fwd_eeg_sphere_model.cpp

Go to the documentation of this file.

//=============================================================================================================
 
//=============================================================================================================
// INCLUDES
//=============================================================================================================
 
#include "math/simplex_algorithm.h"
 
#include "fwd_eeg_sphere_model.h"
#include "fwd_eeg_sphere_model_set.h"
 
#include <qmath.h>
 
#include <Eigen/Core>
 
#include <algorithm>
#include <vector>
#include <Eigen/Dense>
 
//=============================================================================================================
// USED NAMESPACES
//=============================================================================================================
 
using namespace Eigen;
using namespace UTILSLIB;
using namespace FWDLIB;
 
//=============================================================================================================
// Local constants and helpers
//=============================================================================================================
 
namespace
{
constexpr int FAIL = -1;
constexpr int OK = 0;
constexpr int MAXTERMS = 1000;
constexpr double EPS = 1e-10;
constexpr double SIN_EPS = 1e-3;
} // anonymous namespace
 
//=============================================================================================================
// DEFINE MEMBER METHODS
//=============================================================================================================
 
FwdEegSphereModel::FwdEegSphereModel()
: nterms(0)
, nfit(0)
, scale_pos(0)
{
    r0.setZero();
}
 
//=============================================================================================================
 
FwdEegSphereModel::FwdEegSphereModel(const FwdEegSphereModel& p_FwdEegSphereModel) = default;
 
//=============================================================================================================
 
FwdEegSphereModel::UPtr FwdEegSphereModel::fwd_create_eeg_sphere_model(const QString& name,
                                                                       int nlayer,
                                                                       const VectorXf& rads,
                                                                       const VectorXf& sigmas)
/*
      * Produce a new sphere model structure
      */
{
    auto new_model = std::make_unique<FwdEegSphereModel>();
 
    new_model->name = name;
 
    for (int k = 0; k < nlayer; k++) {
        FwdEegSphereLayer layer;
        layer.rad = layer.rel_rad = rads[k];
        layer.sigma = sigmas[k];
        new_model->layers.push_back(layer);
    }
    /*
   * Sort...
   */
    std::sort(new_model->layers.begin(), new_model->layers.end(), FwdEegSphereLayer::comp_layers);
 
    /*
   * Scale the radiuses
   */
    float R = new_model->layers[nlayer - 1].rad;
    float rR = new_model->layers[nlayer - 1].rel_rad;
    for (int k = 0; k < nlayer; k++) {
        new_model->layers[k].rad = new_model->layers[k].rad / R;
        new_model->layers[k].rel_rad = new_model->layers[k].rel_rad / rR;
    }
    return new_model;
}
 
//=============================================================================================================
 
FwdEegSphereModel::~FwdEegSphereModel()
{
}
 
//=============================================================================================================
 
FwdEegSphereModel::UPtr FwdEegSphereModel::setup_eeg_sphere_model(const QString& eeg_model_file, QString eeg_model_name, float eeg_sphere_rad)
{
    if (eeg_model_name.isEmpty())
        eeg_model_name = QString("Default");
 
    FwdEegSphereModelSet::UPtr eeg_models = FwdEegSphereModelSet::fwd_load_eeg_sphere_models(eeg_model_file, nullptr);
    eeg_models->fwd_list_eeg_sphere_models();
 
    FwdEegSphereModel::UPtr eeg_model = eeg_models->fwd_select_eeg_sphere_model(eeg_model_name);
    if (!eeg_model) {
        return nullptr;
    }
 
    if (!eeg_model->fwd_setup_eeg_sphere_model(eeg_sphere_rad, true, 3)) {
        return nullptr;
    }
 
    qInfo("Using EEG sphere model \"%s\" with scalp radius %7.1f mm",
          eeg_model->name.toUtf8().constData(), 1000 * eeg_sphere_rad);
    return eeg_model;
}
 
//=============================================================================================================
 
std::unique_ptr<fitUserRec> FwdEegSphereModel::new_fit_user(int nfit, int nterms)
{
    auto u = std::make_unique<fitUserRec>();
    u->y.resize(nterms - 1);
    u->resi.resize(nterms - 1);
    u->M = MatrixXd::Zero(nterms - 1, nfit - 1);
    u->uu = MatrixXd::Zero(nfit - 1, nterms - 1);
    u->vv = MatrixXd::Zero(nfit - 1, nfit - 1);
    u->sing.resize(nfit);
    u->fn.resize(nterms);
    u->w.resize(nterms);
    u->nfit = nfit;
    u->nterms = nterms;
    return u;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
double FwdEegSphereModel::fwd_eeg_get_multi_sphere_model_coeff(int n)
{
    MatrixXd M, Mn, help, Mm;
    static MatrixXd mat1;
    static MatrixXd mat2;
    static MatrixXd mat3;
    static VectorXd c1;
    static VectorXd c2;
    static VectorXd cr;
    static VectorXd cr_mult;
    double div, div_mult;
    double n1;
#ifdef TEST
    double rel1, rel2;
    double b, c;
#endif
    int k;
 
    if (this->nlayer() == 0 || this->nlayer() == 1)
        return 1.0;
    /*
   * Now follows the tricky case
   */
#ifdef TEST
    if (this->nlayer() == 2) {
        rel1 = layers[0].sigma / layers[1].sigma;
        n1 = n + 1.0;
        div_mult = 2.0 * n + 1;
        b = pow(this->layers[0].rel_rad, div_mult);
        return div_mult / ((n1 + n * rel1) + b * n1 * (rel1 - 1.0));
    } else if (this->nlayer() == 3) {
        rel1 = this->layers[0].sigma / this->layers[1].sigma;
        rel2 = this->layers[1].sigma / this->layers[2].sigma;
        n1 = n + 1.0;
        div_mult = 2.0 * n + 1.0;
        b = pow(this->layers[0].rel_rad, div_mult);
        c = pow(this->layers[1].rel_rad, div_mult);
        div_mult = div_mult * div_mult;
        div = (b * n * n1 * (rel1 - 1.0) * (rel2 - 1.0) + c * (rel1 * n + n1) * (rel2 * n + n1)) / c +
            n1 * (b * (rel1 - 1.0) * (rel2 * n1 + n) + c * (rel1 * n + n1) * (rel2 - 1.0));
        return div_mult / div;
    }
#endif
    if (n == 1) {
        /*
     * Initialize the arrays
     */
        c1.resize(this->nlayer() - 1);
        c2.resize(this->nlayer() - 1);
        cr.resize(this->nlayer() - 1);
        cr_mult.resize(this->nlayer() - 1);
        for (k = 0; k < this->nlayer() - 1; k++) {
            c1[k] = this->layers[k].sigma / this->layers[k + 1].sigma;
            c2[k] = c1[k] - 1.0;
            cr_mult[k] = this->layers[k].rel_rad;
            cr[k] = cr_mult[k];
            cr_mult[k] = cr_mult[k] * cr_mult[k];
        }
        if (mat1.cols() == 0)
            mat1 = MatrixXd(2, 2);
        if (mat2.cols() == 0)
            mat2 = MatrixXd(2, 2);
        if (mat3.cols() == 0)
            mat3 = MatrixXd(2, 2);
    }
    /*
   * Increment the radius coefficients
   */
    for (k = 0; k < this->nlayer() - 1; k++)
        cr[k] = cr[k] * cr_mult[k];
    /*
   * Multiply the matrices
   */
    M = mat1;
    Mn = mat2;
    Mm = mat3;
    M(0, 0) = M(1, 1) = 1.0;
    M(0, 1) = M(1, 0) = 0.0;
    div = 1.0;
    div_mult = 2.0 * n + 1.0;
    n1 = n + 1.0;
 
    for (k = this->nlayer() - 2; k >= 0; k--) {
        Mm(0, 0) = (n + n1 * c1[k]);
        Mm(0, 1) = n1 * c2[k] / cr[k];
        Mm(1, 0) = n * c2[k] * cr[k];
        Mm(1, 1) = n1 + n * c1[k];
 
        Mn(0, 0) = Mm(0, 0) * M(0, 0) + Mm(0, 1) * M(1, 0);
        Mn(0, 1) = Mm(0, 0) * M(0, 1) + Mm(0, 1) * M(1, 1);
        Mn(1, 0) = Mm(1, 0) * M(0, 0) + Mm(1, 1) * M(1, 0);
        Mn(1, 1) = Mm(1, 0) * M(0, 1) + Mm(1, 1) * M(1, 1);
        help = M;
        M = Mn;
        Mn = help;
        div = div * div_mult;
    }
    return n * div / (n * M(1, 1) + n1 * M(1, 0));
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
void FwdEegSphereModel::next_legen(int n, double x, double& p0, double& p01, double& p1, double& p11)
/*
          * Compute the next Legendre polynomials of the
          * first and second kind using the recursion formulas.
          *
          * The routine initializes automatically with the known values
          * when n = 1
          */
{
    double help0, help1;
 
    if (n > 1) {
        help0 = p0;
        help1 = p1;
        p0 = ((2 * n - 1) * x * help0 - (n - 1) * (p01)) / n;
        p1 = ((2 * n - 1) * x * help1 - n * (p11)) / (n - 1);
        p01 = help0;
        p11 = help1;
    } else if (n == 0) {
        p0 = 1.0;
        p1 = 0.0;
    } else if (n == 1) {
        p01 = 1.0;
        p0 = x;
        p11 = 0.0;
        p1 = sqrt(1.0 - x * x);
    }
    return;
}
 
//=============================================================================================================
 
void FwdEegSphereModel::calc_pot_components(double beta, double cgamma, double& Vrp, double& Vtp, const Eigen::VectorXd& fn, int nterms)
{
    double Vt = 0.0;
    double Vr = 0.0;
    double p0, p01, p1, p11;
    double betan, multn;
    int n;
 
    betan = 1.0;
    p0 = p01 = p1 = p11 = 0.0;
    for (n = 1; n <= nterms; n++) {
        if (betan < EPS) {
            break;
        }
        next_legen(n, cgamma, p0, p01, p1, p11);
        multn = betan * fn[n - 1]; /* The 2*n + 1 factor is included in fn */
        Vr = Vr + multn * p0;
        Vt = Vt + multn * p1 / n;
        betan = beta * betan;
    }
    Vrp = Vr;
    Vtp = Vt;
    return;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
int FwdEegSphereModel::fwd_eeg_multi_spherepot(const Eigen::Vector3f& rd_in, const Eigen::Vector3f& Q_in, const Eigen::Matrix<float, Eigen::Dynamic, 3, Eigen::RowMajor>& el, int neeg, Eigen::VectorXf& Vval, void* client)
/*
 * Compute the electric potentials in a set of electrodes in spherically
 * Symmetric head model.
 *
 * The code is based on the formulas presented in
 *
 * Z. Zhang, A fast method to compute surface potentials
 * generated by dipoles within multilayer anisotropic spheres,
 * Phys. Med. Biol., 40, 335 - 349, 1995.
 *
 * and
 *
 * J.C. Moscher, R.M. Leahy, and P.S. Lewis, Matrix Kernels for
 * Modeling of EEG and MEG Data, Los Alamos Technical Report,
 * LA-UR-96-1993, 1996.
 *
 * This version does not use the acceleration with help of equivalent sources
 * in the homogeneous model
 *
 */
{
    auto* m = static_cast<FwdEegSphereModel*>(client);
    Eigen::Vector3f rd = rd_in - m->r0;
    Eigen::Vector3f Q = Q_in;
    Eigen::Vector3f pos;
    int k;
    float pos2, rd_len, pos_len;
    double beta, cos_gamma, Vr, Vt;
    Eigen::Vector3f vec1, vec2;
    float v1, v2;
    float cos_beta, Qr, Qt, Q2, c;
    float pi4_inv = static_cast<float>(0.25 / M_PI);
    float sigmaM_inv;
    /*
       * Precompute the coefficients
       */
    if (m->fn.size() == 0 || m->nterms != MAXTERMS) {
        m->fn.resize(MAXTERMS);
        m->nterms = MAXTERMS;
        for (k = 0; k < MAXTERMS; k++)
            m->fn[k] = (2 * k + 3) * m->fwd_eeg_get_multi_sphere_model_coeff(k + 1);
    }
    /*
       * Move to the sphere coordinates
       */
    rd_len = rd.norm();
    Q2 = Q.dot(Q);
    /*
       * Ignore dipoles outside the innermost sphere
       */
    if (rd_len >= m->layers[0].rad) {
        for (k = 0; k < neeg; k++)
            Vval[k] = 0.0;
        return OK;
    }
    /*
       * Special case: rd and Q are parallel
       */
    c = rd.dot(Q) / (rd_len * sqrt(Q2));
    if ((1.0 - c * c) < SIN_EPS) { /* Almost parallel:
                         * Q is purely radial */
        Qr = sqrt(Q2);
        Qt = 0.0;
        cos_beta = 0.0;
        v1 = 0.0;
        vec1.setZero();
    } else {
        vec1 = rd.cross(Q);
        v1 = vec1.norm();
        cos_beta = 0.0;
        Qr = Qt = 0.0;
    }
    for (k = 0; k < neeg; k++) {
        pos = el.row(k).transpose() - m->r0;
        /*
         * Should the position be scaled or not?
         */
        if (m->scale_pos) {
            pos_len = m->layers[m->nlayer() - 1].rad / pos.norm();
#ifdef DEBUG
            qInfo("%10.4f %10.4f %10.4f %10.4f", pos_len, 1000 * pos[0], 1000 * pos[1], 1000 * pos[2]);
#endif
            pos *= pos_len;
        }
        pos2 = pos.dot(pos);
        pos_len = sqrt(pos2);
        /*
         * Calculate the two ingredients for the final result
         */
        cos_gamma = pos.dot(rd) / (rd_len * pos_len);
        beta = rd_len / pos_len;
        calc_pot_components(beta, cos_gamma, Vr, Vt, m->fn, m->nterms);
        /*
         * Then compute the combined result
         */
        if (v1 > 0.0) {
            vec2 = rd.cross(pos);
            v2 = vec2.norm();
 
            if (v2 > 0.0)
                cos_beta = vec1.dot(vec2) / (v1 * v2);
            else
                cos_beta = 0.0;
 
            Qr = Q.dot(rd) / rd_len;
            Qt = sqrt(Q2 - Qr * Qr);
        }
        Vval[k] = static_cast<double>(pi4_inv) * (static_cast<double>(Qr) * Vr + static_cast<double>(Qt) * cos_beta * Vt) / pos2;
    }
    /*
       * Scale by the conductivity if we have the layers
       * defined
       */
    if (m->nlayer() > 0) {
        sigmaM_inv = 1.0 / m->layers[m->nlayer() - 1].sigma;
        for (k = 0; k < neeg; k++)
            Vval[k] = Vval[k] * sigmaM_inv;
    }
    return OK;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
int FwdEegSphereModel::fwd_eeg_multi_spherepot_coil1(const Eigen::Vector3f& rd, const Eigen::Vector3f& Q, FwdCoilSet& els, Eigen::Ref<Eigen::VectorXf> Vval, void* client) /* Client data will be the sphere model definition */
/*
 * Calculate the EEG in the sphere model using the fwdCoilSet structure
 *
 * This version does not use the acceleration with help of equivalent sources
 * in the homogeneous model
 *
 */
{
    VectorXf vval_one;
    float val;
    int nvval = 0;
    int k, c;
    FwdCoil* el;
 
    Vval.resize(els.ncoil());
    for (k = 0; k < els.ncoil(); k++) {
        el = els.coils[k].get();
        if (el->coil_class == FWD_COILC_EEG) {
            if (el->np > nvval) {
                vval_one.resize(el->np);
                nvval = el->np;
            }
            if (fwd_eeg_multi_spherepot(rd, Q, el->rmag, el->np, vval_one, client) != OK) {
                return FAIL;
            }
            for (c = 0, val = 0.0; c < el->np; c++)
                val += el->w[c] * vval_one[c];
            Vval[k] = val;
        }
    }
    return OK;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
bool FwdEegSphereModel::fwd_eeg_spherepot_vec(const Eigen::Vector3f& rd_in, const Eigen::Matrix<float, Eigen::Dynamic, 3, Eigen::RowMajor>& el, int neeg, Eigen::MatrixXf& Vval_vec, void* client)
{
    auto* m = static_cast<FwdEegSphereModel*>(client);
    float fact = 0.25f / static_cast<float>(M_PI);
    Eigen::Vector3f a_vec;
    float a, a2, a3;
    float rrd, rd2, rd2_inv, r, r2, ra, rda;
    float F;
    float c1, c2, m1, m2;
    int k, eq;
    Eigen::Vector3f orig_rd = rd_in - m->r0;
    Eigen::Vector3f rd;
    Eigen::Vector3f pos;
    float pos_len;
    /*
   * Initialize the arrays
   */
    for (k = 0; k < neeg; k++) {
        Vval_vec(0, k) = 0.0;
        Vval_vec(1, k) = 0.0;
        Vval_vec(2, k) = 0.0;
    }
    /*
   * Ignore dipoles outside the innermost sphere
   */
    if (orig_rd.norm() >= m->layers[0].rad)
        return true;
    /*
   * Default to homogeneous model if no model was previously set
   */
#ifdef FOO
    if (nequiv == 0) /* what to do */
        eeg_set_homog_sphere_model();
#endif
    /*
   * Make a weighted sum over the equivalence parameters
   */
    for (eq = 0; eq < m->nfit; eq++) {
        /*
     * Scale the dipole position
     */
        rd = m->mu[eq] * orig_rd;
 
        rd2 = rd.dot(rd);
        rd2_inv = 1.0 / rd2;
 
        /*
     * Go over all electrodes
     */
        for (k = 0; k < neeg; k++) {
            pos = el.row(k).transpose() - m->r0;
            /*
       * Scale location onto the surface of the sphere
       */
            if (m->scale_pos) {
                pos_len = m->layers[m->nlayer() - 1].rad / pos.norm();
                pos *= pos_len;
            }
 
            /* Vector from dipole to the field point */
 
            a_vec = pos - rd;
 
            /* Compute the dot products needed */
 
            a2 = a_vec.dot(a_vec);
            a = sqrt(a2);
            a3 = 2.0 / (a2 * a);
            r2 = pos.dot(pos);
            r = sqrt(r2);
            rrd = pos.dot(rd);
            ra = r2 - rrd;
            rda = rrd - rd2;
 
            /* The main ingredients */
 
            F = a * (r * a + ra);
            c1 = a3 * rda + 1.0 / a - 1.0 / r;
            c2 = a3 + (a + r) / (r * F);
 
            /* Mix them together and scale by lambda/(rd*rd) */
 
            m1 = (c1 - c2 * rrd);
            m2 = c2 * rd2;
 
            Vval_vec(0, k) = Vval_vec(0, k) + m->lambda[eq] * rd2_inv * (m1 * rd[0] + m2 * pos[0]);
            Vval_vec(1, k) = Vval_vec(1, k) + m->lambda[eq] * rd2_inv * (m1 * rd[1] + m2 * pos[1]);
            Vval_vec(2, k) = Vval_vec(2, k) + m->lambda[eq] * rd2_inv * (m1 * rd[2] + m2 * pos[2]);
        } /* All electrodes done */
    } /* All equivalent dipoles done */
    /*
   * Finish by scaling by 1/(4*M_PI);
   */
    for (k = 0; k < neeg; k++) {
        Vval_vec(0, k) = fact * Vval_vec(0, k);
        Vval_vec(1, k) = fact * Vval_vec(1, k);
        Vval_vec(2, k) = fact * Vval_vec(2, k);
    }
    return true;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
int FwdEegSphereModel::fwd_eeg_spherepot_coil_vec(const Eigen::Vector3f& rd, FwdCoilSet& els, Eigen::Ref<Eigen::MatrixXf> Vval_vec, void* client)
{
    Eigen::MatrixXf vval_one;
    float val;
    int nvval = 0;
    int k, c, p;
    FwdCoil* el;
 
    for (k = 0; k < els.ncoil(); k++) {
        el = els.coils[k].get();
        if (el->coil_class == FWD_COILC_EEG) {
            if (el->np > nvval) {
                vval_one.resize(3, el->np);
                nvval = el->np;
            }
            if (!fwd_eeg_spherepot_vec(rd, el->rmag, el->np, vval_one, client)) {
                return FAIL;
            }
            for (p = 0; p < 3; p++) {
                for (c = 0, val = 0.0; c < el->np; c++)
                    val += el->w[c] * vval_one(p, c);
                Vval_vec(p, k) = val;
            }
        }
    }
    return OK;
}
 
//=============================================================================================================
 
int FwdEegSphereModel::fwd_eeg_spherepot_grad_coil(const Eigen::Vector3f& rd, const Eigen::Vector3f& Q, FwdCoilSet& coils, Eigen::Ref<Eigen::VectorXf> Vval, Eigen::Ref<Eigen::VectorXf> xgrad, Eigen::Ref<Eigen::VectorXf> ygrad, Eigen::Ref<Eigen::VectorXf> zgrad, void* client) /* Client data to be passed to some foward modelling routines */
/*
          * Quick and dirty solution: use differences
          *
          * This routine uses the acceleration with help of equivalent sources
          * in the homogeneous sphere.
          *
          */
{
    Eigen::Vector3f my_rd;
    float step = 0.0005f;
    float step2 = 2 * step;
    int p, q;
 
    Eigen::Ref<Eigen::VectorXf>* grads[3] = {&xgrad, &ygrad, &zgrad};
 
    for (p = 0; p < 3; p++) {
        my_rd = rd;
        my_rd[p] += step;
        if (fwd_eeg_spherepot_coil(my_rd, Q, coils, *grads[p], client) == FAIL)
            return FAIL;
        my_rd = rd;
        my_rd[p] -= step;
        if (fwd_eeg_spherepot_coil(my_rd, Q, coils, Vval, client) == FAIL)
            return FAIL;
        for (q = 0; q < coils.ncoil(); q++)
            (*grads[p])[q] = ((*grads[p])[q] - Vval[q]) / step2;
    }
    if (fwd_eeg_spherepot_coil(rd, Q, coils, Vval, client) == FAIL)
        return FAIL;
    return OK;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
int FwdEegSphereModel::fwd_eeg_spherepot(const Eigen::Vector3f& rd_in,
                                         const Eigen::Vector3f& Q_in,
                                         const Eigen::Matrix<float, Eigen::Dynamic, 3, Eigen::RowMajor>& el,
                                         int neeg,
                                         VectorXf& Vval,
                                         void* client)
/*
      * This routine calculates the potentials for a specific dipole direction
      *
      * This routine uses the acceleration with help of equivalent sources
      * in the homogeneous sphere.
      */
{
    auto* m = static_cast<FwdEegSphereModel*>(client);
    float fact = static_cast<float>(0.25 / M_PI);
    Eigen::Vector3f a_vec;
    float a, a2, a3;
    float rrd, rd2, rd2_inv, r, r2, ra, rda;
    float F;
    float c1, c2, m1, m2, f1, f2;
    int k, eq;
    Eigen::Vector3f orig_rd = rd_in - m->r0;
    Eigen::Vector3f rd;
    Eigen::Vector3f Q = Q_in;
    Eigen::Vector3f pos;
    float pos_len;
    /*
   * Initialize the arrays
   */
    for (k = 0; k < neeg; k++)
        Vval[k] = 0.0;
    /*
   * Ignore dipoles outside the innermost sphere
   */
    if (orig_rd.norm() >= m->layers[0].rad)
        return true;
    /*
   * Default to homogeneous model if no model was previously set
   */
#ifdef FOO
    if (nequiv == 0) /* what to do */
        eeg_set_homog_sphere_model();
#endif
    /*
   * Make a weighted sum over the equivalence parameters
   */
    for (eq = 0; eq < m->nfit; eq++) {
        /*
     * Scale the dipole position
     */
        rd = m->mu[eq] * orig_rd;
 
        rd2 = rd.dot(rd);
        rd2_inv = 1.0 / rd2;
 
        f1 = rd.dot(Q);
        /*
     * Go over all electrodes
     */
        for (k = 0; k < neeg; k++) {
            pos = el.row(k).transpose() - m->r0;
            /*
       * Scale location onto the surface of the sphere
       */
            if (m->scale_pos) {
                pos_len = m->layers[m->nlayer() - 1].rad / pos.norm();
                pos *= pos_len;
            }
 
            /* Vector from dipole to the field point */
 
            a_vec = pos - rd;
 
            /* Compute the dot products needed */
 
            a2 = a_vec.dot(a_vec);
            a = sqrt(a2);
            a3 = 2.0 / (a2 * a);
            r2 = pos.dot(pos);
            r = sqrt(r2);
            rrd = pos.dot(rd);
            ra = r2 - rrd;
            rda = rrd - rd2;
 
            /* The main ingredients */
 
            F = a * (r * a + ra);
            c1 = a3 * rda + 1.0 / a - 1.0 / r;
            c2 = a3 + (a + r) / (r * F);
 
            /* Mix them together and scale by lambda/(rd*rd) */
 
            m1 = (c1 - c2 * rrd);
            m2 = c2 * rd2;
 
            f2 = pos.dot(Q);
            Vval[k] = Vval[k] + m->lambda[eq] * rd2_inv * (m1 * f1 + m2 * f2);
        } /* All electrodes done */
    } /* All equivalent dipoles done */
    /*
   * Finish by scaling by 1/(4*M_PI);
   */
    for (k = 0; k < neeg; k++)
        Vval[k] = fact * Vval[k];
    return OK;
}
 
//=============================================================================================================
// fwd_multi_spherepot.c
int FwdEegSphereModel::fwd_eeg_spherepot_coil(const Eigen::Vector3f& rd, const Eigen::Vector3f& Q, FwdCoilSet& els, Eigen::Ref<Eigen::VectorXf> Vval, void* client)
{
    VectorXf vval_one;
    float val;
    int nvval = 0;
    int k, c;
    FwdCoil* el;
 
    for (k = 0; k < els.ncoil(); k++) {
        el = els.coils[k].get();
        if (el->coil_class == FWD_COILC_EEG) {
            if (el->np > nvval) {
                vval_one.resize(el->np);
                nvval = el->np;
            }
            if (fwd_eeg_spherepot(rd, Q, el->rmag, el->np, vval_one, client) != OK) {
                return FAIL;
            }
            for (c = 0, val = 0.0; c < el->np; c++)
                val += el->w[c] * vval_one[c];
            Vval[k] = val;
        }
    }
    return OK;
}
 
//=============================================================================================================
// fwd_eeg_sphere_models.c
bool FwdEegSphereModel::fwd_setup_eeg_sphere_model(float rad, bool fit_berg_scherg, int nFit)
{
    int nTerms = 200;
    float rv;
 
    /*
     * Scale the relative radiuses
     */
    for (int k = 0; k < this->nlayer(); k++)
        this->layers[k].rad = rad * this->layers[k].rel_rad;
 
    if (fit_berg_scherg) {
        if (this->fwd_eeg_fit_berg_scherg(nTerms, nFit, rv)) {
            qInfo("Equiv. model fitting -> RV = %g %%", 100 * rv);
            for (int k = 0; k < nFit; k++)
                qInfo("mu%d = %g\tlambda%d = %g", k + 1, this->mu[k], k + 1, this->layers[this->nlayer() - 1].sigma * this->lambda[k]);
        } else
            return false;
    }
 
    qInfo("Defined EEG sphere model with rad = %7.2f mm", 1000.0 * rad);
    return true;
}
 
static void compute_svd(Eigen::MatrixXd& mat,
                        Eigen::VectorXd& sing,
                        Eigen::MatrixXd& uu,
                        Eigen::MatrixXd* vv)
/*
 * Compute the SVD of mat.
 * Results are stored in sing, uu, and optionally vv.
 * mat is not modified.
 */
{
    int udim = std::min(static_cast<int>(mat.rows()), static_cast<int>(mat.cols()));
 
    Eigen::JacobiSVD<Eigen::MatrixXd> svd(mat, Eigen::ComputeFullU | Eigen::ComputeFullV);
 
    sing = svd.singularValues();
    uu = svd.matrixU().transpose().topRows(udim);
 
    if (vv != nullptr)
        *vv = svd.matrixV().transpose();
}
 
/*
 * Include the simplex and SVD code here.
 * It is not too much of a problem
 */
 
 
namespace FWDLIB
{

struct BergSchergPar
{
    double lambda; 
    double mu;     
};
 
} // namespace
 
static void sort_parameters(VectorXd& mu, VectorXd& lambda, int nfit)
{
    std::vector<BergSchergPar> pars(nfit);
 
    for (int k = 0; k < nfit; k++) {
        pars[k].mu = mu[k];
        pars[k].lambda = lambda[k];
    }
 
    std::sort(pars.begin(), pars.end(), [](const BergSchergPar& a, const BergSchergPar& b) {
        return a.mu > b.mu;
    });
 
    for (int k = 0; k < nfit; k++) {
        mu[k] = pars[k].mu;
        lambda[k] = pars[k].lambda;
    }
}
 
static bool report_fit([[maybe_unused]] int loop,
                       [[maybe_unused]] const VectorXd& fitpar,
                       [[maybe_unused]] double Smin,
                       double /*fval_hi*/,
                       double /*par_diff*/)
{
#ifdef LOG_FIT
    for (int k = 0; k < fitpar.size(); k++)
        qInfo("%g ", mu[k]);
    qInfo("%g", Smin);
#endif
    return true;
}
 
static MatrixXd get_initial_simplex(const VectorXd& pars,
                                    double simplex_size)
 
{
    int npar = pars.size();
 
    MatrixXd simplex = MatrixXd::Zero(npar + 1, npar);
 
    simplex.rowwise() += pars.transpose();
 
    for (int k = 1; k < npar + 1; k++)
        simplex(k, k - 1) += simplex_size;
 
    return simplex;
}
 
void FwdEegSphereModel::compose_linear_fitting_data(const VectorXd& mu, fitUser u)
{
    double mu1n, k1;
    int k, p;
    /*
   * y is the data to be fitted (nterms-1 x 1)
   * M is the model matrix      (nterms-1 x nfit-1)
   */
    for (k = 0; k < u->nterms - 1; k++) {
        k1 = k + 1;
        mu1n = pow(mu[0], k1);
        u->y[k] = u->w[k] * (u->fn[k + 1] - mu1n * u->fn[0]);
        for (p = 0; p < u->nfit - 1; p++)
            u->M(k, p) = u->w[k] * (pow(mu[p + 1], k1) - mu1n);
    }
}
 
// fwd_fit_berg_scherg.c
double FwdEegSphereModel::compute_linear_parameters(const VectorXd& mu,
                                                    VectorXd& lambda,
                                                    fitUser u)
/*
      * Compute the best-fitting linear parameters
      * Return the corresponding RV
      */
{
    int k, p, q;
    VectorXd vec(u->nfit - 1);
    double sum;
 
    compose_linear_fitting_data(mu, u);
 
    compute_svd(u->M, u->sing, u->uu, &u->vv);
    /*
   * Compute the residuals
   */
    for (k = 0; k < u->nterms - 1; k++)
        u->resi[k] = u->y[k];
 
    for (p = 0; p < u->nfit - 1; p++) {
        vec[p] = u->uu.row(p).head(u->nterms - 1).dot(u->y.head(u->nterms - 1));
        for (k = 0; k < u->nterms - 1; k++)
            u->resi[k] = u->resi[k] - u->uu(p, k) * vec[p];
        vec[p] = vec[p] / u->sing[p];
    }
 
    for (p = 0; p < u->nfit - 1; p++) {
        for (q = 0, sum = 0.0; q < u->nfit - 1; q++)
            sum += u->vv(q, p) * vec[q];
        lambda[p + 1] = sum;
    }
    for (p = 1, sum = 0.0; p < u->nfit; p++)
        sum += lambda[p];
    lambda[0] = u->fn[0] - sum;
    return u->resi.head(u->nterms - 1).squaredNorm() / u->y.head(u->nterms - 1).squaredNorm();
}
 
// fwd_fit_berg_scherg.c
double FwdEegSphereModel::one_step(const VectorXd& mu, fitUserRec& user)
/*
      * Evaluate the residual sum of squares fit for one set of
      * mu values
      */
{
    int k, p;
    double dot;
    fitUser u = &user;
 
    for (k = 0; k < u->nfit; k++) {
        if (std::fabs(mu[k]) > 1.0)
            return 1.0;
    }
    /*
   * Compose the data for the linear fitting
   */
    compose_linear_fitting_data(mu, u);
    /*
   * Compute SVD
   */
    compute_svd(u->M, u->sing, u->uu, nullptr);
    /*
   * Compute the residuals
   */
    for (k = 0; k < u->nterms - 1; k++)
        u->resi[k] = u->y[k];
    for (p = 0; p < u->nfit - 1; p++) {
        dot = u->uu.row(p).head(u->nterms - 1).dot(u->y.head(u->nterms - 1));
        for (k = 0; k < u->nterms - 1; k++)
            u->resi[k] = u->resi[k] - u->uu(p, k) * dot;
    }
    /*
   * Return their sum of squares
   */
    return u->resi.head(u->nterms - 1).squaredNorm();
}
 
// fwd_fit_berg_scherg.c
bool FwdEegSphereModel::fwd_eeg_fit_berg_scherg(int nTerms, /* Number of terms to use in the series expansion
                                                                                    * when fitting the parameters */
                                                int nFit,   /* Number of equivalent dipoles to fit */
                                                float& rv)
/*
      * This routine fits the Berg-Scherg equivalent spherical model
      * dipole parameters by minimizing the difference between the
      * actual and approximative series expansions
      */
{
    bool res = false;
    int k;
    double rd, R, f;
    double simplex_size = 0.01;
    MatrixXd simplex;
    VectorXd func_val;
    double ftol = 1e-9;
    VectorXd lambdaFit;
    VectorXd muFit;
    int neval;
    int max_eval = 1000;
    int report = 1;
    const auto u = new_fit_user(nFit, nTerms);
 
    if (nFit < 2) {
        qWarning("fwd_fit_berg_scherg does not work with less than two equivalent sources.");
        return false;
    }
 
    /*
   * (1) Calculate the coefficients of the true expansion
   */
    for (k = 0; k < nTerms; k++)
        u->fn[k] = this->fwd_eeg_get_multi_sphere_model_coeff(k + 1);
 
    /*
   * (2) Calculate the weighting
   */
    rd = R = this->layers[0].rad;
    for (k = 1; k < this->nlayer(); k++) {
        if (this->layers[k].rad > R)
            R = this->layers[k].rad;
        if (this->layers[k].rad < rd)
            rd = this->layers[k].rad;
    }
    f = rd / R;
 
#ifdef ZHANG
    /*
   * This is the Zhang weighting
   */
    for (k = 1; k < nTerms; k++)
        u->w[k - 1] = pow(f, k);
#else
    /*
   * This is the correct weighting
   */
    for (k = 1; k < nTerms; k++)
        u->w[k - 1] = sqrt((2.0 * k + 1) * (3.0 * k + 1.0) / k) * pow(f, (k - 1.0));
#endif
 
    /*
   * (3) Prepare for simplex minimization
   */
    func_val = VectorXd(nFit + 1);
    lambdaFit = VectorXd(nFit);
    muFit = VectorXd(nFit);
    /*
   * (4) Rather arbitrary initial guess
   */
    for (k = 0; k < nFit; k++) {
        muFit[k] = (k + 1) * 0.1 * f;
    }
 
    simplex = get_initial_simplex(muFit, simplex_size);
    for (k = 0; k < nFit + 1; k++)
        func_val[k] = one_step(VectorXd(simplex.row(k).transpose()), *u);
 
    // Capture user data in type-safe lambdaFit — no void* needed
    auto cost = [&u](const VectorXd& x) -> double {
        return one_step(x, *u);
    };
 
    /*
   * (5) Do the nonlinear minimization
   */
    res = SimplexAlgorithm::simplex_minimize<double>(simplex,
                                                     func_val,
                                                     ftol,
                                                     0.0,
                                                     cost,
                                                     max_eval,
                                                     neval,
                                                     report,
                                                     report_fit);
 
    if (res) {
        for (k = 0; k < nFit; k++)
            muFit[k] = simplex(0, k);
 
        /*
       * (6) Do the final step: calculation of the linear parameters
       */
        rv = compute_linear_parameters(muFit, lambdaFit, u.get());
 
        sort_parameters(muFit, lambdaFit, nFit);
#ifdef LOG_FIT
        qInfo("RV = %g %%", 100 * rv);
#endif
        this->mu.resize(nFit);
        this->lambda.resize(nFit);
        this->nfit = nFit;
        for (k = 0; k < nFit; k++) {
            this->mu[k] = muFit[k];
            /*
         * This division takes into account the actual conductivities
         */
            this->lambda[k] = lambdaFit[k] / this->layers[this->nlayer() - 1].sigma;
#ifdef LOG_FIT
            qInfo("lambdaFit%d = %g\tmu%d = %g", k + 1, lambdaFit[k], k + 1, muFit[k]);
#endif
        }
    }
 
    return res;
}