0;
       for j=2:J
          term0 = term0 + lam_berg(j)*(mu_berg(j)^(n-1) - mu_berg(1)^(n-1));
       end
       term1 = ( (R(1)/R(NL))^(n-1) )* ...
         ( f(n)-f(1)*mu_berg(1)^(n-1) - term0);
       delta = delta + term1*term1;
    end
    %
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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