libigl_Martin/include/igl/flip_avoiding_line_search.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 "flip_avoiding_line_search.h"
#include "line_search.h"

#include <Eigen/Dense>
#include <vector>

//---------------------------------------------------------------------------
// x - array of size 3
// In case 3 real roots: => x[0], x[1], x[2], return 3
//         2 real roots: x[0], x[1],          return 2
//         1 real root : x[0], x[1] ± i*x[2], return 1
// http://math.ivanovo.ac.ru/dalgebra/Khashin/poly/index.html
int SolveP3(std::vector<double>& x,double a,double b,double c)
{ // solve cubic equation x^3 + a*x^2 + b*x + c
  using namespace std;
  double a2 = a*a;
    double q  = (a2 - 3*b)/9;
  double r  = (a*(2*a2-9*b) + 27*c)/54;
    double r2 = r*r;
  double q3 = q*q*q;
  double A,B;
    if(r2<q3)
    {
      double t=r/sqrt(q3);
      if( t<-1) t=-1;
      if( t> 1) t= 1;
      t=acos(t);
      a/=3; q=-2*sqrt(q);
      x[0]=q*cos(t/3)-a;
      x[1]=q*cos((t+(2*M_PI))/3)-a;
      x[2]=q*cos((t-(2*M_PI))/3)-a;
      return(3);
    }
    else
    {
      A =-pow(fabs(r)+sqrt(r2-q3),1./3);
      if( r<0 ) A=-A;
      B = A==0? 0 : B=q/A;

      a/=3;
      x[0] =(A+B)-a;
      x[1] =-0.5*(A+B)-a;
      x[2] = 0.5*sqrt(3.)*(A-B);
      if(fabs(x[2])<1e-14)
      {
        x[2]=x[1]; return(2);
      }
      return(1);
    }
}

double get_smallest_pos_quad_zero(double a,double b, double c)
{
  using namespace std;
  double t1,t2;
  if (a != 0)
  {
    double delta_in = pow(b,2) - 4*a*c;
    if (delta_in < 0)
    {
      return INFINITY;
    }
    double delta = sqrt(delta_in);
    t1 = (-b + delta)/ (2*a);
    t2 = (-b - delta)/ (2*a);
  }
  else
  {
    t1 = t2 = -b/c;
  }
  assert (std::isfinite(t1));
  assert (std::isfinite(t2));

  double tmp_n = min(t1,t2);
  t1 = max(t1,t2); t2 = tmp_n;
  if (t1 == t2)
  {
    return INFINITY; // means the orientation flips twice = doesn't flip
  }
  // return the smallest negative root if it exists, otherwise return infinity
  if (t1 > 0)
  {
    if (t2 > 0)
    {
      return t2;
    }
    else
    {
      return t1;
    }
  }
  else
  {
    return INFINITY;
  }
}

 double get_min_pos_root_2D(const Eigen::MatrixXd& uv,const Eigen::MatrixXi& F,
            Eigen::MatrixXd& d, int f)
{
  using namespace std;
/*
      Finding the smallest timestep t s.t a triangle get degenerated (<=> det = 0)
      The following code can be derived by a symbolic expression in matlab:

      Symbolic matlab:
      U11 = sym('U11');
      U12 = sym('U12');
      U21 = sym('U21');
      U22 = sym('U22');
      U31 = sym('U31');
      U32 = sym('U32');

      V11 = sym('V11');
      V12 = sym('V12');
      V21 = sym('V21');
      V22 = sym('V22');
      V31 = sym('V31');
      V32 = sym('V32');

      t = sym('t');

      U1 = [U11,U12];
      U2 = [U21,U22];
      U3 = [U31,U32];

      V1 = [V11,V12];
      V2 = [V21,V22];
      V3 = [V31,V32];

      A = [(U2+V2*t) - (U1+ V1*t)];
      B = [(U3+V3*t) - (U1+ V1*t)];
      C = [A;B];

      solve(det(C), t);
      cf = coeffs(det(C),t); % Now cf(1),cf(2),cf(3) holds the coefficients for the polynom. at order c,b,a
    */

  int v1 = F(f,0); int v2 = F(f,1); int v3 = F(f,2);
  // get quadratic coefficients (ax^2 + b^x + c)
  const double& U11 = uv(v1,0);
  const double& U12 = uv(v1,1);
  const double& U21 = uv(v2,0);
  const double& U22 = uv(v2,1);
  const double& U31 = uv(v3,0);
  const double& U32 = uv(v3,1);

  const double& V11 = d(v1,0);
  const double& V12 = d(v1,1);
  const double& V21 = d(v2,0);
  const double& V22 = d(v2,1);
  const double& V31 = d(v3,0);
  const double& V32 = d(v3,1);

  double a = V11*V22 - V12*V21 - V11*V32 + V12*V31 + V21*V32 - V22*V31;
  double b = U11*V22 - U12*V21 - U21*V12 + U22*V11 - U11*V32 + U12*V31 + U31*V12 - U32*V11 + U21*V32 - U22*V31 - U31*V22 + U32*V21;
  double c = U11*U22 - U12*U21 - U11*U32 + U12*U31 + U21*U32 - U22*U31;

  return get_smallest_pos_quad_zero(a,b,c);
}

double get_min_pos_root_3D(const Eigen::MatrixXd& uv,const Eigen::MatrixXi& F,
            Eigen::MatrixXd& direc, int f)
{
  using namespace std;
  /*
      Searching for the roots of:
        +-1/6 * |ax ay az 1|
                |bx by bz 1|
                |cx cy cz 1|
                |dx dy dz 1|
      Every point ax,ay,az has a search direction a_dx,a_dy,a_dz, and so we add those to the matrix, and solve the cubic to find the step size t for a 0 volume
      Symbolic matlab:
        syms a_x a_y a_z a_dx a_dy a_dz % tetrahedera point and search direction
        syms b_x b_y b_z b_dx b_dy b_dz
        syms c_x c_y c_z c_dx c_dy c_dz
        syms d_x d_y d_z d_dx d_dy d_dz
        syms t % Timestep var, this is what we're looking for


        a_plus_t = [a_x,a_y,a_z] + t*[a_dx,a_dy,a_dz];
        b_plus_t = [b_x,b_y,b_z] + t*[b_dx,b_dy,b_dz];
        c_plus_t = [c_x,c_y,c_z] + t*[c_dx,c_dy,c_dz];
        d_plus_t = [d_x,d_y,d_z] + t*[d_dx,d_dy,d_dz];

        vol_mat = [a_plus_t,1;b_plus_t,1;c_plus_t,1;d_plus_t,1]
        //cf = coeffs(det(vol_det),t); % Now cf(1),cf(2),cf(3),cf(4) holds the coefficients for the polynom
        [coefficients,terms] = coeffs(det(vol_det),t); % terms = [ t^3, t^2, t, 1], Coefficients hold the coeff we seek
  */
  int v1 = F(f,0); int v2 = F(f,1); int v3 = F(f,2); int v4 = F(f,3);
  const double& a_x = uv(v1,0);
  const double& a_y = uv(v1,1);
  const double& a_z = uv(v1,2);
  const double& b_x = uv(v2,0);
  const double& b_y = uv(v2,1);
  const double& b_z = uv(v2,2);
  const double& c_x = uv(v3,0);
  const double& c_y = uv(v3,1);
  const double& c_z = uv(v3,2);
  const double& d_x = uv(v4,0);
  const double& d_y = uv(v4,1);
  const double& d_z = uv(v4,2);

  const double& a_dx = direc(v1,0);
  const double& a_dy = direc(v1,1);
  const double& a_dz = direc(v1,2);
  const double& b_dx = direc(v2,0);
  const double& b_dy = direc(v2,1);
  const double& b_dz = direc(v2,2);
  const double& c_dx = direc(v3,0);
  const double& c_dy = direc(v3,1);
  const double& c_dz = direc(v3,2);
  const double& d_dx = direc(v4,0);
  const double& d_dy = direc(v4,1);
  const double& d_dz = direc(v4,2);

  // Find solution for: a*t^3 + b*t^2 + c*d +d = 0
  double a = a_dx*b_dy*c_dz - a_dx*b_dz*c_dy - a_dy*b_dx*c_dz + a_dy*b_dz*c_dx + a_dz*b_dx*c_dy - a_dz*b_dy*c_dx - a_dx*b_dy*d_dz + a_dx*b_dz*d_dy + a_dy*b_dx*d_dz - a_dy*b_dz*d_dx - a_dz*b_dx*d_dy + a_dz*b_dy*d_dx + a_dx*c_dy*d_dz - a_dx*c_dz*d_dy - a_dy*c_dx*d_dz + a_dy*c_dz*d_dx + a_dz*c_dx*d_dy - a_dz*c_dy*d_dx - b_dx*c_dy*d_dz + b_dx*c_dz*d_dy + b_dy*c_dx*d_dz - b_dy*c_dz*d_dx - b_dz*c_dx*d_dy + b_dz*c_dy*d_dx;

  double b = a_dy*b_dz*c_x - a_dy*b_x*c_dz - a_dz*b_dy*c_x + a_dz*b_x*c_dy + a_x*b_dy*c_dz - a_x*b_dz*c_dy - a_dx*b_dz*c_y + a_dx*b_y*c_dz + a_dz*b_dx*c_y - a_dz*b_y*c_dx - a_y*b_dx*c_dz + a_y*b_dz*c_dx + a_dx*b_dy*c_z - a_dx*b_z*c_dy - a_dy*b_dx*c_z + a_dy*b_z*c_dx + a_z*b_dx*c_dy - a_z*b_dy*c_dx - a_dy*b_dz*d_x + a_dy*b_x*d_dz + a_dz*b_dy*d_x - a_dz*b_x*d_dy - a_x*b_dy*d_dz + a_x*b_dz*d_dy + a_dx*b_dz*d_y - a_dx*b_y*d_dz - a_dz*b_dx*d_y + a_dz*b_y*d_dx + a_y*b_dx*d_dz - a_y*b_dz*d_dx - a_dx*b_dy*d_z + a_dx*b_z*d_dy + a_dy*b_dx*d_z - a_dy*b_z*d_dx - a_z*b_dx*d_dy + a_z*b_dy*d_dx + a_dy*c_dz*d_x - a_dy*c_x*d_dz - a_dz*c_dy*d_x + a_dz*c_x*d_dy + a_x*c_dy*d_dz - a_x*c_dz*d_dy - a_dx*c_dz*d_y + a_dx*c_y*d_dz + a_dz*c_dx*d_y - a_dz*c_y*d_dx - a_y*c_dx*d_dz + a_y*c_dz*d_dx + a_dx*c_dy*d_z - a_dx*c_z*d_dy - a_dy*c_dx*d_z + a_dy*c_z*d_dx + a_z*c_dx*d_dy - a_z*c_dy*d_dx - b_dy*c_dz*d_x + b_dy*c_x*d_dz + b_dz*c_dy*d_x - b_dz*c_x*d_dy - b_x*c_dy*d_dz + b_x*c_dz*d_dy + b_dx*c_dz*d_y - b_dx*c_y*d_dz - b_dz*c_dx*d_y + b_dz*c_y*d_dx + b_y*c_dx*d_dz - b_y*c_dz*d_dx - b_dx*c_dy*d_z + b_dx*c_z*d_dy + b_dy*c_dx*d_z - b_dy*c_z*d_dx - b_z*c_dx*d_dy + b_z*c_dy*d_dx;

  double c = a_dz*b_x*c_y - a_dz*b_y*c_x - a_x*b_dz*c_y + a_x*b_y*c_dz + a_y*b_dz*c_x - a_y*b_x*c_dz - a_dy*b_x*c_z + a_dy*b_z*c_x + a_x*b_dy*c_z - a_x*b_z*c_dy - a_z*b_dy*c_x + a_z*b_x*c_dy + a_dx*b_y*c_z - a_dx*b_z*c_y - a_y*b_dx*c_z + a_y*b_z*c_dx + a_z*b_dx*c_y - a_z*b_y*c_dx - a_dz*b_x*d_y + a_dz*b_y*d_x + a_x*b_dz*d_y - a_x*b_y*d_dz - a_y*b_dz*d_x + a_y*b_x*d_dz + a_dy*b_x*d_z - a_dy*b_z*d_x - a_x*b_dy*d_z + a_x*b_z*d_dy + a_z*b_dy*d_x - a_z*b_x*d_dy - a_dx*b_y*d_z + a_dx*b_z*d_y + a_y*b_dx*d_z - a_y*b_z*d_dx - a_z*b_dx*d_y + a_z*b_y*d_dx + a_dz*c_x*d_y - a_dz*c_y*d_x - a_x*c_dz*d_y + a_x*c_y*d_dz + a_y*c_dz*d_x - a_y*c_x*d_dz - a_dy*c_x*d_z + a_dy*c_z*d_x + a_x*c_dy*d_z - a_x*c_z*d_dy - a_z*c_dy*d_x + a_z*c_x*d_dy + a_dx*c_y*d_z - a_dx*c_z*d_y - a_y*c_dx*d_z + a_y*c_z*d_dx + a_z*c_dx*d_y - a_z*c_y*d_dx - b_dz*c_x*d_y + b_dz*c_y*d_x + b_x*c_dz*d_y - b_x*c_y*d_dz - b_y*c_dz*d_x + b_y*c_x*d_dz + b_dy*c_x*d_z - b_dy*c_z*d_x - b_x*c_dy*d_z + b_x*c_z*d_dy + b_z*c_dy*d_x - b_z*c_x*d_dy - b_dx*c_y*d_z + b_dx*c_z*d_y + b_y*c_dx*d_z - b_y*c_z*d_dx - b_z*c_dx*d_y + b_z*c_y*d_dx;

  double d = a_x*b_y*c_z - a_x*b_z*c_y - a_y*b_x*c_z + a_y*b_z*c_x + a_z*b_x*c_y - a_z*b_y*c_x - a_x*b_y*d_z + a_x*b_z*d_y + a_y*b_x*d_z - a_y*b_z*d_x - a_z*b_x*d_y + a_z*b_y*d_x + a_x*c_y*d_z - a_x*c_z*d_y - a_y*c_x*d_z + a_y*c_z*d_x + a_z*c_x*d_y - a_z*c_y*d_x - b_x*c_y*d_z + b_x*c_z*d_y + b_y*c_x*d_z - b_y*c_z*d_x - b_z*c_x*d_y + b_z*c_y*d_x;

  if (a==0)
  {
    return get_smallest_pos_quad_zero(b,c,d);
  }
  b/=a; c/=a; d/=a; // normalize it all
  std::vector<double> res(3);
  int real_roots_num = SolveP3(res,b,c,d);
  switch (real_roots_num)
  {
    case 1:
      return (res[0] >= 0) ? res[0]:INFINITY;
    case 2:
    {
      double max_root = max(res[0],res[1]); double min_root = min(res[0],res[1]);
      if (min_root > 0) return min_root;
      if (max_root > 0) return max_root;
      return INFINITY;
    }
    case 3:
    default:
    {
      std::sort(res.begin(),res.end());
      if (res[0] > 0) return res[0];
      if (res[1] > 0) return res[1];
      if (res[2] > 0) return res[2];
      return INFINITY;
    }
  }
}

double compute_max_step_from_singularities(const Eigen::MatrixXd& uv,
                                            const Eigen::MatrixXi& F,
                                            Eigen::MatrixXd& d)
{
  using namespace std;
  double max_step = INFINITY;

  // The if statement is outside the for loops to avoid branching/ease parallelizing
  if (uv.cols() == 2)
  {
    for (int f = 0; f < F.rows(); f++)
    {
      double min_positive_root = get_min_pos_root_2D(uv,F,d,f);
      max_step = min(max_step, min_positive_root);
    }
  }
  else
  { // volumetric deformation
    for (int f = 0; f < F.rows(); f++)
    {
      double min_positive_root = get_min_pos_root_3D(uv,F,d,f);
      max_step = min(max_step, min_positive_root);
    }
  }
  return max_step;
}

IGL_INLINE double igl::flip_avoiding_line_search(
  const Eigen::MatrixXi F,
  Eigen::MatrixXd& cur_v,
  Eigen::MatrixXd& dst_v,
  std::function<double(Eigen::MatrixXd&)> energy,
  double cur_energy)
{
  using namespace std;
  Eigen::MatrixXd d = dst_v - cur_v;

  double min_step_to_singularity = compute_max_step_from_singularities(cur_v,F,d);
  double max_step_size = min(1., min_step_to_singularity*0.8);

  return igl::line_search(cur_v,d,max_step_size, energy, cur_energy);
}

#ifdef IGL_STATIC_LIBRARY
#endif