libigl/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 "volume.h"
#include "grad.h"
#include "local_basis.h"
#include "per_face_normals.h"
#include "volume.h"
#include "polar_svd.h"
#include "flip_avoiding_line_search.h"
#include "mapping_energy_with_jacobians.h"

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

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

#include "Timer.h"
#include "sparse_cached.h"
#include "AtA_cached.h"

#ifdef CHOLMOD
#include <Eigen/CholmodSupport>
#endif

namespace igl
{
  namespace slim
  {
    // Definitions of internal functions
    IGL_INLINE void buildRhs(igl::SLIMData& s, const Eigen::SparseMatrix<double> &A);
    IGL_INLINE void add_soft_constraints(igl::SLIMData& s, Eigen::SparseMatrix<double> &L);
    IGL_INLINE double compute_energy(igl::SLIMData& s, const Eigen::MatrixXd &V_new);
    IGL_INLINE double compute_soft_const_energy(igl::SLIMData& s,
                                                const Eigen::MatrixXd &V,
                                                const Eigen::MatrixXi &F,
                                                const Eigen::MatrixXd &V_o);

    IGL_INLINE void solve_weighted_arap(igl::SLIMData& s,
                                        const Eigen::MatrixXd &V,
                                        const Eigen::MatrixXi &F,
                                        Eigen::MatrixXd &uv,
                                        Eigen::VectorXi &soft_b_p,
                                        Eigen::MatrixXd &soft_bc_p);
    IGL_INLINE void update_weights_and_closest_rotations( igl::SLIMData& s,
                                                          Eigen::MatrixXd &uv);
    IGL_INLINE void compute_jacobians(igl::SLIMData& s, const Eigen::MatrixXd &uv);
    IGL_INLINE void build_linear_system(igl::SLIMData& s, Eigen::SparseMatrix<double> &L);
    IGL_INLINE void pre_calc(igl::SLIMData& s);

    // Implementation

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

    IGL_INLINE void update_weights_and_closest_rotations(igl::SLIMData& s, Eigen::MatrixXd &uv)
    {
      compute_jacobians(s, uv);
      slim_update_weights_and_closest_rotations_with_jacobians(s.Ji, s.slim_energy, s.exp_factor, s.W, s.Ri);
    }
    



    IGL_INLINE void solve_weighted_arap(
      igl::SLIMData& s,
      const Eigen::MatrixXd & /*V*/,
      const Eigen::MatrixXi & /*F*/,
      Eigen::MatrixXd &uv,
      Eigen::VectorXi & /*soft_b_p*/,
      Eigen::MatrixXd & /*soft_bc_p*/)
    {
      Eigen::SparseMatrix<double> L;
      build_linear_system(s,L);

      igl::Timer t;
      
      //t.start();
      // solve
      Eigen::VectorXd Uc;
#ifndef CHOLMOD
      if (s.dim == 2)
      {
        Eigen::SimplicialLDLT<Eigen::SparseMatrix<double> > solver;
        Uc = solver.compute(L).solve(s.rhs);
      }
      else
      { // seems like CG performs much worse for 2D and way better for 3D
        Eigen::VectorXd guess(uv.rows() * s.dim);
        for (int i = 0; i < s.v_num; i++) for (int j = 0; j < s.dim; j++) guess(uv.rows() * j + i) = uv(i, j); // flatten vector
        Eigen::ConjugateGradient<Eigen::SparseMatrix<double>, Eigen::Lower | Eigen::Upper> cg;
        cg.setTolerance(1e-8);
        cg.compute(L);
        Uc = cg.solveWithGuess(s.rhs, guess);
      }
#else
        CholmodSimplicialLDLT<Eigen::SparseMatrix<double> > solver;
        Uc = solver.compute(L).solve(s.rhs);
#endif
      for (int i = 0; i < s.dim; i++)
        uv.col(i) = Uc.block(i * s.v_n, 0, s.v_n, 1);

    }


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

        if (s.F.cols() == 3)
        {
          s.dim = 2;
          Eigen::MatrixXd F1, F2, F3;
          igl::local_basis(s.V, s.F, F1, F2, F3);
          Eigen::SparseMatrix<double> G;
          igl::grad(s.V, s.F, G);
          Eigen::SparseMatrix<double> Face_Proj;

          auto face_proj = [](Eigen::MatrixXd& F){
            std::vector<Eigen::Triplet<double> >IJV;
            int f_num = F.rows();
            for(int i=0; i<F.rows(); i++) {
              IJV.push_back(Eigen::Triplet<double>(i, i, F(i,0)));
              IJV.push_back(Eigen::Triplet<double>(i, i+f_num, F(i,1)));
              IJV.push_back(Eigen::Triplet<double>(i, i+2*f_num, F(i,2)));
            }
            Eigen::SparseMatrix<double> P(f_num, 3*f_num);
            P.setFromTriplets(IJV.begin(), IJV.end());
            return P;
          };
          
          s.Dx = face_proj(F1) * G;
          s.Dy = face_proj(F2) * G;
        }
        else
        {
          s.dim = 3;
          Eigen::SparseMatrix<double> G;
          igl::grad(s.V, s.F, G,
                    s.mesh_improvement_3d /*use normal gradient, or one from a "regular" tet*/);
          s.Dx = G.block(0, 0, s.F.rows(), s.V.rows());
          s.Dy = G.block(s.F.rows(), 0, s.F.rows(), s.V.rows());
          s.Dz = G.block(2 * s.F.rows(), 0, s.F.rows(), s.V.rows());
        }

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

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

        s.first_solve = true;
        s.has_pre_calc = true;
      }
    }

    IGL_INLINE void build_linear_system(igl::SLIMData& s, Eigen::SparseMatrix<double> &L)
    {
      // formula (35) in paper
      std::vector<Eigen::Triplet<double> > IJV;
      
      #ifdef SLIM_CACHED
      slim_buildA(s.Dx, s.Dy, s.Dz, s.W, IJV);
      if (s.A.rows() == 0)
      {
        s.A = Eigen::SparseMatrix<double>(s.dim * s.dim * s.f_n, s.dim * s.v_n);
        igl::sparse_cached_precompute(IJV,s.A_data,s.A);
      }
      else
        igl::sparse_cached(IJV,s.A_data,s.A);
      #else
      Eigen::SparseMatrix<double> A(s.dim * s.dim * s.f_n, s.dim * s.v_n);
      slim_buildA(s.Dx, s.Dy, s.Dz, s.W, IJV);
      A.setFromTriplets(IJV.begin(),IJV.end());
      A.makeCompressed();
      #endif

      #ifdef SLIM_CACHED
      #else
      Eigen::SparseMatrix<double> At = A.transpose();
      At.makeCompressed();
      #endif

      #ifdef SLIM_CACHED
      Eigen::SparseMatrix<double> id_m(s.A.cols(), s.A.cols());
      #else
      Eigen::SparseMatrix<double> id_m(A.cols(), A.cols());
      #endif

      id_m.setIdentity();

      // add proximal penalty
      #ifdef SLIM_CACHED
      s.AtA_data.W = s.WGL_M;
      if (s.AtA.rows() == 0)
        igl::AtA_cached_precompute(s.A,s.AtA_data,s.AtA);
      else
        igl::AtA_cached(s.A,s.AtA_data,s.AtA);

      L = s.AtA + s.proximal_p * id_m; //add also a proximal 
      L.makeCompressed();

      #else
      L = At * s.WGL_M.asDiagonal() * A + s.proximal_p * id_m; //add also a proximal term
      L.makeCompressed();
      #endif

      #ifdef SLIM_CACHED
      buildRhs(s, s.A);
      #else
      buildRhs(s, A);
      #endif

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

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

    IGL_INLINE double compute_energy(igl::SLIMData& s, const Eigen::MatrixXd &V_new)
    {
      compute_jacobians(s,V_new);
      return mapping_energy_with_jacobians(s.Ji, s.M, s.slim_energy, s.exp_factor) +
             compute_soft_const_energy(s, s.V, s.F, V_new);
    }

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



    IGL_INLINE void buildRhs(igl::SLIMData& s, const Eigen::SparseMatrix<double> &A)
    {
      Eigen::VectorXd f_rhs(s.dim * s.dim * s.f_n);
      f_rhs.setZero();
      if (s.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 < s.f_n; i++)
        {
          f_rhs(i + 0 * s.f_n) = s.W(i, 0) * s.Ri(i, 0) + s.W(i, 1) * s.Ri(i, 1);
          f_rhs(i + 1 * s.f_n) = s.W(i, 0) * s.Ri(i, 2) + s.W(i, 1) * s.Ri(i, 3);
          f_rhs(i + 2 * s.f_n) = s.W(i, 2) * s.Ri(i, 0) + s.W(i, 3) * s.Ri(i, 1);
          f_rhs(i + 3 * s.f_n) = s.W(i, 2) * s.Ri(i, 2) + s.W(i, 3) * s.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 < s.f_n; i++)
        {
          f_rhs(i + 0 * s.f_n) = s.W(i, 0) * s.Ri(i, 0) + s.W(i, 1) * s.Ri(i, 1) + s.W(i, 2) * s.Ri(i, 2);
          f_rhs(i + 1 * s.f_n) = s.W(i, 0) * s.Ri(i, 3) + s.W(i, 1) * s.Ri(i, 4) + s.W(i, 2) * s.Ri(i, 5);
          f_rhs(i + 2 * s.f_n) = s.W(i, 0) * s.Ri(i, 6) + s.W(i, 1) * s.Ri(i, 7) + s.W(i, 2) * s.Ri(i, 8);
          f_rhs(i + 3 * s.f_n) = s.W(i, 3) * s.Ri(i, 0) + s.W(i, 4) * s.Ri(i, 1) + s.W(i, 5) * s.Ri(i, 2);
          f_rhs(i + 4 * s.f_n) = s.W(i, 3) * s.Ri(i, 3) + s.W(i, 4) * s.Ri(i, 4) + s.W(i, 5) * s.Ri(i, 5);
          f_rhs(i + 5 * s.f_n) = s.W(i, 3) * s.Ri(i, 6) + s.W(i, 4) * s.Ri(i, 7) + s.W(i, 5) * s.Ri(i, 8);
          f_rhs(i + 6 * s.f_n) = s.W(i, 6) * s.Ri(i, 0) + s.W(i, 7) * s.Ri(i, 1) + s.W(i, 8) * s.Ri(i, 2);
          f_rhs(i + 7 * s.f_n) = s.W(i, 6) * s.Ri(i, 3) + s.W(i, 7) * s.Ri(i, 4) + s.W(i, 8) * s.Ri(i, 5);
          f_rhs(i + 8 * s.f_n) = s.W(i, 6) * s.Ri(i, 6) + s.W(i, 7) * s.Ri(i, 7) + s.W(i, 8) * s.Ri(i, 8);
        }
      }
      Eigen::VectorXd uv_flat(s.dim *s.v_n);
      for (int i = 0; i < s.dim; i++)
        for (int j = 0; j < s.v_n; j++)
          uv_flat(s.v_n * i + j) = s.V_o(j, i);

      s.rhs = (f_rhs.transpose() * s.WGL_M.asDiagonal() * A).transpose() + s.proximal_p * uv_flat;
    }

  }
}

IGL_INLINE void igl::slim_update_weights_and_closest_rotations_with_jacobians(const Eigen::MatrixXd &Ji,
                                          igl::MappingEnergyType slim_energy,
                                          double exp_factor,
                                          Eigen::MatrixXd &W,
                                          Eigen::MatrixXd &Ri)
{
  const double eps = 1e-8;
  double exp_f = exp_factor;
  const int dim = (Ji.cols()==4? 2:3);

  if (dim == 2)
  {
    for (int i = 0; i < Ji.rows(); ++i)
    {
      typedef Eigen::Matrix2d Mat2;
      typedef Eigen::Matrix<double, 2, 2, Eigen::RowMajor> RMat2;
      typedef Eigen::Vector2d Vec2;
      Mat2 ji, ri, ti, ui, vi;
      Vec2 sing;
      Vec2 closest_sing_vec;
      RMat2 mat_W;
      Vec2 m_sing_new;
      double s1, s2;

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

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

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

      // Update Weights according to energy
      switch (slim_energy)
      {
        case igl::MappingEnergyType::ARAP:
        {
          m_sing_new << 1, 1;
          break;
        }
        case igl::MappingEnergyType::SYMMETRIC_DIRICHLET:
        {
          double s1_g = 2 * (s1 - pow(s1, -3));
          double s2_g = 2 * (s2 - pow(s2, -3));
          // Limit is 4 if s==1 according to Equation (32) in Rabinovich et al.
          // [2017]
          m_sing_new << 
            (s1==1?4:sqrt(s1_g / (2 * (s1 - 1)))), 
            (s2==1?4:sqrt(s2_g / (2 * (s2 - 1))));
          break;
        }
        case igl::MappingEnergyType::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::MappingEnergyType::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::MappingEnergyType::EXP_CONFORMAL:
        {
          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(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::MappingEnergyType::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;
        }
        default: assert(false);
      }

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

      W.row(i) = Eigen::Map<Eigen::Matrix<double, 1, 4, Eigen::RowMajor>>(mat_W.data());
      // 2) Update local step (doesn't have to be a rotation, for instance in case of conformal energy)
      Ri.row(i) = Eigen::Map<Eigen::Matrix<double, 1,4,Eigen::RowMajor>>(ri.data());
    }
  }
  else
  {
    typedef Eigen::Matrix<double, 3, 1> Vec3;
    typedef Eigen::Matrix<double, 3, 3, Eigen::ColMajor> Mat3;
    typedef Eigen::Matrix<double, 3, 3, Eigen::RowMajor> RMat3;
    Mat3 ji;
    Vec3 m_sing_new;
    Vec3 closest_sing_vec;
    const double sqrt_2 = sqrt(2);
    for (int i = 0; i < Ji.rows(); ++i)
    {
      ji << Ji(i,0), Ji(i,1), Ji(i,2), 
      Ji(i,3), Ji(i,4), Ji(i,5), 
      Ji(i,6), Ji(i,7), 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 (slim_energy)
      {
        case igl::MappingEnergyType::ARAP:
        {
          m_sing_new << 1, 1, 1;
          break;
        }
        case igl::MappingEnergyType::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::MappingEnergyType::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::MappingEnergyType::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::MappingEnergyType::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::MappingEnergyType::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();
          break;
        }
        default: assert(false);
      }
      if (std::abs(s1 - 1) < eps) m_sing_new(0) = 1;
      if (std::abs(s2 - 1) < eps) m_sing_new(1) = 1;
      if (std::abs(s3 - 1) < eps) m_sing_new(2) = 1;
      RMat3 mat_W;
      mat_W = ui * m_sing_new.asDiagonal() * ui.transpose();

      W.row(i) = Eigen::Map<Eigen::Matrix<double, 1,9,Eigen::RowMajor>>(mat_W.data());
      // 2) Update closest rotations (not rotations in case of conformal energy)
      Ri.row(i) = Eigen::Map<Eigen::Matrix<double, 1,9,Eigen::RowMajor>>(ri.data());
    } // for loop end

  } // if dim end

}

IGL_INLINE void igl::slim_buildA(const Eigen::SparseMatrix<double> &Dx,
          const Eigen::SparseMatrix<double> &Dy,
          const Eigen::SparseMatrix<double> &Dz,
          const Eigen::MatrixXd &W,
std::vector<Eigen::Triplet<double> > & IJV)
{
  const int dim = (W.cols() == 4) ? 2 : 3;
  const int f_n = W.rows();
  const int v_n = Dx.cols();

  // formula (35) in paper
  if (dim == 2)
  {
    IJV.reserve(4 * (Dx.outerSize() + Dy.outerSize()));

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

        IJV.push_back(Eigen::Triplet<double>(dx_r, dx_c, val * W(dx_r, 0)));
        IJV.push_back(Eigen::Triplet<double>(dx_r, v_n + dx_c, val * W(dx_r, 1)));

        IJV.push_back(Eigen::Triplet<double>(2 * f_n + dx_r, dx_c, val * W(dx_r, 2)));
        IJV.push_back(Eigen::Triplet<double>(2 * f_n + dx_r, v_n + dx_c, val * W(dx_r, 3)));
      }
    }

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

        IJV.push_back(Eigen::Triplet<double>(f_n + dy_r, dy_c, val * W(dy_r, 0)));
        IJV.push_back(Eigen::Triplet<double>(f_n + dy_r, v_n + dy_c, val * W(dy_r, 1)));

        IJV.push_back(Eigen::Triplet<double>(3 * f_n + dy_r, dy_c, val * W(dy_r, 2)));
        IJV.push_back(Eigen::Triplet<double>(3 * f_n + dy_r, v_n + dy_c, val * W(dy_r, 3)));
      }
    }
  }
  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 * (Dx.outerSize() + Dy.outerSize() + Dz.outerSize()));
    for (int k = 0; k < Dx.outerSize(); k++)
    {
      for (Eigen::SparseMatrix<double>::InnerIterator it(Dx, k); it; ++it)
      {
        int dx_r = it.row();
        int dx_c = it.col();
        double val = it.value();

        IJV.push_back(Eigen::Triplet<double>(dx_r, dx_c, val * W(dx_r, 0)));
        IJV.push_back(Eigen::Triplet<double>(dx_r, v_n + dx_c, val * W(dx_r, 1)));
        IJV.push_back(Eigen::Triplet<double>(dx_r, 2 * v_n + dx_c, val * W(dx_r, 2)));

        IJV.push_back(Eigen::Triplet<double>(3 * f_n + dx_r, dx_c, val * W(dx_r, 3)));
        IJV.push_back(Eigen::Triplet<double>(3 * f_n + dx_r, v_n + dx_c, val * W(dx_r, 4)));
        IJV.push_back(Eigen::Triplet<double>(3 * f_n + dx_r, 2 * v_n + dx_c, val * W(dx_r, 5)));

        IJV.push_back(Eigen::Triplet<double>(6 * f_n + dx_r, dx_c, val * W(dx_r, 6)));
        IJV.push_back(Eigen::Triplet<double>(6 * f_n + dx_r, v_n + dx_c, val * W(dx_r, 7)));
        IJV.push_back(Eigen::Triplet<double>(6 * f_n + dx_r, 2 * v_n + dx_c, val * W(dx_r, 8)));
      }
    }

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

        IJV.push_back(Eigen::Triplet<double>(f_n + dy_r, dy_c, val * W(dy_r, 0)));
        IJV.push_back(Eigen::Triplet<double>(f_n + dy_r, v_n + dy_c, val * W(dy_r, 1)));
        IJV.push_back(Eigen::Triplet<double>(f_n + dy_r, 2 * v_n + dy_c, val * W(dy_r, 2)));

        IJV.push_back(Eigen::Triplet<double>(4 * f_n + dy_r, dy_c, val * W(dy_r, 3)));
        IJV.push_back(Eigen::Triplet<double>(4 * f_n + dy_r, v_n + dy_c, val * W(dy_r, 4)));
        IJV.push_back(Eigen::Triplet<double>(4 * f_n + dy_r, 2 * v_n + dy_c, val * W(dy_r, 5)));

        IJV.push_back(Eigen::Triplet<double>(7 * f_n + dy_r, dy_c, val * W(dy_r, 6)));
        IJV.push_back(Eigen::Triplet<double>(7 * f_n + dy_r, v_n + dy_c, val * W(dy_r, 7)));
        IJV.push_back(Eigen::Triplet<double>(7 * f_n + dy_r, 2 * v_n + dy_c, val * W(dy_r, 8)));
      }
    }

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

        IJV.push_back(Eigen::Triplet<double>(2 * f_n + dz_r, dz_c, val * W(dz_r, 0)));
        IJV.push_back(Eigen::Triplet<double>(2 * f_n + dz_r, v_n + dz_c, val * W(dz_r, 1)));
        IJV.push_back(Eigen::Triplet<double>(2 * f_n + dz_r, 2 * v_n + dz_c, val * W(dz_r, 2)));

        IJV.push_back(Eigen::Triplet<double>(5 * f_n + dz_r, dz_c, val * W(dz_r, 3)));
        IJV.push_back(Eigen::Triplet<double>(5 * f_n + dz_r, v_n + dz_c, val * W(dz_r, 4)));
        IJV.push_back(Eigen::Triplet<double>(5 * f_n + dz_r, 2 * v_n + dz_c, val * W(dz_r, 5)));

        IJV.push_back(Eigen::Triplet<double>(8 * f_n + dz_r, dz_c, val * W(dz_r, 6)));
        IJV.push_back(Eigen::Triplet<double>(8 * f_n + dz_r, v_n + dz_c, val * W(dz_r, 7)));
        IJV.push_back(Eigen::Triplet<double>(8 * f_n + dz_r, 2 * v_n + dz_c, val * W(dz_r, 8)));
      }
    }
  }
}
/// Slim Implementation

IGL_INLINE void igl::slim_precompute(
  const Eigen::MatrixXd &V, 
  const Eigen::MatrixXi &F, 
  const Eigen::MatrixXd &V_init, 
  igl::SLIMData &data,
  igl::MappingEnergyType slim_energy, 
  const Eigen::VectorXi &b,
  const 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;

  if(F.cols() == 3)
  {
    igl::doublearea(V, F, data.M);
    data.mesh_area = data.M.sum()/2;
  }else 
  {
    assert(F.cols() == 4);
    igl::volume(V, F, data.M);
    // actually volume.
    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(igl::SLIMData &data, int iter_num)
{
  for (int i = 0; i < iter_num; i++)
  {
    Eigen::MatrixXd dest_res;
    dest_res = data.V_o;

    // Solve Weighted Proxy
    igl::slim::update_weights_and_closest_rotations(data, dest_res);
    igl::slim::solve_weighted_arap(data,data.V, data.F, dest_res, data.b, data.bc);

    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;
}


#ifdef IGL_STATIC_LIBRARY
// Explicit template instantiation
#endif