void assemble_cortical(const Geometry& geo, Matrix& mat, const Head2EEGMat& M, const std::string& domain_name, const unsigned gauss_order, double alpha, double beta, const std::string &filename)
{
// Following the article: M. Clerc, J. Kybic "Cortical mapping by Laplace–Cauchy transmission using a boundary element method".
// Assumptions:
// - domain_name: the domain containing the sources is an innermost domain (defined as the interior of only one interface (called Cortex)
// - Cortex interface is composed of one mesh only (no shared vertices)
// TODO check orders of MxM products for efficiency ... delete intermediate matrices
const Domain& SourceDomain = geo.domain(domain_name);
const Interface& Cortex = SourceDomain.begin()->interface();
const Mesh& cortex = Cortex.begin()->mesh();
// test the assumption
assert(SourceDomain.size() == 1);
assert(Cortex.size() == 1);
// shape of the new matrix:
unsigned Nl = geo.size()-geo.outermost_interface().nb_triangles()-Cortex.nb_vertices()-Cortex.nb_triangles();
unsigned Nc = geo.size()-geo.outermost_interface().nb_triangles();
std::fstream f(filename.c_str());
Matrix P;
if ( !f ) {
// build the HeadMat:
// The following is the same as assemble_HM except N_11, D_11 and S_11 are not computed.
SymMatrix mat_temp(Nc);
mat_temp.set(0.0);
double K = 1.0 / (4.0 * M_PI);
// We iterate over the meshes (or pair of domains) to fill the lower half of the HeadMat (since its symmetry)
for ( Geometry::const_iterator mit1 = geo.begin(); mit1 != geo.end(); ++mit1) {
for ( Geometry::const_iterator mit2 = geo.begin(); (mit2 != (mit1+1)); ++mit2) {
// if mit1 and mit2 communicate, i.e they are used for the definition of a common domain
const int orientation = geo.oriented(*mit1, *mit2); // equals 0, if they don't have any domains in common
// equals 1, if they are both oriented toward the same domain
// equals -1, if they are not
if ( orientation != 0) {
double Scoeff = orientation * geo.sigma_inv(*mit1, *mit2) * K;
double Dcoeff = - orientation * geo.indicator(*mit1, *mit2) * K;
double Ncoeff;
if ( !(mit1->outermost() || mit2->outermost()) && ( (*mit1 != *mit2)||( *mit1 != cortex) ) ) {
// Computing S block first because it's needed for the corresponding N block
operatorS(*mit1, *mit2, mat_temp, Scoeff, gauss_order);
Ncoeff = geo.sigma(*mit1, *mit2)/geo.sigma_inv(*mit1, *mit2);
} else {
Ncoeff = orientation * geo.sigma(*mit1, *mit2) * K;
}
if ( !mit1->outermost() && (( (*mit1 != *mit2)||( *mit1 != cortex) )) ) {
// Computing D block
operatorD(*mit1, *mit2, mat_temp, Dcoeff, gauss_order);
}
if ( ( *mit1 != *mit2 ) && ( !mit2->outermost() ) ) {
// Computing D* block
operatorD(*mit1, *mit2, mat_temp, Dcoeff, gauss_order, true);
}
// Computing N block
if ( (*mit1 != *mit2)||( *mit1 != cortex) ) {
operatorN(*mit1, *mit2, mat_temp, Ncoeff, gauss_order);
}
}
}
}
// Deflate the diagonal block (N33) of 'mat' : (in order to have a zero-mean potential for the outermost interface)
const Interface i = geo.outermost_interface();
unsigned i_first = (*i.begin()->mesh().vertex_begin())->index();
deflat(mat_temp, i, mat_temp(i_first, i_first) / (geo.outermost_interface().nb_vertices()));
mat = Matrix(Nl, Nc);
mat.set(0.0);
// copy mat_temp into mat except the lines for cortex vertices [i_vb_c, i_ve_c] and cortex triangles [i_tb_c, i_te_c].
unsigned iNl = 0;
unsigned i_vb_c = (*cortex.vertex_begin())->index();
unsigned i_ve_c = (*cortex.vertex_rbegin())->index();
unsigned i_tb_c = cortex.begin()->index();
unsigned i_te_c = cortex.rbegin()->index();
for ( unsigned i = 0; i < Nc; ++i) {
if ( !(i_vb_c<=i && i<=i_ve_c) && !(i_tb_c<=i && i<=i_te_c) ) {
mat.setlin(iNl, mat_temp.getlin(i));
++iNl;
}
}
// ** Construct P: the null-space projector **
Matrix W;
{
Matrix U, s;
mat.svd(U, s, W);
}
SparseMatrix S(Nc,Nc);
// we set S to 0 everywhere, except in the last part of the diag:
for ( unsigned i = Nl; i < Nc; ++i) {
S(i, i) = 1.0;
}
P = (W * S) * W.transpose(); // P is a projector: P^2 = P and mat*P*X = 0
if ( filename.length() != 0 ) {
std::cout << "Saving projector P (" << filename << ")." << std::endl;
P.save(filename);
}
} else {
std::cout << "Loading projector P (" << filename << ")." << std::endl;
P.load(filename);
}
// ** Get the gradient of P1&P0 elements on the meshes **
Matrix MM(M.transpose() * M);
SymMatrix RR(Nc, Nc); RR.set(0.);
for ( Geometry::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
mit->gradient_norm2(RR);
}
// ** Choose Regularization parameter **
SparseMatrix alphas(Nc,Nc); // diagonal matrix
Matrix Z;
if ( alpha < 0 ) { // try an automatic method... TODO find better estimation
double nRR_v = RR.submat(0, geo.nb_vertices(), 0, geo.nb_vertices()).frobenius_norm();
alphas.set(0.);
alpha = MM.frobenius_norm() / (1.e3*nRR_v);
beta = alpha * 50000.;
for ( Vertices::const_iterator vit = geo.vertex_begin(); vit != geo.vertex_end(); ++vit) {
alphas(vit->index(), vit->index()) = alpha;
}
for ( Meshes::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
if ( !mit->outermost() ) {
for ( Mesh::const_iterator tit = mit->begin(); tit != mit->end(); ++tit) {
alphas(tit->index(), tit->index()) = beta;
}
}
}
std::cout << "AUTOMATIC alphas = " << alpha << "\tbeta = " << beta << std::endl;
} else {
for ( Vertices::const_iterator vit = geo.vertex_begin(); vit != geo.vertex_end(); ++vit) {
alphas(vit->index(), vit->index()) = alpha;
}
for ( Meshes::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
if ( !mit->outermost() ) {
for ( Mesh::const_iterator tit = mit->begin(); tit != mit->end(); ++tit) {
alphas(tit->index(), tit->index()) = beta;
}
}
}
std::cout << "alphas = " << alpha << "\tbeta = " << beta << std::endl;
}
Z = P.transpose() * (MM + alphas*RR) * P;
// ** PseudoInverse and return **
// X = P * { (M*P)' * (M*P) + (R*P)' * (R*P) }¡(-1) * (M*P)'m
// X = P * { P'*M'*M*P + P'*R'*R*P }¡(-1) * P'*M'm
// X = P * { P'*(MM + a*RR)*P }¡(-1) * P'*M'm
// X = P * Z¡(-1) * P' * M'm
Matrix rhs = P.transpose() * M.transpose();
mat = P * Z.pinverse() * rhs;
}
void assemble_cortical2(const Geometry& geo, Matrix& mat, const Head2EEGMat& M, const std::string& domain_name, const unsigned gauss_order, double gamma, const std::string &filename)
{
// Re-writting of the optimization problem in M. Clerc, J. Kybic "Cortical mapping by Laplace–Cauchy transmission using a boundary element method".
// with a Lagrangian formulation as in see http://www.math.uh.edu/~rohop/fall_06/Chapter3.pdf eq3.3
// find argmin(norm(gradient(X)) under constraints:
// H * X = 0 and M * X = m
// let G be the gradient norm matrix, l1, l2 the lagrange parameters
//
// [ G H' M'] [ X ] [ 0 ]
// | H 0 | | l1 | = | 0 |
// [ M 0 ] [ l2 ] [ m ]
//
// {----,----}
// K
// we want a submat of the inverse of K (using blockwise inversion, (TODO maybe iterative solution better ?)).
// Assumptions:
// - domain_name: the domain containing the sources is an innermost domain (defined as the interior of only one interface (called Cortex)
// - Cortex interface is composed of one mesh only (no shared vertices)
const Domain& SourceDomain = geo.domain(domain_name);
const Interface& Cortex = SourceDomain.begin()->interface();
const Mesh& cortex = Cortex.begin()->mesh();
om_error(SourceDomain.size()==1);
om_error(Cortex.size()==1);
// shape of the new matrix:
unsigned Nl = geo.size()-geo.outermost_interface().nb_triangles()-Cortex.nb_vertices()-Cortex.nb_triangles();
unsigned Nc = geo.size()-geo.outermost_interface().nb_triangles();
std::fstream f(filename.c_str());
Matrix H;
if ( !f ) {
// build the HeadMat:
// The following is the same as assemble_HM except N_11, D_11 and S_11 are not computed.
SymMatrix mat_temp(Nc);
mat_temp.set(0.0);
double K = 1.0 / (4.0 * M_PI);
// We iterate over the meshes (or pair of domains) to fill the lower half of the HeadMat (since its symmetry)
for ( Geometry::const_iterator mit1 = geo.begin(); mit1 != geo.end(); ++mit1) {
for ( Geometry::const_iterator mit2 = geo.begin(); (mit2 != (mit1+1)); ++mit2) {
// if mit1 and mit2 communicate, i.e they are used for the definition of a common domain
const int orientation = geo.oriented(*mit1, *mit2); // equals 0, if they don't have any domains in common
// equals 1, if they are both oriented toward the same domain
// equals -1, if they are not
if ( orientation != 0) {
double Scoeff = orientation * geo.sigma_inv(*mit1, *mit2) * K;
double Dcoeff = - orientation * geo.indicator(*mit1, *mit2) * K;
double Ncoeff;
if ( !(mit1->outermost() || mit2->outermost()) && ( (*mit1 != *mit2)||( *mit1 != cortex) ) ) {
// Computing S block first because it's needed for the corresponding N block
operatorS(*mit1, *mit2, mat_temp, Scoeff, gauss_order);
Ncoeff = geo.sigma(*mit1, *mit2)/geo.sigma_inv(*mit1, *mit2);
} else {
Ncoeff = orientation * geo.sigma(*mit1, *mit2) * K;
}
if ( !mit1->outermost() && (( (*mit1 != *mit2)||( *mit1 != cortex) )) ) {
// Computing D block
operatorD(*mit1, *mit2, mat_temp, Dcoeff, gauss_order);
}
if ( ( *mit1 != *mit2 ) && ( !mit2->outermost() ) ) {
// Computing D* block
operatorD(*mit1, *mit2, mat_temp, Dcoeff, gauss_order, true);
}
// Computing N block
if ( (*mit1 != *mit2)||( *mit1 != cortex) ) {
operatorN(*mit1, *mit2, mat_temp, Ncoeff, gauss_order);
}
}
}
}
// Deflate the diagonal block (N33) of 'mat' : (in order to have a zero-mean potential for the outermost interface)
const Interface i = geo.outermost_interface();
unsigned i_first = (*i.begin()->mesh().vertex_begin())->index();
deflat(mat_temp, i, mat_temp(i_first, i_first) / (geo.outermost_interface().nb_vertices()));
H = Matrix(Nl + M.nlin(), Nc);
H.set(0.0);
// copy mat_temp into H except the lines for cortex vertices [i_vb_c, i_ve_c] and cortex triangles [i_tb_c, i_te_c].
unsigned iNl = 0;
for ( Geometry::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
if ( *mit != cortex ) {
for ( Mesh::const_vertex_iterator vit = mit->vertex_begin(); vit != mit->vertex_end(); ++vit) {
H.setlin(iNl, mat_temp.getlin((*vit)->index()));
++iNl;
}
if ( !mit->outermost() ) {
for ( Mesh::const_iterator tit = mit->begin(); tit != mit->end(); ++tit) {
H.setlin(iNl, mat_temp.getlin(tit->index()));
++iNl;
}
}
}
}
if ( filename.length() != 0 ) {
std::cout << "Saving matrix H (" << filename << ")." << std::endl;
H.save(filename);
}
} else {
std::cout << "Loading matrix H (" << filename << ")." << std::endl;
H.load(filename);
}
// concat M to H
for ( unsigned i = Nl; i < Nl + M.nlin(); ++i) {
for ( unsigned j = 0; j < Nc; ++j) {
H(i, j) = M(i-Nl, j);
}
}
// ** Get the gradient of P1&P0 elements on the meshes **
SymMatrix G(Nc);
G.set(0.);
for ( Geometry::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
mit->gradient_norm2(G);
}
// multiply by gamma the submat of current gradient norm2
for ( Meshes::const_iterator mit = geo.begin(); mit != geo.end(); ++mit) {
if ( !mit->outermost() ) {
for ( Mesh::const_iterator tit1 = mit->begin(); tit1 != mit->end(); ++tit1) {
for ( Mesh::const_iterator tit2 = mit->begin(); tit2 != mit->end(); ++tit2) {
G(tit1->index(), tit2->index()) *= gamma;
}
}
}
}
std::cout << "gamma = " << gamma << std::endl;
G.invert();
mat = (G * H.transpose() * (H * G * H.transpose()).inverse()).submat(0, Nc, Nl, M.nlin());
}