bool solveLinearSystem()
{
X.resize (L.rows (), B.cols ());
// Nothing to solve
if (L.rows () == 0 || B.cols () == 0)
return true;
Eigen::SimplicialCholesky<SparseMatrix, Eigen::Lower> cg;
cg.compute (L);
bool succeeded = true;
for (int i = 0; i < B.cols (); ++i)
{
Vector b = B.col (i);
X.col (i) = cg.solve (b);
if (cg.info () != Eigen::Success)
succeeded = false;
}
assignColors ();
return succeeded;
}
static void
_solve_linear_system(TriMesh& src, const TriMesh& dst,
const std::vector<std::vector<int>>& tri_neighbours,
const std::vector<std::pair<int, int> >& corres,
double weights[3]
) {
int rows = 0; int cols = src.vert_num + src.poly_num - corres.size();
assert (tri_neighbours.size() == src.poly_num);
std::vector<std::pair<int, int>> soft_corres;
_setup_kd_correspondence(src, dst, soft_corres);
#if 0
std::ofstream ofs("E:/data/ScapeOriginal/build/data_ming/dt_pairs.txt");
ofs << soft_corres.size() << endl;
for (size_t i=0; i<soft_corres.size(); ++i)
ofs << soft_corres[i].first << "\t" << soft_corres[i].second << std::endl;
printf("check soft pair\n");
getchar();
#endif
for (int i=0; i<src.poly_num; ++i)
rows += 3*tri_neighbours[i].size(); //smooth part
rows += src.poly_num*3; //identity part
static std::vector<bool> vertex_flag; //indicate whether the vertex is hard constrainted by corres
if (vertex_flag.empty()) {
vertex_flag.resize(src.vert_num, false);
for (size_t i=0; i<corres.size(); ++i)
vertex_flag[corres[i].first] = true;
}
for (int i=0; i<soft_corres.size(); ++i) { //soft constraints part
if (vertex_flag[soft_corres[i].first]) continue;
++rows;
}
//vertex_real_col stores two information : unknow vertex's col(offset by
//poly_num) in X and know vertexs' corresponding index in dst mesh.
//there is no need to compute this in each iteration
static std::vector<int> vertex_real_col;
if (vertex_real_col.empty()) {
vertex_real_col.resize(src.vert_num, -1);
std::map<int, int> corres_map;
for (int i=0; i<corres.size(); ++i) corres_map.insert(std::make_pair(corres[i].first, corres[i].second));
int real_col = 0;
for (int i=0; i<src.vert_num; ++i) {
if (vertex_flag[i]) {
vertex_real_col[i] = corres_map[i];
} else {
vertex_real_col[i] = real_col;
real_col++;
}
}
assert (real_col == src.vert_num - corres.size()); //make sure indexes in corres are different from each other
}
SparseMatrix<double> A(rows, cols);
A.reserve(Eigen::VectorXi::Constant(cols, 200));
std::vector<VectorXd> Y(3, VectorXd(rows));
Y[0].setZero();Y[1].setZero();Y[2].setZero();
//precompute Q_hat^-1 in Q_hat_inverse [n v2-v1 v3-v1]^-1
src.updateNorm();
assert (!src.face_norm.empty());
std::vector<Matrix3d> Q_hat_inverse(src.poly_num);
Eigen::Matrix3d _inverse;
for (int i=0; i<src.poly_num; ++i) {
unsigned int* index_i = src.polyIndex[i].vert_index;
Vector3d v[3];
v[0] = src.face_norm[i];
v[1] = src.vertex_coord[index_i[1]] - src.vertex_coord[index_i[0]];//v2-v1
v[2] = src.vertex_coord[index_i[2]] - src.vertex_coord[index_i[0]];//v3-v1
for (int k=0; k<3; ++k)
for (int j=0; j<3; ++j)
Q_hat_inverse[i](k, j) = v[j][k];
_inverse = Q_hat_inverse[i].inverse();
Q_hat_inverse[i] = _inverse;
}
int energy_size[3] = {0, 0, 0}; //each energy part's starting index
//start establishing the large linear sparse system
double weight_smooth = weights[0];
int row = 0;
for (int i=0; i<src.poly_num; ++i) {
Eigen::Matrix3d& Q_i_hat = Q_hat_inverse[i];
unsigned int* index_i = src.polyIndex[i].vert_index;
for (size_t _j=0; _j<tri_neighbours[i].size(); ++_j) {
int j = tri_neighbours[i][_j]; //triangle index
Eigen::Matrix3d& Q_j_hat = Q_hat_inverse[j];
unsigned int* index_j = src.polyIndex[j].vert_index;
for (int k=0; k<3; ++k)
for (int dim = 0; dim < 3; ++dim)
Y[dim](row+k) = 0.0;
for (int k=0; k<3; ++k) {
A.coeffRef(row+k, i) = weight_smooth*Q_i_hat(0, k); //n
A.coeffRef(row+k, j) = -weight_smooth*Q_j_hat(0, k); //n
}
for (int k=0; k<3; ++k)
for (int p=0; p<3; ++p)
if (!vertex_flag[index_i[p]])
A.coeffRef(row+k, src.poly_num+vertex_real_col[index_i[p]]) = 0.0;
if (vertex_flag[index_i[0]]) {
for (int k=0; k<3; ++k) {
for (int dim = 0; dim < 3; ++dim)
Y[dim](row + k) += weight_smooth*(Q_i_hat(1, k)+Q_i_hat(2, k))*dst.vertex_coord[vertex_real_col[index_i[0]]][dim];
}
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num + vertex_real_col[index_i[0]]) +=
-weight_smooth*(Q_i_hat(1, k)+Q_i_hat(2, k));
}
if (vertex_flag[index_j[0]]) {
for (int k=0; k<3; ++k) {
for (int dim = 0; dim < 3; ++dim)
Y[dim](row + k) += -weight_smooth*(Q_j_hat(1, k)+Q_j_hat(2, k))*dst.vertex_coord[vertex_real_col[index_j[0]]][dim];
}
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num + vertex_real_col[index_j[0]]) +=
weight_smooth*(Q_j_hat(1, k)+Q_j_hat(2, k));
}
for (int p=1; p<3; ++p) {//v2 or v3
if (vertex_flag[index_i[p]]) {
for (int k=0; k<3; ++k)
for (int dim=0; dim<3; ++dim)
Y[dim](row + k) += -weight_smooth*Q_i_hat(p, k)*dst.vertex_coord[vertex_real_col[index_i[p]]][dim];
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num + vertex_real_col[index_i[p]]) += weight_smooth*Q_i_hat(p, k);
}
}
for (int p=1; p<3; ++p) {
if (vertex_flag[index_j[p]]) {
for (int k=0; k<3; ++k)
for (int dim=0; dim < 3; ++dim)
Y[dim](row + k) += weight_smooth*Q_j_hat(p, k)*dst.vertex_coord[vertex_real_col[index_j[p]]][dim];
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num + vertex_real_col[index_j[p]]) += -weight_smooth*Q_j_hat(p, k);
}
}
row += 3;
}
}
energy_size[0] = row;
double weight_identity = weights[1];
for (int i=0; i<src.poly_num; ++i) {
Eigen::Matrix3d& Q_i_hat = Q_hat_inverse[i];
unsigned int* index_i = src.polyIndex[i].vert_index;
Y[0](row) = weight_identity; Y[0](row+1) = 0.0; Y[0](row+2) = 0.0;
Y[1](row) = 0.0; Y[1](row+1) = weight_identity; Y[1](row+2) = 0.0;
Y[2](row) = 0.0; Y[2](row+1) = 0.0; Y[2](row+2) = weight_identity;
for (int k=0; k<3; ++k)
A.coeffRef(row+k, i) = weight_identity*Q_i_hat(0, k); //n
if (vertex_flag[index_i[0]]) {
for (int k=0; k<3; ++k)
for (int dim = 0; dim < 3; ++dim)
Y[dim](row+k) += weight_identity*(Q_i_hat(1, k)+Q_i_hat(2,k))*dst.vertex_coord[vertex_real_col[index_i[0]]][dim];
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num+vertex_real_col[index_i[0]]) = -weight_identity*(Q_i_hat(1, k)+Q_i_hat(2,k));
}
for (int p=1; p<3; ++p) {
if (vertex_flag[index_i[p]]) {
for (int k=0; k<3; ++k)
for (int dim=0; dim<3; ++dim)
Y[dim](row + k) += -weight_identity*Q_i_hat(p, k)*dst.vertex_coord[vertex_real_col[index_i[p]]][dim];
} else {
for (int k=0; k<3; ++k)
A.coeffRef(row+k, src.poly_num + vertex_real_col[index_i[p]]) = weight_identity*Q_i_hat(p, k);
}
}
row += 3;
}
energy_size[1] = row;
double weight_soft_constraint = weights[2];
for (int i=0; i<soft_corres.size(); ++i) {
if (vertex_flag[soft_corres[i].first]) continue;
A.coeffRef(row, src.poly_num + vertex_real_col[soft_corres[i].first]) = weight_soft_constraint;
for (int dim=0; dim<3; ++dim)
Y[dim](row) += weight_soft_constraint*dst.vertex_coord[soft_corres[i].second][dim];
++row;
}
energy_size[2] = row;
assert (row == rows);
//start solving the least-square problem
fprintf(stdout, "finished filling matrix\n");
Eigen::SparseMatrix<double> At = A.transpose();
Eigen::SparseMatrix<double> AtA = At*A;
Eigen::SimplicialCholesky<SparseMatrix<double>> solver;
solver.compute(AtA);
if (solver.info() != Eigen::Success) {
fprintf(stdout, "unable to defactorize AtA\n");
exit(-1);
}
VectorXd X[3];
for (int i=0; i<3; ++i) {
VectorXd AtY = At*Y[i];
X[i] = solver.solve(AtY);
Eigen::VectorXd Energy = A*X[i] - Y[i];
Eigen::VectorXd smoothEnergy = Energy.head(energy_size[0]);
Eigen::VectorXd identityEnergy = Energy.segment(energy_size[0], energy_size[1]-energy_size[0]);
Eigen::VectorXd softRegularEnergy = Energy.tail(energy_size[2]-energy_size[1]);
fprintf(stdout, "\t%lf = %lf + %lf + %lf\n",
Energy.dot(Energy), smoothEnergy.dot(smoothEnergy),
identityEnergy.dot(identityEnergy), softRegularEnergy.dot(softRegularEnergy));
}
//fill data back to src
for (int i=0; i<src.poly_num; ++i)
for (int d=0; d<3; ++d)
src.face_norm[i][d] = X[d](i);
for (size_t i=0; i<corres.size(); ++i) src.vertex_coord[corres[i].first] = dst.vertex_coord[corres[i].second];
int p = 0;
for (int i=0; i<src.vert_num; ++i) {
if (vertex_flag[i]) continue;
for (int d=0; d<3; ++d)
src.vertex_coord[i][d] = X[d](src.poly_num+p);
++p;
}
return;
}