libigl_Martin/include/igl/slim.cpp

// This file is part of libigl, a simple c++ geometry processing library.
//
// Copyright (C) 2016 Michael Rabinovich
//
// This Source Code Form is subject to the terms of the Mozilla Public License
// v. 2.0. If a copy of the MPL was not distributed with this file, You can
// obtain one at http://mozilla.org/MPL/2.0/.
#include "slim.h"

#include "boundary_loop.h"
#include "cotmatrix.h"
#include "edge_lengths.h"
#include "grad.h"
#include "local_basis.h"
#include "repdiag.h"
#include "vector_area_matrix.h"
#include "arap.h"
#include "cat.h"
#include "doublearea.h"
#include "grad.h"
#include "local_basis.h"
#include "per_face_normals.h"
#include "slice_into.h"
#include "volume.h"
#include "polar_svd.h"
#include "flip_avoiding_line_search.h"

#include <iostream>
#include <map>
#include <set>
#include <vector>

#include <Eigen/IterativeLinearSolvers>
#include <Eigen/SparseCholesky>
#include <Eigen/IterativeLinearSolvers>


using namespace std;
using namespace Eigen;

///////// Helper functions to compute gradient matrices

namespace igl
{
  namespace slim
  {
    // Definitions of internal functions
    void compute_surface_gradient_matrix(const Eigen::MatrixXd &V, const Eigen::MatrixXi &F,
                                         const Eigen::MatrixXd &F1, const Eigen::MatrixXd &F2,
                                         Eigen::SparseMatrix<double> &D1, Eigen::SparseMatrix<double> &D2);
    void buildA(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &A);
    void buildRhs(igl::SLIMData& m_state, const Eigen::SparseMatrix<double> &At);
    void add_soft_constraints(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &L);
    double compute_energy(igl::SLIMData& m_state, Eigen::MatrixXd &V_new);
    double compute_soft_const_energy(igl::SLIMData& m_state,
                                     const Eigen::MatrixXd &V,
                                     const Eigen::MatrixXi &F,
                                     Eigen::MatrixXd &V_o);
    double compute_energy_with_jacobians(igl::SLIMData& m_state,
                                         const Eigen::MatrixXd &V,
                                         const Eigen::MatrixXi &F, const Eigen::MatrixXd &Ji,
                                         Eigen::MatrixXd &uv, Eigen::VectorXd &areas);

    void solve_weighted_arap(igl::SLIMData& m_state,
                             const Eigen::MatrixXd &V,
                             const Eigen::MatrixXi &F,
                             Eigen::MatrixXd &uv,
                             Eigen::VectorXi &soft_b_p,
                             Eigen::MatrixXd &soft_bc_p);

    void update_weights_and_closest_rotations(igl::SLIMData& m_state,
                                              const Eigen::MatrixXd &V,
                                              const Eigen::MatrixXi &F,
                                              Eigen::MatrixXd &uv);
    void solve_weighted_proxy(igl::SLIMData& m_state, Eigen::MatrixXd &V_new);
    void compute_jacobians(igl::SLIMData& m_state, const Eigen::MatrixXd &uv);
    void build_linear_system(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &L);

    // Implementation

    void compute_surface_gradient_matrix(const Eigen::MatrixXd &V, const Eigen::MatrixXi &F,
                                         const Eigen::MatrixXd &F1, const Eigen::MatrixXd &F2,
                                         Eigen::SparseMatrix<double> &D1, Eigen::SparseMatrix<double> &D2)
    {

      Eigen::SparseMatrix<double> G;
      igl::grad(V, F, G);
      Eigen::SparseMatrix<double> Dx = G.block(0, 0, F.rows(), V.rows());
      Eigen::SparseMatrix<double> Dy = G.block(F.rows(), 0, F.rows(), V.rows());
      Eigen::SparseMatrix<double> Dz = G.block(2 * F.rows(), 0, F.rows(), V.rows());

      D1 = F1.col(0).asDiagonal() * Dx + F1.col(1).asDiagonal() * Dy + F1.col(2).asDiagonal() * Dz;
      D2 = F2.col(0).asDiagonal() * Dx + F2.col(1).asDiagonal() * Dy + F2.col(2).asDiagonal() * Dz;
    }

    void solve_weighted_proxy(igl::SLIMData& m_state, Eigen::MatrixXd &V_new)
    {

      update_weights_and_closest_rotations(m_state,m_state.V, m_state.F, V_new);
      solve_weighted_arap(m_state,m_state.V, m_state.F, V_new, m_state.b, m_state.bc);
    }

    void compute_jacobians(igl::SLIMData& m_state, const Eigen::MatrixXd &uv)
    {
      if (m_state.F.cols() == 3)
      {
        // Ji=[D1*u,D2*u,D1*v,D2*v];
        m_state.Ji.col(0) = m_state.Dx * uv.col(0);
        m_state.Ji.col(1) = m_state.Dy * uv.col(0);
        m_state.Ji.col(2) = m_state.Dx * uv.col(1);
        m_state.Ji.col(3) = m_state.Dy * uv.col(1);
      }
      else /*tet mesh*/{
        // Ji=[D1*u,D2*u,D3*u, D1*v,D2*v, D3*v, D1*w,D2*w,D3*w];
        m_state.Ji.col(0) = m_state.Dx * uv.col(0);
        m_state.Ji.col(1) = m_state.Dy * uv.col(0);
        m_state.Ji.col(2) = m_state.Dz * uv.col(0);
        m_state.Ji.col(3) = m_state.Dx * uv.col(1);
        m_state.Ji.col(4) = m_state.Dy * uv.col(1);
        m_state.Ji.col(5) = m_state.Dz * uv.col(1);
        m_state.Ji.col(6) = m_state.Dx * uv.col(2);
        m_state.Ji.col(7) = m_state.Dy * uv.col(2);
        m_state.Ji.col(8) = m_state.Dz * uv.col(2);
      }
    }

    void update_weights_and_closest_rotations(igl::SLIMData& m_state,
                                              const Eigen::MatrixXd &V,
                                              const Eigen::MatrixXi &F,
                                              Eigen::MatrixXd &uv)
    {
      compute_jacobians(m_state, uv);

      const double eps = 1e-8;
      double exp_f = m_state.exp_factor;

      if (m_state.dim == 2)
      {
        for (int i = 0; i < m_state.Ji.rows(); ++i)
        {
          typedef Eigen::Matrix<double, 2, 2> Mat2;
          typedef Eigen::Matrix<double, 2, 1> Vec2;
          Mat2 ji, ri, ti, ui, vi;
          Vec2 sing;
          Vec2 closest_sing_vec;
          Mat2 mat_W;
          Vec2 m_sing_new;
          double s1, s2;

          ji(0, 0) = m_state.Ji(i, 0);
          ji(0, 1) = m_state.Ji(i, 1);
          ji(1, 0) = m_state.Ji(i, 2);
          ji(1, 1) = m_state.Ji(i, 3);

          igl::polar_svd(ji, ri, ti, ui, sing, vi);

          s1 = sing(0);
          s2 = sing(1);

          // Update Weights according to energy
          switch (m_state.slim_energy)
          {
            case igl::SLIMData::ARAP:
            {
              m_sing_new << 1, 1;
              break;
            }
            case igl::SLIMData::SYMMETRIC_DIRICHLET:
            {
              double s1_g = 2 * (s1 - pow(s1, -3));
              double s2_g = 2 * (s2 - pow(s2, -3));
              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1)));
              break;
            }
            case igl::SLIMData::LOG_ARAP:
            {
              double s1_g = 2 * (log(s1) / s1);
              double s2_g = 2 * (log(s2) / s2);
              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1)));
              break;
            }
            case igl::SLIMData::CONFORMAL:
            {
              double s1_g = 1 / (2 * s2) - s2 / (2 * pow(s1, 2));
              double s2_g = 1 / (2 * s1) - s1 / (2 * pow(s2, 2));

              double geo_avg = sqrt(s1 * s2);
              double s1_min = geo_avg;
              double s2_min = geo_avg;

              m_sing_new << sqrt(s1_g / (2 * (s1 - s1_min))), sqrt(s2_g / (2 * (s2 - s2_min)));

              // change local step
              closest_sing_vec << s1_min, s2_min;
              ri = ui * closest_sing_vec.asDiagonal() * vi.transpose();
              break;
            }
            case igl::SLIMData::EXP_CONFORMAL:
            {
              double s1_g = 2 * (s1 - pow(s1, -3));
              double s2_g = 2 * (s2 - pow(s2, -3));

              double geo_avg = sqrt(s1 * s2);
              double s1_min = geo_avg;
              double s2_min = geo_avg;

              double in_exp = exp_f * ((pow(s1, 2) + pow(s2, 2)) / (2 * s1 * s2));
              double exp_thing = exp(in_exp);

              s1_g *= exp_thing * exp_f;
              s2_g *= exp_thing * exp_f;

              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1)));
              break;
            }
            case igl::SLIMData::EXP_SYMMETRIC_DIRICHLET:
            {
              double s1_g = 2 * (s1 - pow(s1, -3));
              double s2_g = 2 * (s2 - pow(s2, -3));

              double in_exp = exp_f * (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2));
              double exp_thing = exp(in_exp);

              s1_g *= exp_thing * exp_f;
              s2_g *= exp_thing * exp_f;

              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1)));
              break;
            }
          }

          if (abs(s1 - 1) < eps) m_sing_new(0) = 1;
          if (abs(s2 - 1) < eps) m_sing_new(1) = 1;
          mat_W = ui * m_sing_new.asDiagonal() * ui.transpose();

          m_state.W_11(i) = mat_W(0, 0);
          m_state.W_12(i) = mat_W(0, 1);
          m_state.W_21(i) = mat_W(1, 0);
          m_state.W_22(i) = mat_W(1, 1);

          // 2) Update local step (doesn't have to be a rotation, for instance in case of conformal energy)
          m_state.Ri(i, 0) = ri(0, 0);
          m_state.Ri(i, 1) = ri(1, 0);
          m_state.Ri(i, 2) = ri(0, 1);
          m_state.Ri(i, 3) = ri(1, 1);
        }
      }
      else
      {
        typedef Eigen::Matrix<double, 3, 1> Vec3;
        typedef Eigen::Matrix<double, 3, 3> Mat3;
        Mat3 ji;
        Vec3 m_sing_new;
        Vec3 closest_sing_vec;
        const double sqrt_2 = sqrt(2);
        for (int i = 0; i < m_state.Ji.rows(); ++i)
        {
          ji(0, 0) = m_state.Ji(i, 0);
          ji(0, 1) = m_state.Ji(i, 1);
          ji(0, 2) = m_state.Ji(i, 2);
          ji(1, 0) = m_state.Ji(i, 3);
          ji(1, 1) = m_state.Ji(i, 4);
          ji(1, 2) = m_state.Ji(i, 5);
          ji(2, 0) = m_state.Ji(i, 6);
          ji(2, 1) = m_state.Ji(i, 7);
          ji(2, 2) = m_state.Ji(i, 8);

          Mat3 ri, ti, ui, vi;
          Vec3 sing;
          igl::polar_svd(ji, ri, ti, ui, sing, vi);

          double s1 = sing(0);
          double s2 = sing(1);
          double s3 = sing(2);

          // 1) Update Weights
          switch (m_state.slim_energy)
          {
            case igl::SLIMData::ARAP:
            {
              m_sing_new << 1, 1, 1;
              break;
            }
            case igl::SLIMData::LOG_ARAP:
            {
              double s1_g = 2 * (log(s1) / s1);
              double s2_g = 2 * (log(s2) / s2);
              double s3_g = 2 * (log(s3) / s3);
              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1))), sqrt(s3_g / (2 * (s3 - 1)));
              break;
            }
            case igl::SLIMData::SYMMETRIC_DIRICHLET:
            {
              double s1_g = 2 * (s1 - pow(s1, -3));
              double s2_g = 2 * (s2 - pow(s2, -3));
              double s3_g = 2 * (s3 - pow(s3, -3));
              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1))), sqrt(s3_g / (2 * (s3 - 1)));
              break;
            }
            case igl::SLIMData::EXP_SYMMETRIC_DIRICHLET:
            {
              double s1_g = 2 * (s1 - pow(s1, -3));
              double s2_g = 2 * (s2 - pow(s2, -3));
              double s3_g = 2 * (s3 - pow(s3, -3));
              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1))), sqrt(s3_g / (2 * (s3 - 1)));

              double in_exp = exp_f * (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2) + pow(s3, 2) + pow(s3, -2));
              double exp_thing = exp(in_exp);

              s1_g *= exp_thing * exp_f;
              s2_g *= exp_thing * exp_f;
              s3_g *= exp_thing * exp_f;

              m_sing_new << sqrt(s1_g / (2 * (s1 - 1))), sqrt(s2_g / (2 * (s2 - 1))), sqrt(s3_g / (2 * (s3 - 1)));

              break;
            }
            case igl::SLIMData::CONFORMAL:
            {
              double common_div = 9 * (pow(s1 * s2 * s3, 5. / 3.));

              double s1_g = (-2 * s2 * s3 * (pow(s2, 2) + pow(s3, 2) - 2 * pow(s1, 2))) / common_div;
              double s2_g = (-2 * s1 * s3 * (pow(s1, 2) + pow(s3, 2) - 2 * pow(s2, 2))) / common_div;
              double s3_g = (-2 * s1 * s2 * (pow(s1, 2) + pow(s2, 2) - 2 * pow(s3, 2))) / common_div;

              double closest_s = sqrt(pow(s1, 2) + pow(s3, 2)) / sqrt_2;
              double s1_min = closest_s;
              double s2_min = closest_s;
              double s3_min = closest_s;

              m_sing_new << sqrt(s1_g / (2 * (s1 - s1_min))), sqrt(s2_g / (2 * (s2 - s2_min))), sqrt(
                  s3_g / (2 * (s3 - s3_min)));

              // change local step
              closest_sing_vec << s1_min, s2_min, s3_min;
              ri = ui * closest_sing_vec.asDiagonal() * vi.transpose();
              break;
            }
            case igl::SLIMData::EXP_CONFORMAL:
            {
              // E_conf = (s1^2 + s2^2 + s3^2)/(3*(s1*s2*s3)^(2/3) )
              // dE_conf/ds1 = (-2*(s2*s3)*(s2^2+s3^2 -2*s1^2) ) / (9*(s1*s2*s3)^(5/3))
              // Argmin E_conf(s1): s1 = sqrt(s1^2+s2^2)/sqrt(2)
              double common_div = 9 * (pow(s1 * s2 * s3, 5. / 3.));

              double s1_g = (-2 * s2 * s3 * (pow(s2, 2) + pow(s3, 2) - 2 * pow(s1, 2))) / common_div;
              double s2_g = (-2 * s1 * s3 * (pow(s1, 2) + pow(s3, 2) - 2 * pow(s2, 2))) / common_div;
              double s3_g = (-2 * s1 * s2 * (pow(s1, 2) + pow(s2, 2) - 2 * pow(s3, 2))) / common_div;

              double in_exp = exp_f * ((pow(s1, 2) + pow(s2, 2) + pow(s3, 2)) / (3 * pow((s1 * s2 * s3), 2. / 3)));;
              double exp_thing = exp(in_exp);

              double closest_s = sqrt(pow(s1, 2) + pow(s3, 2)) / sqrt_2;
              double s1_min = closest_s;
              double s2_min = closest_s;
              double s3_min = closest_s;

              s1_g *= exp_thing * exp_f;
              s2_g *= exp_thing * exp_f;
              s3_g *= exp_thing * exp_f;

              m_sing_new << sqrt(s1_g / (2 * (s1 - s1_min))), sqrt(s2_g / (2 * (s2 - s2_min))), sqrt(
                  s3_g / (2 * (s3 - s3_min)));

              // change local step
              closest_sing_vec << s1_min, s2_min, s3_min;
              ri = ui * closest_sing_vec.asDiagonal() * vi.transpose();
            }
          }
          if (abs(s1 - 1) < eps) m_sing_new(0) = 1;
          if (abs(s2 - 1) < eps) m_sing_new(1) = 1;
          if (abs(s3 - 1) < eps) m_sing_new(2) = 1;
          Mat3 mat_W;
          mat_W = ui * m_sing_new.asDiagonal() * ui.transpose();

          m_state.W_11(i) = mat_W(0, 0);
          m_state.W_12(i) = mat_W(0, 1);
          m_state.W_13(i) = mat_W(0, 2);
          m_state.W_21(i) = mat_W(1, 0);
          m_state.W_22(i) = mat_W(1, 1);
          m_state.W_23(i) = mat_W(1, 2);
          m_state.W_31(i) = mat_W(2, 0);
          m_state.W_32(i) = mat_W(2, 1);
          m_state.W_33(i) = mat_W(2, 2);

          // 2) Update closest rotations (not rotations in case of conformal energy)
          m_state.Ri(i, 0) = ri(0, 0);
          m_state.Ri(i, 1) = ri(1, 0);
          m_state.Ri(i, 2) = ri(2, 0);
          m_state.Ri(i, 3) = ri(0, 1);
          m_state.Ri(i, 4) = ri(1, 1);
          m_state.Ri(i, 5) = ri(2, 1);
          m_state.Ri(i, 6) = ri(0, 2);
          m_state.Ri(i, 7) = ri(1, 2);
          m_state.Ri(i, 8) = ri(2, 2);
        } // for loop end

      } // if dim end

    }

    void solve_weighted_arap(igl::SLIMData& m_state,
                             const Eigen::MatrixXd &V,
                             const Eigen::MatrixXi &F,
                             Eigen::MatrixXd &uv,
                             Eigen::VectorXi &soft_b_p,
                             Eigen::MatrixXd &soft_bc_p)
    {
      using namespace Eigen;

      Eigen::SparseMatrix<double> L;
      build_linear_system(m_state,L);

      // solve
      Eigen::VectorXd Uc;
      if (m_state.dim == 2)
      {
        SimplicialLDLT<SparseMatrix<double> > solver;
        Uc = solver.compute(L).solve(m_state.rhs);
      }
      else
      { // seems like CG performs much worse for 2D and way better for 3D
        Eigen::VectorXd guess(uv.rows() * m_state.dim);
        for (int i = 0; i < m_state.dim; i++) for (int j = 0; j < m_state.dim; j++) guess(uv.rows() * i + j) = uv(i, j); // flatten vector
        ConjugateGradient<SparseMatrix<double>, Eigen::Lower | Upper> solver;
        solver.setTolerance(1e-8);
        Uc = solver.compute(L).solveWithGuess(m_state.rhs, guess);
      }

      for (int i = 0; i < m_state.dim; i++)
        uv.col(i) = Uc.block(i * m_state.v_n, 0, m_state.v_n, 1);
    }


    void pre_calc(igl::SLIMData& m_state)
    {
      if (!m_state.has_pre_calc)
      {
        m_state.v_n = m_state.v_num;
        m_state.f_n = m_state.f_num;

        if (m_state.F.cols() == 3)
        {
          m_state.dim = 2;
          Eigen::MatrixXd F1, F2, F3;
          igl::local_basis(m_state.V, m_state.F, F1, F2, F3);
          compute_surface_gradient_matrix(m_state.V, m_state.F, F1, F2, m_state.Dx, m_state.Dy);

          m_state.W_11.resize(m_state.f_n);
          m_state.W_12.resize(m_state.f_n);
          m_state.W_21.resize(m_state.f_n);
          m_state.W_22.resize(m_state.f_n);
        }
        else
        {
          m_state.dim = 3;
          Eigen::SparseMatrix<double> G;
          igl::grad(m_state.V, m_state.F, G,
                    m_state.mesh_improvement_3d /*use normal gradient, or one from a "regular" tet*/);
          m_state.Dx = G.block(0, 0, m_state.F.rows(), m_state.V.rows());
          m_state.Dy = G.block(m_state.F.rows(), 0, m_state.F.rows(), m_state.V.rows());
          m_state.Dz = G.block(2 * m_state.F.rows(), 0, m_state.F.rows(), m_state.V.rows());


          m_state.W_11.resize(m_state.f_n);
          m_state.W_12.resize(m_state.f_n);
          m_state.W_13.resize(m_state.f_n);
          m_state.W_21.resize(m_state.f_n);
          m_state.W_22.resize(m_state.f_n);
          m_state.W_23.resize(m_state.f_n);
          m_state.W_31.resize(m_state.f_n);
          m_state.W_32.resize(m_state.f_n);
          m_state.W_33.resize(m_state.f_n);
        }

        m_state.Dx.makeCompressed();
        m_state.Dy.makeCompressed();
        m_state.Dz.makeCompressed();
        m_state.Ri.resize(m_state.f_n, m_state.dim * m_state.dim);
        m_state.Ji.resize(m_state.f_n, m_state.dim * m_state.dim);
        m_state.rhs.resize(m_state.dim * m_state.v_num);

        // flattened weight matrix
        m_state.WGL_M.resize(m_state.dim * m_state.dim * m_state.f_n);
        for (int i = 0; i < m_state.dim * m_state.dim; i++)
          for (int j = 0; j < m_state.f_n; j++)
            m_state.WGL_M(i * m_state.f_n + j) = m_state.M(j);

        m_state.first_solve = true;
        m_state.has_pre_calc = true;
      }
    }

    void build_linear_system(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &L)
    {
      // formula (35) in paper
      Eigen::SparseMatrix<double> A(m_state.dim * m_state.dim * m_state.f_n, m_state.dim * m_state.v_n);
      buildA(m_state,A);

      Eigen::SparseMatrix<double> At = A.transpose();
      At.makeCompressed();

      Eigen::SparseMatrix<double> id_m(At.rows(), At.rows());
      id_m.setIdentity();

      // add proximal penalty
      L = At * m_state.WGL_M.asDiagonal() * A + m_state.proximal_p * id_m; //add also a proximal term
      L.makeCompressed();

      buildRhs(m_state, At);
      Eigen::SparseMatrix<double> OldL = L;
      add_soft_constraints(m_state,L);
      L.makeCompressed();
    }

    void add_soft_constraints(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &L)
    {
      int v_n = m_state.v_num;
      for (int d = 0; d < m_state.dim; d++)
      {
        for (int i = 0; i < m_state.b.rows(); i++)
        {
          int v_idx = m_state.b(i);
          m_state.rhs(d * v_n + v_idx) += m_state.soft_const_p * m_state.bc(i, d); // rhs
          L.coeffRef(d * v_n + v_idx, d * v_n + v_idx) += m_state.soft_const_p; // diagonal of matrix
        }
      }
    }

    double compute_energy(igl::SLIMData& m_state, Eigen::MatrixXd &V_new)
    {
      compute_jacobians(m_state,V_new);
      return compute_energy_with_jacobians(m_state, m_state.V, m_state.F, m_state.Ji, V_new, m_state.M) +
             compute_soft_const_energy(m_state, m_state.V, m_state.F, V_new);
    }

    double compute_soft_const_energy(igl::SLIMData& m_state,
                                     const Eigen::MatrixXd &V,
                                     const Eigen::MatrixXi &F,
                                     Eigen::MatrixXd &V_o)
    {
      double e = 0;
      for (int i = 0; i < m_state.b.rows(); i++)
      {
        e += m_state.soft_const_p * (m_state.bc.row(i) - V_o.row(m_state.b(i))).squaredNorm();
      }
      return e;
    }

    double compute_energy_with_jacobians(igl::SLIMData& m_state,
                                         const Eigen::MatrixXd &V,
                                         const Eigen::MatrixXi &F, const Eigen::MatrixXd &Ji,
                                         Eigen::MatrixXd &uv, Eigen::VectorXd &areas)
    {

      double energy = 0;
      if (m_state.dim == 2)
      {
        Eigen::Matrix<double, 2, 2> ji;
        for (int i = 0; i < m_state.f_n; i++)
        {
          ji(0, 0) = Ji(i, 0);
          ji(0, 1) = Ji(i, 1);
          ji(1, 0) = Ji(i, 2);
          ji(1, 1) = Ji(i, 3);

          typedef Eigen::Matrix<double, 2, 2> Mat2;
          typedef Eigen::Matrix<double, 2, 1> Vec2;
          Mat2 ri, ti, ui, vi;
          Vec2 sing;
          igl::polar_svd(ji, ri, ti, ui, sing, vi);
          double s1 = sing(0);
          double s2 = sing(1);

          switch (m_state.slim_energy)
          {
            case igl::SLIMData::ARAP:
            {
              energy += areas(i) * (pow(s1 - 1, 2) + pow(s2 - 1, 2));
              break;
            }
            case igl::SLIMData::SYMMETRIC_DIRICHLET:
            {
              energy += areas(i) * (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2));
              break;
            }
            case igl::SLIMData::EXP_SYMMETRIC_DIRICHLET:
            {
              energy += areas(i) * exp(m_state.exp_factor * (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2)));
              break;
            }
            case igl::SLIMData::LOG_ARAP:
            {
              energy += areas(i) * (pow(log(s1), 2) + pow(log(s2), 2));
              break;
            }
            case igl::SLIMData::CONFORMAL:
            {
              energy += areas(i) * ((pow(s1, 2) + pow(s2, 2)) / (2 * s1 * s2));
              break;
            }
            case igl::SLIMData::EXP_CONFORMAL:
            {
              energy += areas(i) * exp(m_state.exp_factor * ((pow(s1, 2) + pow(s2, 2)) / (2 * s1 * s2)));
              break;
            }

          }

        }
      }
      else
      {
        Eigen::Matrix<double, 3, 3> ji;
        for (int i = 0; i < m_state.f_n; i++)
        {
          ji(0, 0) = Ji(i, 0);
          ji(0, 1) = Ji(i, 1);
          ji(0, 2) = Ji(i, 2);
          ji(1, 0) = Ji(i, 3);
          ji(1, 1) = Ji(i, 4);
          ji(1, 2) = Ji(i, 5);
          ji(2, 0) = Ji(i, 6);
          ji(2, 1) = Ji(i, 7);
          ji(2, 2) = Ji(i, 8);

          typedef Eigen::Matrix<double, 3, 3> Mat3;
          typedef Eigen::Matrix<double, 3, 1> Vec3;
          Mat3 ri, ti, ui, vi;
          Vec3 sing;
          igl::polar_svd(ji, ri, ti, ui, sing, vi);
          double s1 = sing(0);
          double s2 = sing(1);
          double s3 = sing(2);

          switch (m_state.slim_energy)
          {
            case igl::SLIMData::ARAP:
            {
              energy += areas(i) * (pow(s1 - 1, 2) + pow(s2 - 1, 2) + pow(s3 - 1, 2));
              break;
            }
            case igl::SLIMData::SYMMETRIC_DIRICHLET:
            {
              energy += areas(i) * (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2) + pow(s3, 2) + pow(s3, -2));
              break;
            }
            case igl::SLIMData::EXP_SYMMETRIC_DIRICHLET:
            {
              energy += areas(i) * exp(m_state.exp_factor *
                                       (pow(s1, 2) + pow(s1, -2) + pow(s2, 2) + pow(s2, -2) + pow(s3, 2) + pow(s3, -2)));
              break;
            }
            case igl::SLIMData::LOG_ARAP:
            {
              energy += areas(i) * (pow(log(s1), 2) + pow(log(abs(s2)), 2) + pow(log(abs(s3)), 2));
              break;
            }
            case igl::SLIMData::CONFORMAL:
            {
              energy += areas(i) * ((pow(s1, 2) + pow(s2, 2) + pow(s3, 2)) / (3 * pow(s1 * s2 * s3, 2. / 3.)));
              break;
            }
            case igl::SLIMData::EXP_CONFORMAL:
            {
              energy += areas(i) * exp((pow(s1, 2) + pow(s2, 2) + pow(s3, 2)) / (3 * pow(s1 * s2 * s3, 2. / 3.)));
              break;
            }
          }
        }
      }

      return energy;
    }

    void buildA(igl::SLIMData& m_state, Eigen::SparseMatrix<double> &A)
    {
      // formula (35) in paper
      std::vector<Triplet<double> > IJV;
      if (m_state.dim == 2)
      {
        IJV.reserve(4 * (m_state.Dx.outerSize() + m_state.Dy.outerSize()));

        /*A = [W11*Dx, W12*Dx;
             W11*Dy, W12*Dy;
             W21*Dx, W22*Dx;
             W21*Dy, W22*Dy];*/
        for (int k = 0; k < m_state.Dx.outerSize(); ++k)
        {
          for (SparseMatrix<double>::InnerIterator it(m_state.Dx, k); it; ++it)
          {
            int dx_r = it.row();
            int dx_c = it.col();
            double val = it.value();

            IJV.push_back(Triplet<double>(dx_r, dx_c, val * m_state.W_11(dx_r)));
            IJV.push_back(Triplet<double>(dx_r, m_state.v_n + dx_c, val * m_state.W_12(dx_r)));

            IJV.push_back(Triplet<double>(2 * m_state.f_n + dx_r, dx_c, val * m_state.W_21(dx_r)));
            IJV.push_back(Triplet<double>(2 * m_state.f_n + dx_r, m_state.v_n + dx_c, val * m_state.W_22(dx_r)));
          }
        }

        for (int k = 0; k < m_state.Dy.outerSize(); ++k)
        {
          for (SparseMatrix<double>::InnerIterator it(m_state.Dy, k); it; ++it)
          {
            int dy_r = it.row();
            int dy_c = it.col();
            double val = it.value();

            IJV.push_back(Triplet<double>(m_state.f_n + dy_r, dy_c, val * m_state.W_11(dy_r)));
            IJV.push_back(Triplet<double>(m_state.f_n + dy_r, m_state.v_n + dy_c, val * m_state.W_12(dy_r)));

            IJV.push_back(Triplet<double>(3 * m_state.f_n + dy_r, dy_c, val * m_state.W_21(dy_r)));
            IJV.push_back(Triplet<double>(3 * m_state.f_n + dy_r, m_state.v_n + dy_c, val * m_state.W_22(dy_r)));
          }
        }
      }
      else
      {

        /*A = [W11*Dx, W12*Dx, W13*Dx;
               W11*Dy, W12*Dy, W13*Dy;
               W11*Dz, W12*Dz, W13*Dz;
               W21*Dx, W22*Dx, W23*Dx;
               W21*Dy, W22*Dy, W23*Dy;
               W21*Dz, W22*Dz, W23*Dz;
               W31*Dx, W32*Dx, W33*Dx;
               W31*Dy, W32*Dy, W33*Dy;
               W31*Dz, W32*Dz, W33*Dz;];*/
        IJV.reserve(9 * (m_state.Dx.outerSize() + m_state.Dy.outerSize() + m_state.Dz.outerSize()));
        for (int k = 0; k < m_state.Dx.outerSize(); k++)
        {
          for (SparseMatrix<double>::InnerIterator it(m_state.Dx, k); it; ++it)
          {
            int dx_r = it.row();
            int dx_c = it.col();
            double val = it.value();

            IJV.push_back(Triplet<double>(dx_r, dx_c, val * m_state.W_11(dx_r)));
            IJV.push_back(Triplet<double>(dx_r, m_state.v_n + dx_c, val * m_state.W_12(dx_r)));
            IJV.push_back(Triplet<double>(dx_r, 2 * m_state.v_n + dx_c, val * m_state.W_13(dx_r)));

            IJV.push_back(Triplet<double>(3 * m_state.f_n + dx_r, dx_c, val * m_state.W_21(dx_r)));
            IJV.push_back(Triplet<double>(3 * m_state.f_n + dx_r, m_state.v_n + dx_c, val * m_state.W_22(dx_r)));
            IJV.push_back(Triplet<double>(3 * m_state.f_n + dx_r, 2 * m_state.v_n + dx_c, val * m_state.W_23(dx_r)));

            IJV.push_back(Triplet<double>(6 * m_state.f_n + dx_r, dx_c, val * m_state.W_31(dx_r)));
            IJV.push_back(Triplet<double>(6 * m_state.f_n + dx_r, m_state.v_n + dx_c, val * m_state.W_32(dx_r)));
            IJV.push_back(Triplet<double>(6 * m_state.f_n + dx_r, 2 * m_state.v_n + dx_c, val * m_state.W_33(dx_r)));
          }
        }

        for (int k = 0; k < m_state.Dy.outerSize(); k++)
        {
          for (SparseMatrix<double>::InnerIterator it(m_state.Dy, k); it; ++it)
          {
            int dy_r = it.row();
            int dy_c = it.col();
            double val = it.value();

            IJV.push_back(Triplet<double>(m_state.f_n + dy_r, dy_c, val * m_state.W_11(dy_r)));
            IJV.push_back(Triplet<double>(m_state.f_n + dy_r, m_state.v_n + dy_c, val * m_state.W_12(dy_r)));
            IJV.push_back(Triplet<double>(m_state.f_n + dy_r, 2 * m_state.v_n + dy_c, val * m_state.W_13(dy_r)));

            IJV.push_back(Triplet<double>(4 * m_state.f_n + dy_r, dy_c, val * m_state.W_21(dy_r)));
            IJV.push_back(Triplet<double>(4 * m_state.f_n + dy_r, m_state.v_n + dy_c, val * m_state.W_22(dy_r)));
            IJV.push_back(Triplet<double>(4 * m_state.f_n + dy_r, 2 * m_state.v_n + dy_c, val * m_state.W_23(dy_r)));

            IJV.push_back(Triplet<double>(7 * m_state.f_n + dy_r, dy_c, val * m_state.W_31(dy_r)));
            IJV.push_back(Triplet<double>(7 * m_state.f_n + dy_r, m_state.v_n + dy_c, val * m_state.W_32(dy_r)));
            IJV.push_back(Triplet<double>(7 * m_state.f_n + dy_r, 2 * m_state.v_n + dy_c, val * m_state.W_33(dy_r)));
          }
        }

        for (int k = 0; k < m_state.Dz.outerSize(); k++)
        {
          for (SparseMatrix<double>::InnerIterator it(m_state.Dz, k); it; ++it)
          {
            int dz_r = it.row();
            int dz_c = it.col();
            double val = it.value();

            IJV.push_back(Triplet<double>(2 * m_state.f_n + dz_r, dz_c, val * m_state.W_11(dz_r)));
            IJV.push_back(Triplet<double>(2 * m_state.f_n + dz_r, m_state.v_n + dz_c, val * m_state.W_12(dz_r)));
            IJV.push_back(Triplet<double>(2 * m_state.f_n + dz_r, 2 * m_state.v_n + dz_c, val * m_state.W_13(dz_r)));

            IJV.push_back(Triplet<double>(5 * m_state.f_n + dz_r, dz_c, val * m_state.W_21(dz_r)));
            IJV.push_back(Triplet<double>(5 * m_state.f_n + dz_r, m_state.v_n + dz_c, val * m_state.W_22(dz_r)));
            IJV.push_back(Triplet<double>(5 * m_state.f_n + dz_r, 2 * m_state.v_n + dz_c, val * m_state.W_23(dz_r)));

            IJV.push_back(Triplet<double>(8 * m_state.f_n + dz_r, dz_c, val * m_state.W_31(dz_r)));
            IJV.push_back(Triplet<double>(8 * m_state.f_n + dz_r, m_state.v_n + dz_c, val * m_state.W_32(dz_r)));
            IJV.push_back(Triplet<double>(8 * m_state.f_n + dz_r, 2 * m_state.v_n + dz_c, val * m_state.W_33(dz_r)));
          }
        }
      }
      A.setFromTriplets(IJV.begin(), IJV.end());
    }

    void buildRhs(igl::SLIMData& m_state, const Eigen::SparseMatrix<double> &At)
    {
      VectorXd f_rhs(m_state.dim * m_state.dim * m_state.f_n);
      f_rhs.setZero();
      if (m_state.dim == 2)
      {
        /*b = [W11*R11 + W12*R21; (formula (36))
             W11*R12 + W12*R22;
             W21*R11 + W22*R21;
             W21*R12 + W22*R22];*/
        for (int i = 0; i < m_state.f_n; i++)
        {
          f_rhs(i + 0 * m_state.f_n) = m_state.W_11(i) * m_state.Ri(i, 0) + m_state.W_12(i) * m_state.Ri(i, 1);
          f_rhs(i + 1 * m_state.f_n) = m_state.W_11(i) * m_state.Ri(i, 2) + m_state.W_12(i) * m_state.Ri(i, 3);
          f_rhs(i + 2 * m_state.f_n) = m_state.W_21(i) * m_state.Ri(i, 0) + m_state.W_22(i) * m_state.Ri(i, 1);
          f_rhs(i + 3 * m_state.f_n) = m_state.W_21(i) * m_state.Ri(i, 2) + m_state.W_22(i) * m_state.Ri(i, 3);
        }
      }
      else
      {
        /*b = [W11*R11 + W12*R21 + W13*R31;
             W11*R12 + W12*R22 + W13*R32;
             W11*R13 + W12*R23 + W13*R33;
             W21*R11 + W22*R21 + W23*R31;
             W21*R12 + W22*R22 + W23*R32;
             W21*R13 + W22*R23 + W23*R33;
             W31*R11 + W32*R21 + W33*R31;
             W31*R12 + W32*R22 + W33*R32;
             W31*R13 + W32*R23 + W33*R33;];*/
        for (int i = 0; i < m_state.f_n; i++)
        {
          f_rhs(i + 0 * m_state.f_n) = m_state.W_11(i) * m_state.Ri(i, 0) + m_state.W_12(i) * m_state.Ri(i, 1) + m_state.W_13(i) * m_state.Ri(i, 2);
          f_rhs(i + 1 * m_state.f_n) = m_state.W_11(i) * m_state.Ri(i, 3) + m_state.W_12(i) * m_state.Ri(i, 4) + m_state.W_13(i) * m_state.Ri(i, 5);
          f_rhs(i + 2 * m_state.f_n) = m_state.W_11(i) * m_state.Ri(i, 6) + m_state.W_12(i) * m_state.Ri(i, 7) + m_state.W_13(i) * m_state.Ri(i, 8);
          f_rhs(i + 3 * m_state.f_n) = m_state.W_21(i) * m_state.Ri(i, 0) + m_state.W_22(i) * m_state.Ri(i, 1) + m_state.W_23(i) * m_state.Ri(i, 2);
          f_rhs(i + 4 * m_state.f_n) = m_state.W_21(i) * m_state.Ri(i, 3) + m_state.W_22(i) * m_state.Ri(i, 4) + m_state.W_23(i) * m_state.Ri(i, 5);
          f_rhs(i + 5 * m_state.f_n) = m_state.W_21(i) * m_state.Ri(i, 6) + m_state.W_22(i) * m_state.Ri(i, 7) + m_state.W_23(i) * m_state.Ri(i, 8);
          f_rhs(i + 6 * m_state.f_n) = m_state.W_31(i) * m_state.Ri(i, 0) + m_state.W_32(i) * m_state.Ri(i, 1) + m_state.W_33(i) * m_state.Ri(i, 2);
          f_rhs(i + 7 * m_state.f_n) = m_state.W_31(i) * m_state.Ri(i, 3) + m_state.W_32(i) * m_state.Ri(i, 4) + m_state.W_33(i) * m_state.Ri(i, 5);
          f_rhs(i + 8 * m_state.f_n) = m_state.W_31(i) * m_state.Ri(i, 6) + m_state.W_32(i) * m_state.Ri(i, 7) + m_state.W_33(i) * m_state.Ri(i, 8);
        }
      }
      VectorXd uv_flat(m_state.dim *m_state.v_n);
      for (int i = 0; i < m_state.dim; i++)
        for (int j = 0; j < m_state.v_n; j++)
          uv_flat(m_state.v_n * i + j) = m_state.V_o(j, i);

      m_state.rhs = (At * m_state.WGL_M.asDiagonal() * f_rhs + m_state.proximal_p * uv_flat);
    }

  }
}

/// Slim Implementation

IGL_INLINE void igl::slim_precompute(Eigen::MatrixXd &V, Eigen::MatrixXi &F, Eigen::MatrixXd &V_init, SLIMData &data,
                                     SLIMData::SLIM_ENERGY slim_energy, Eigen::VectorXi &b, Eigen::MatrixXd &bc,
                                     double soft_p)
{

  data.V = V;
  data.F = F;
  data.V_o = V_init;

  data.v_num = V.rows();
  data.f_num = F.rows();

  data.slim_energy = slim_energy;

  data.b = b;
  data.bc = bc;
  data.soft_const_p = soft_p;

  data.proximal_p = 0.0001;

  igl::doublearea(V, F, data.M);
  data.M /= 2.;
  data.mesh_area = data.M.sum();
  data.mesh_improvement_3d = false; // whether to use a jacobian derived from a real mesh or an abstract regular mesh (used for mesh improvement)
  data.exp_factor = 1.0; // param used only for exponential energies (e.g exponential symmetric dirichlet)

  assert (F.cols() == 3 || F.cols() == 4);

  igl::slim::pre_calc(data);
  data.energy = igl::slim::compute_energy(data,data.V_o) / data.mesh_area;
}

IGL_INLINE Eigen::MatrixXd igl::slim_solve(SLIMData &data, int iter_num)
{
  for (int i = 0; i < iter_num; i++)
  {
    Eigen::MatrixXd dest_res;
    dest_res = data.V_o;
    igl::slim::solve_weighted_proxy(data,dest_res);

    double old_energy = data.energy;

    std::function<double(Eigen::MatrixXd &)> compute_energy = [&](
        Eigen::MatrixXd &aaa) { return igl::slim::compute_energy(data,aaa); };

    data.energy = igl::flip_avoiding_line_search(data.F, data.V_o, dest_res, compute_energy,
                                                 data.energy * data.mesh_area) / data.mesh_area;
  }
  return data.V_o;
}