text: /*
   * Copyright (C) 2011 The Android Open Source Project
   *
   * Licensed under the Apache License, Version 2.0 (the "License");
   * you may not use this file except in compliance with the License.
   * You may obtain a copy of the License at
   *
   *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  
  /* $Id: db_rob_image_homography.cpp,v 1.2 2011/06/17 14:03:31 mbansal Exp $ */
  
  #include "db_utilities.h"
  #include "db_rob_image_homography.h"
  #include "db_bundle.h"
  
  
  
  /*****************************************************************
  *    Lean and mean begins here                                   *
  *****************************************************************/
  
  #include "db_image_homography.h"
  
  #ifdef _VERBOSE_
  #include <iostream>
  using namespace std;
  #endif /*VERBOSE*/
  
  inline double db_RobImageHomography_Cost(double H[9],int point_count,double *x_i,double *xp_i,double one_over_scale2)
  {
      int c;
      double back,acc,*x_i_temp,*xp_i_temp;
  
      for(back=0.0,c=0;c<point_count;)
      {
          /*Take log of product of ten reprojection
          errors to reduce nr of expensive log operations*/
          if(c+9<point_count)
          {
              x_i_temp=x_i+(c<<1);
              xp_i_temp=xp_i+(c<<1);
  
              acc=db_ExpCauchyInhomogenousHomographyError(xp_i_temp,H,x_i_temp,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+2,H,x_i_temp+2,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+4,H,x_i_temp+4,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+6,H,x_i_temp+6,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+8,H,x_i_temp+8,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+10,H,x_i_temp+10,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+12,H,x_i_temp+12,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+14,H,x_i_temp+14,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+16,H,x_i_temp+16,one_over_scale2);
              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+18,H,x_i_temp+18,one_over_scale2);
              c+=10;
          }
          else
          {
              for(acc=1.0;c<point_count;c++)
              {
                  acc*=db_ExpCauchyInhomogenousHomographyError(xp_i+(c<<1),H,x_i+(c<<1),one_over_scale2);
              }
          }
          back+=log(acc);
      }
      return(back);
  }
  
  inline double db_RobImageHomography_Statistics(double H[9],int point_count,double *x_i,double *xp_i,double one_over_scale2,db_Statistics *stat,double thresh=DB_OUTLIER_THRESHOLD)
  {
      int c,i;
      double t2,frac;
  
      t2=thresh*thresh;
      for(i=0,c=0;c<point_count;c++)
      {
          i+=(db_SquaredInhomogenousHomographyError(xp_i+(c<<1),H,x_i+(c<<1))*one_over_scale2<=t2)?1:0;
      }
      frac=((double)i)/((double)(db_maxi(point_count,1)));
  
  #ifdef _VERBOSE_
      std::cout << "Inlier Percentage RobImageHomography: " << frac*100.0 << "% out of " << point_count << " constraints" << std::endl;
  #endif /*_VERBOSE_*/
  
      if(stat)
      {
          stat->nr_points=point_count;
          stat->one_over_scale2=one_over_scale2;
          stat->nr_inliers=i;
          stat->inlier_fraction=frac;
  
          stat->cost=db_RobImageHomography_Cost(H,point_count,x_i,xp_i,one_over_scale2);
          stat->model_dimension=0;
          /*stat->nr_parameters=;*/
  
          stat->lambda1=log(4.0);
          stat->lambda2=log(4.0*((double)db_maxi(1,stat->nr_points)));
          stat->lambda3=10.0;
          stat->gric=stat->cost+stat->lambda1*stat->model_dimension*((double)stat->nr_points)+stat->lambda2*((double)stat->nr_parameters);
          stat->inlier_evidence=((double)stat->nr_inliers)-stat->lambda3*((double)stat->nr_parameters);
      }
  
      return(frac);
  }
  
  /*Compute min_Jtf and upper right of JtJ. Return cost.*/
  inline double db_RobImageHomography_Jacobians(double JtJ[81],double min_Jtf[9],double H[9],int point_count,double *x_i,double *xp_i,double one_over_scale2)
  {
      double back,Jf_dx[18],f[2],temp,temp2;
      int i;
  
      db_Zero(JtJ,81);
      db_Zero(min_Jtf,9);
      for(back=0.0,i=0;i<point_count;i++)
      {
          /*Compute reprojection error vector and its Jacobian
          for this point*/
          db_DerivativeCauchyInhomHomographyReprojection(Jf_dx,f,xp_i+(i<<1),H,x_i+(i<<1),one_over_scale2);
          /*Perform
          min_Jtf-=Jf_dx*f[0] and
          min_Jtf-=(Jf_dx+9)*f[1] to accumulate -Jt%f*/
          db_RowOperation9(min_Jtf,Jf_dx,f[0]);
          db_RowOperation9(min_Jtf,Jf_dx+9,f[1]);
          /*Accumulate upper right of JtJ with outer product*/
          temp=Jf_dx[0]; temp2=Jf_dx[9];
          JtJ[0]+=temp*Jf_dx[0]+temp2*Jf_dx[9];
          JtJ[1]+=temp*Jf_dx[1]+temp2*Jf_dx[10];
          JtJ[2]+=temp*Jf_dx[2]+temp2*Jf_dx[11];
          JtJ[3]+=temp*Jf_dx[3]+temp2*Jf_dx[12];
          JtJ[4]+=temp*Jf_dx[4]+temp2*Jf_dx[13];
          JtJ[5]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[6]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[7]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[8]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[1]; temp2=Jf_dx[10];
          JtJ[10]+=temp*Jf_dx[1]+temp2*Jf_dx[10];
          JtJ[11]+=temp*Jf_dx[2]+temp2*Jf_dx[11];
          JtJ[12]+=temp*Jf_dx[3]+temp2*Jf_dx[12];
          JtJ[13]+=temp*Jf_dx[4]+temp2*Jf_dx[13];
          JtJ[14]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[15]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[16]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[17]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[2]; temp2=Jf_dx[11];
          JtJ[20]+=temp*Jf_dx[2]+temp2*Jf_dx[11];
          JtJ[21]+=temp*Jf_dx[3]+temp2*Jf_dx[12];
          JtJ[22]+=temp*Jf_dx[4]+temp2*Jf_dx[13];
          JtJ[23]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[24]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[25]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[26]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[3]; temp2=Jf_dx[12];
          JtJ[30]+=temp*Jf_dx[3]+temp2*Jf_dx[12];
          JtJ[31]+=temp*Jf_dx[4]+temp2*Jf_dx[13];
          JtJ[32]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[33]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[34]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[35]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[4]; temp2=Jf_dx[13];
          JtJ[40]+=temp*Jf_dx[4]+temp2*Jf_dx[13];
          JtJ[41]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[42]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[43]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[44]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[5]; temp2=Jf_dx[14];
          JtJ[50]+=temp*Jf_dx[5]+temp2*Jf_dx[14];
          JtJ[51]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[52]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[53]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[6]; temp2=Jf_dx[15];
          JtJ[60]+=temp*Jf_dx[6]+temp2*Jf_dx[15];
          JtJ[61]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[62]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[7]; temp2=Jf_dx[16];
          JtJ[70]+=temp*Jf_dx[7]+temp2*Jf_dx[16];
          JtJ[71]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
          temp=Jf_dx[8]; temp2=Jf_dx[17];
          JtJ[80]+=temp*Jf_dx[8]+temp2*Jf_dx[17];
  
          /*Add square-sum to cost*/
          back+=db_sqr(f[0])+db_sqr(f[1]);
      }
  
      return(back);
  }
  
  /*Compute min_Jtf and upper right of JtJ. Return cost*/
  inline double db_RobCamRotation_Jacobians(double JtJ[9],double min_Jtf[3],double H[9],int point_count,double *x_i,double *xp_i,double one_over_scale2)
  {
      double back,Jf_dx[6],f[2];
      int i,j;
  
      db_Zero(JtJ,9);
      db_Zero(min_Jtf,3);
      for(back=0.0,i=0;i<point_count;i++)
      {
          /*Compute reprojection error vector and its Jacobian
          for this point*/
          j=(i<<1);
          db_DerivativeCauchyInhomRotationReprojection(Jf_dx,f,xp_i+j,H,x_i+j,one_over_scale2);
          /*Perform
          min_Jtf-=Jf_dx*f[0] and
          min_Jtf-=(Jf_dx+3)*f[1] to accumulate -Jt%f*/
          db_RowOperation3(min_Jtf,Jf_dx,f[0]);
          db_RowOperation3(min_Jtf,Jf_dx+3,f[1]);
          /*Accumulate upper right of JtJ with outer product*/
          JtJ[0]+=Jf_dx[0]*Jf_dx[0]+Jf_dx[3]*Jf_dx[3];
          JtJ[1]+=Jf_dx[0]*Jf_dx[1]+Jf_dx[3]*Jf_dx[4];
          JtJ[2]+=Jf_dx[0]*Jf_dx[2]+Jf_dx[3]*Jf_dx[5];
          JtJ[4]+=Jf_dx[1]*Jf_dx[1]+Jf_dx[4]*Jf_dx[4];
          JtJ[5]+=Jf_dx[1]*Jf_dx[2]+Jf_dx[4]*Jf_dx[5];
          JtJ[8]+=Jf_dx[2]*Jf_dx[2]+Jf_dx[5]*Jf_dx[5];
  
          /*Add square-sum to cost*/
          back+=db_sqr(f[0])+db_sqr(f[1]);
      }
  
      return(back);
  }
  
  void db_RobCamRotation_Polish(double H[9],int point_count,double *x_i,double *xp_i,double one_over_scale2,
                                 int max_iterations,double improvement_requirement)
  {
      int i,update,stop;
      double lambda,cost,current_cost;
      double JtJ[9],min_Jtf[3],dx[3],H_p_dx[9];
  
      lambda=0.001;
      for(update=1,stop=0,i=0;(stop<2) && (i<max_iterations);i++)
      {
          /*if first time since improvement, compute Jacobian and residual*/
          if(update)
          {
              current_cost=db_RobCamRotation_Jacobians(JtJ,min_Jtf,H,point_count,x_i,xp_i,one_over_scale2);
              update=0;
          }
  
  #ifdef _VERBOSE_
          /*std::cout << "Cost:" << current_cost << " ";*/
  #endif /*_VERBOSE_*/
  
          /*Come up with a hypothesis dx
          based on the current lambda*/
          db_Compute_dx_3x3(dx,JtJ,min_Jtf,lambda);
  
          /*Compute Cost(x+dx)*/
          db_UpdateRotation(H_p_dx,H,dx);
          cost=db_RobImageHomography_Cost(H_p_dx,point_count,x_i,xp_i,one_over_scale2);
  
          /*Is there an improvement?*/
          if(cost<current_cost)
          {
              /*improvement*/
              if(current_cost-cost<current_cost*improvement_requirement) stop++;
              else stop=0;
              lambda*=0.1;
              /*Move to the hypothesised position x+dx*/
              current_cost=cost;
              db_Copy9(H,H_p_dx);
              db_OrthonormalizeRotation(H);
              update=1;
  
  #ifdef _VERBOSE_
          std::cout << "Step" << i << "Imp,Lambda=" << lambda << "Cost:" << current_cost << std::endl;
  #endif /*_VERBOSE_*/
          }
          else
          {
              /*no improvement*/
              lambda*=10.0;
              stop=0;
          }
      }
  }
  
  inline void db_RobImageHomographyFetchJacobian(double **JtJ_ref,double *min_Jtf,double **JtJ_temp_ref,double *min_Jtf_temp,int n,int *fetch_vector)
  {
      int i,j,t;
      double *t1,*t2;
  
      for(i=0;i<n;i++)
      {
          t=fetch_vector[i];
          min_Jtf[i]=min_Jtf_temp[t];
          t1=JtJ_ref[i];
          t2=JtJ_temp_ref[t];
          for(j=i;j<n;j++)
          {
              t1[j]=t2[fetch_vector[j]];
          }
      }
  }
  
  inline void db_RobImageHomographyMultiplyJacobian(double **JtJ_ref,double *min_Jtf,double **JtJ_temp_ref,double *min_Jtf_temp,double **JE_dx_ref,int n)
  {
      double JtJ_JE[72],*JtJ_JE_ref[9];
  
      db_SetupMatrixRefs(JtJ_JE_ref,9,8,JtJ_JE);
  
      db_SymmetricExtendUpperToLower(JtJ_temp_ref,9,9);
      db_MultiplyMatricesAB(JtJ_JE_ref,JtJ_temp_ref,JE_dx_ref,9,9,n);
      db_UpperMultiplyMatricesAtB(JtJ_ref,JE_dx_ref,JtJ_JE_ref,n,9,n);
      db_MultiplyMatrixVectorAtb(min_Jtf,JE_dx_ref,min_Jtf_temp,n,9);
  }
  
  inline void db_RobImageHomographyJH_Js(double **JE_dx_ref,int j,double H[9])
  {
      /*Update of upper 2x2 is multiplication by
      [s 0][ cos(theta) sin(theta)]
      [0 s][-sin(theta) cos(theta)]*/
      JE_dx_ref[0][j]=H[0];
      JE_dx_ref[1][j]=H[1];
      JE_dx_ref[2][j]=0;
      JE_dx_ref[3][j]=H[2];
      JE_dx_ref[4][j]=H[3];
      JE_dx_ref[5][j]=0;
      JE_dx_ref[6][j]=0;
      JE_dx_ref[7][j]=0;
      JE_dx_ref[8][j]=0;
  }
  
  inline void db_RobImageHomographyJH_JR(double **JE_dx_ref,int j,double H[9])
  {
      /*Update of upper 2x2 is multiplication by
      [s 0][ cos(theta) sin(theta)]
      [0 s][-sin(theta) cos(theta)]*/
      JE_dx_ref[0][j]=  H[3];
      JE_dx_ref[1][j]=  H[4];
      JE_dx_ref[2][j]=0;
      JE_dx_ref[3][j]= -H[0];
      JE_dx_ref[4][j]= -H[1];
      JE_dx_ref[5][j]=0;
      JE_dx_ref[6][j]=0;
      JE_dx_ref[7][j]=0;
      JE_dx_ref[8][j]=0;
  }
  
  inline void db_RobImageHomographyJH_Jt(double **JE_dx_ref,int j,int k,double H[9])
  {
      JE_dx_ref[0][j]=0;
      JE_dx_ref[1][j]=0;
      JE_dx_ref[2][j]=1.0;
      JE_dx_ref[3][j]=0;
      JE_dx_ref[4][j]=0;
      JE_dx_ref[5][j]=0;
      JE_dx_ref[6][j]=0;
      JE_dx_ref[7][j]=0;
      JE_dx_ref[8][j]=0;
  
      JE_dx_ref[0][k]=0;
      JE_dx_ref[1][k]=0;
      JE_dx_ref[2][k]=0;
      JE_dx_ref[3][k]=0;
      JE_dx_ref[4][k]=0;
      JE_dx_ref[5][k]=1.0;
      JE_dx_ref[6][k]=0;
      JE_dx_ref[7][k]=0;
      JE_dx_ref[8][k]=0;
  }
  
  inline void db_RobImageHomographyJH_dRotFocal(double **JE_dx_ref,int j,int k,int l,int m,double H[9])
  {
      double f,fi,fi2;
      double R[9],J[9];
  
      /*Updated matrix is diag(f+df,f+df)*dR*R*diag(1/(f+df),1/(f+df),1)*/
      f=db_FocalAndRotFromCamRotFocalHomography(R,H);
      fi=db_SafeReciprocal(f);
      fi2=db_sqr(fi);
      db_JacobianOfRotatedPointStride(J,R,3);
      JE_dx_ref[0][j]=   J[0];
      JE_dx_ref[1][j]=   J[1];
      JE_dx_ref[2][j]=f* J[2];
      JE_dx_ref[3][j]=   J[3];
      JE_dx_ref[4][j]=   J[4];
      JE_dx_ref[5][j]=f* J[5];
      JE_dx_ref[6][j]=fi*J[6];
      JE_dx_ref[7][j]=fi*J[7];
      JE_dx_ref[8][j]=   J[8];
      db_JacobianOfRotatedPointStride(J,R+1,3);
      JE_dx_ref[0][k]=   J[0];
      JE_dx_ref[1][k]=   J[1];
      JE_dx_ref[2][k]=f* J[2];
      JE_dx_ref[3][k]=   J[3];
      JE_dx_ref[4][k]=   J[4];
      JE_dx_ref[5][k]=f* J[5];
      JE_dx_ref[6][k]=fi*J[6];
      JE_dx_ref[7][k]=fi*J[7];
      JE_dx_ref[8][k]=   J[8];
      db_JacobianOfRotatedPointStride(J,R+2,3);
      JE_dx_ref[0][l]=   J[0];
      JE_dx_ref[1][l]=   J[1];
      JE_dx_ref[2][l]=f* J[2];
      JE_dx_ref[3][l]=   J[3];
      JE_dx_ref[4][l]=   J[4];
      JE_dx_ref[5][l]=f* J[5];
      JE_dx_ref[6][l]=fi*J[6];
      JE_dx_ref[7][l]=fi*J[7];
      JE_dx_ref[8][l]=   J[8];
  
      JE_dx_ref[0][m]=0;
      JE_dx_ref[1][m]=0;
      JE_dx_ref[2][m]=H[2];
      JE_dx_ref[3][m]=0;
      JE_dx_ref[4][m]=0;
      JE_dx_ref[5][m]=H[5];
      JE_dx_ref[6][m]= -fi2*H[6];
      JE_dx_ref[7][m]= -fi2*H[7];
      JE_dx_ref[8][m]=0;
  }
  
  inline double db_RobImageHomography_Jacobians_Generic(double *JtJ_ref[8],double min_Jtf[8],int *num_param,int *frozen_coord,double H[9],int point_count,double *x_i,double *xp_i,int homography_type,double one_over_scale2)
  {
      double back;
      int i,j,fetch_vector[8],n;
      double JtJ_temp[81],min_Jtf_temp[9],JE_dx[72];
      double *JE_dx_ref[9],*JtJ_temp_ref[9];
  
      /*Compute cost and JtJ,min_Jtf with respect to H*/
      back=db_RobImageHomography_Jacobians(JtJ_temp,min_Jtf_temp,H,point_count,x_i,xp_i,one_over_scale2);
  
      /*Compute JtJ,min_Jtf with respect to the right parameters
      The formulas are
      JtJ=transpose(JE_dx)*JtJ*JE_dx and
      min_Jtf=transpose(JE_dx)*min_Jtf,
      where the 9xN matrix JE_dx is the Jacobian of H with respect
      to the update*/
      db_SetupMatrixRefs(JtJ_temp_ref,9,9,JtJ_temp);
      db_SetupMatrixRefs(JE_dx_ref,9,8,JE_dx);
      switch(homography_type)
      {
          case DB_HOMOGRAPHY_TYPE_SIMILARITY:
          case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
              n=4;
              db_RobImageHomographyJH_Js(JE_dx_ref,0,H);
              db_RobImageHomographyJH_JR(JE_dx_ref,1,H);
              db_RobImageHomographyJH_Jt(JE_dx_ref,2,3,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
          case DB_HOMOGRAPHY_TYPE_ROTATION:
          case DB_HOMOGRAPHY_TYPE_ROTATION_U:
              n=1;
              db_RobImageHomographyJH_JR(JE_dx_ref,0,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
          case DB_HOMOGRAPHY_TYPE_SCALING:
              n=1;
              db_RobImageHomographyJH_Js(JE_dx_ref,0,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
          case DB_HOMOGRAPHY_TYPE_S_T:
              n=3;
              db_RobImageHomographyJH_Js(JE_dx_ref,0,H);
              db_RobImageHomographyJH_Jt(JE_dx_ref,1,2,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
          case DB_HOMOGRAPHY_TYPE_R_T:
              n=3;
              db_RobImageHomographyJH_JR(JE_dx_ref,0,H);
              db_RobImageHomographyJH_Jt(JE_dx_ref,1,2,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
          case DB_HOMOGRAPHY_TYPE_R_S:
              n=2;
              db_RobImageHomographyJH_Js(JE_dx_ref,0,H);
              db_RobImageHomographyJH_JR(JE_dx_ref,1,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
  
          case DB_HOMOGRAPHY_TYPE_TRANSLATION:
              n=2;
              fetch_vector[0]=2;
              fetch_vector[1]=5;
              db_RobImageHomographyFetchJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,n,fetch_vector);
              break;
          case DB_HOMOGRAPHY_TYPE_AFFINE:
              n=6;
              fetch_vector[0]=0;
              fetch_vector[1]=1;
              fetch_vector[2]=2;
              fetch_vector[3]=3;
              fetch_vector[4]=4;
              fetch_vector[5]=5;
              db_RobImageHomographyFetchJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,n,fetch_vector);
              break;
          case DB_HOMOGRAPHY_TYPE_PROJECTIVE:
              n=8;
              *frozen_coord=db_MaxAbsIndex9(H);
              for(j=0,i=0;i<9;i++) if(i!=(*frozen_coord))
              {
                  fetch_vector[j]=i;
                  j++;
              }
              db_RobImageHomographyFetchJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,n,fetch_vector);
              break;
          case DB_HOMOGRAPHY_TYPE_CAMROTATION_F:
          case DB_HOMOGRAPHY_TYPE_CAMROTATION_F_UD:
              n=4;
              db_RobImageHomographyJH_dRotFocal(JE_dx_ref,0,1,2,3,H);
              db_RobImageHomographyMultiplyJacobian(JtJ_ref,min_Jtf,JtJ_temp_ref,min_Jtf_temp,JE_dx_ref,n);
              break;
      }
      *num_param=n;
  
      return(back);
  }
  
  inline void db_ImageHomographyUpdateGeneric(double H_p_dx[9],double H[9],double *dx,int homography_type,int frozen_coord)
  {
      switch(homography_type)
      {
          case DB_HOMOGRAPHY_TYPE_SIMILARITY:
          case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
              db_Copy9(H_p_dx,H);
              db_MultiplyScaleOntoImageHomography(H,1.0+dx[0]);
              db_MultiplyRotationOntoImageHomography(H,dx[1]);
              H_p_dx[2]+=dx[2];
              H_p_dx[5]+=dx[3];
              break;
          case DB_HOMOGRAPHY_TYPE_ROTATION:
          case DB_HOMOGRAPHY_TYPE_ROTATION_U:
              db_MultiplyRotationOntoImageHomography(H,dx[0]);
              break;
          case DB_HOMOGRAPHY_TYPE_SCALING:
              db_MultiplyScaleOntoImageHomography(H,1.0+dx[0]);
              break;
          case DB_HOMOGRAPHY_TYPE_S_T:
              db_Copy9(H_p_dx,H);
              db_MultiplyScaleOntoImageHomography(H,1.0+dx[0]);
              H_p_dx[2]+=dx[1];
              H_p_dx[5]+=dx[2];
              break;
          case DB_HOMOGRAPHY_TYPE_R_T:
              db_Copy9(H_p_dx,H);
              db_MultiplyRotationOntoImageHomography(H,dx[0]);
              H_p_dx[2]+=dx[1];
              H_p_dx[5]+=dx[2];
              break;
          case DB_HOMOGRAPHY_TYPE_R_S:
              db_Copy9(H_p_dx,H);
              db_MultiplyScaleOntoImageHomography(H,1.0+dx[0]);
              db_MultiplyRotationOntoImageHomography(H,dx[1]);
              break;
          case DB_HOMOGRAPHY_TYPE_TRANSLATION:
              db_Copy9(H_p_dx,H);
              H_p_dx[2]+=dx[0];
              H_p_dx[5]+=dx[1];
              break;
          case DB_HOMOGRAPHY_TYPE_AFFINE:
              db_UpdateImageHomographyAffine(H_p_dx,H,dx);
              break;
          case DB_HOMOGRAPHY_TYPE_PROJECTIVE:
              db_UpdateImageHomographyProjective(H_p_dx,H,dx,frozen_coord);
              break;
          case DB_HOMOGRAPHY_TYPE_CAMROTATION_F:
          case DB_HOMOGRAPHY_TYPE_CAMROTATION_F_UD:
              db_UpdateRotFocalHomography(H_p_dx,H,dx);
              break;
      }
  }
  
  void db_RobCamRotation_Polish_Generic(double H[9],int point_count,int homography_type,double *x_i,double *xp_i,double one_over_scale2,
                                 int max_iterations,double improvement_requirement)
  {
      int i,update,stop,n;
      int frozen_coord = 0;
      double lambda,cost,current_cost;
      double JtJ[72],min_Jtf[9],dx[8],H_p_dx[9];
      double *JtJ_ref[9],d[8];
  
      lambda=0.001;
      for(update=1,stop=0,i=0;(stop<2) && (i<max_iterations);i++)
      {
          /*if first time since improvement, compute Jacobian and residual*/
          if(update)
          {
              db_SetupMatrixRefs(JtJ_ref,9,8,JtJ);
              current_cost=db_RobImageHomography_Jacobians_Generic(JtJ_ref,min_Jtf,&n,&frozen_coord,H,point_count,x_i,xp_i,homography_type,one_over_scale2);
              update=0;
          }
  
  #ifdef _VERBOSE_
          /*std::cout << "Cost:" << current_cost << " ";*/
  #endif /*_VERBOSE_*/
  
          /*Come up with a hypothesis dx
          based on the current lambda*/
          db_Compute_dx(dx,JtJ_ref,min_Jtf,lambda,d,n);
  
          /*Compute Cost(x+dx)*/
          db_ImageHomographyUpdateGeneric(H_p_dx,H,dx,homography_type,frozen_coord);
          cost=db_RobImageHomography_Cost(H_p_dx,point_count,x_i,xp_i,one_over_scale2);
  
          /*Is there an improvement?*/
          if(cost<current_cost)
          {
              /*improvement*/
              if(current_cost-cost<current_cost*improvement_requirement) stop++;
              else stop=0;
              lambda*=0.1;
              /*Move to the hypothesised position x+dx*/
              current_cost=cost;
              db_Copy9(H,H_p_dx);
              update=1;
  
  #ifdef _VERBOSE_
          std::cout << "Step" << i << "Imp,Lambda=" << lambda << "Cost:" << current_cost << std::endl;
  #endif /*_VERBOSE_*/
          }
          else
          {
              /*no improvement*/
              lambda*=10.0;
              stop=0;
          }
      }
  }
  void db_RobImageHomography(
                                /*Best homography*/
                                double H[9],
                                /*2DPoint to 2DPoint constraints
                                Points are assumed to be given in
                                homogenous coordinates*/
                                double *im, double *im_p,
                                /*Nr of points in total*/
                                int nr_points,
                                /*Calibration matrices
                                used to normalize the points*/
                                double K[9],
                                double Kp[9],
                                /*Pre-allocated space temp_d
                                should point to at least
                                12*nr_samples+10*nr_points
                                allocated positions*/
                                double *temp_d,
                                /*Pre-allocated space temp_i
                                should point to at least
                                max(nr_samples,nr_points)
                                allocated positions*/
                                int *temp_i,
                                int homography_type,
                                db_Statistics *stat,
                                int max_iterations,
                                int max_points,
                                double scale,
                                int nr_samples,
                                int chunk_size,
                                /////////////////////////////////////////////
                                // regular use: set outlierremoveflagE =0;
                                // flag for the outlier removal
                                int outlierremoveflagE,
                                // if flag is 1, then the following variables
                                // need the input
                                //////////////////////////////////////
                                // 3D coordinates
                                double *wp,
                                // its corresponding stereo pair's points
                                double *im_r,
                                // raw image coordinates
                                double *im_raw, double *im_raw_p,
                                // final matches
                                int *finalNumE)
  {
      /*Random seed*/
      int r_seed;
  
      int point_count_new;
      /*Counters*/
      int i,j,c,point_count,hyp_count;
      int last_hyp,new_last_hyp,last_corr;
      int pos,point_pos,last_point;
      /*Accumulator*/
      double acc;
      /*Hypothesis pointer*/
      double *hyp_point;
      /*Random sample*/
      int s[4];
      /*Pivot for hypothesis pruning*/
      double pivot;
      /*Best hypothesis position*/
      int best_pos;
      /*Best score*/
      double lowest_cost;
      /*One over the squared scale of
      Cauchy distribution*/
      double one_over_scale2;
      /*temporary pointers*/
      double *x_i_temp,*xp_i_temp;
      /*Temporary space for inverse calibration matrices*/
      double K_inv[9];
      double Kp_inv[9];
      /*Temporary space for homography*/
      double H_temp[9],H_temp2[9];
      /*Pointers to homogenous coordinates*/
      double *x_h_point,*xp_h_point;
      /*Array of pointers to inhomogenous coordinates*/
      double *X[3],*Xp[3];
      /*Similarity parameters*/
      int orientation_preserving,allow_scaling,allow_rotation,allow_translation,sample_size;
  
      /*Homogenous coordinates of image points in first image*/
      double *x_h;
      /*Homogenous coordinates of image points in second image*/
      double *xp_h;
      /*Inhomogenous coordinates of image points in first image*/
      double *x_i;
      /*Inhomogenous coordinates of image points in second image*/
      double *xp_i;
      /*Homography hypotheses*/
      double *hyp_H_array;
      /*Cost array*/
      double *hyp_cost_array;
      /*Permutation of the hypotheses*/
      int *hyp_perm;
      /*Sample of the points*/
      int *point_perm;
      /*Temporary space for quick-select
      2*nr_samples*/
      double *temp_select;
  
      /*Get inverse calibration matrices*/
      db_InvertCalibrationMatrix(K_inv,K);
      db_InvertCalibrationMatrix(Kp_inv,Kp);
      /*Compute scale coefficient*/
      one_over_scale2=1.0/(scale*scale);
      /*Initialize random seed*/
      r_seed=12345;
      /*Set pointers to pre-allocated space*/
      hyp_cost_array=temp_d;
      hyp_H_array=temp_d+nr_samples;
      temp_select=temp_d+10*nr_samples;
      x_h=temp_d+12*nr_samples;
      xp_h=temp_d+12*nr_samples+3*nr_points;
      x_i=temp_d+12*nr_samples+6*nr_points;
      xp_i=temp_d+12*nr_samples+8*nr_points;
      hyp_perm=temp_i;
      point_perm=temp_i;
  
      /*Prepare a randomly permuted subset of size
      point_count from the input points*/
  
      point_count=db_mini(nr_points,(int)(chunk_size*log((double)nr_samples)/DB_LN2));
  
      point_count_new = point_count;
  
      for(i=0;i<nr_points;i++) point_perm[i]=i;
  
      for(last_point=nr_points-1,i=0;i<point_count;i++,last_point--)
      {
          pos=db_RandomInt(r_seed,last_point);
          point_pos=point_perm[pos];
          point_perm[pos]=point_perm[last_point];
  
          /*Normalize image points with calibration
          matrices and move them to x_h and xp_h*/
          c=3*point_pos;
          j=3*i;
          x_h_point=x_h+j;
          xp_h_point=xp_h+j;
          db_Multiply3x3_3x1(x_h_point,K_inv,im+c);
          db_Multiply3x3_3x1(xp_h_point,Kp_inv,im_p+c);
  
          db_HomogenousNormalize3(x_h_point);
          db_HomogenousNormalize3(xp_h_point);
  
          /*Dehomogenize image points and move them
          to x_i and xp_i*/
          c=(i<<1);
          db_DeHomogenizeImagePoint(x_i+c,x_h_point); // 2-dimension
          db_DeHomogenizeImagePoint(xp_i+c,xp_h_point); //2-dimension
      }
  
  
      /*Generate Hypotheses*/
      hyp_count=0;
      switch(homography_type)
      {
      case DB_HOMOGRAPHY_TYPE_SIMILARITY:
      case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
      case DB_HOMOGRAPHY_TYPE_TRANSLATION:
      case DB_HOMOGRAPHY_TYPE_ROTATION:
      case DB_HOMOGRAPHY_TYPE_ROTATION_U:
      case DB_HOMOGRAPHY_TYPE_SCALING:
      case DB_HOMOGRAPHY_TYPE_S_T:
      case DB_HOMOGRAPHY_TYPE_R_T:
      case DB_HOMOGRAPHY_TYPE_R_S:
  
          switch(homography_type)
          {
          case DB_HOMOGRAPHY_TYPE_SIMILARITY:
              orientation_preserving=1;
              allow_scaling=1;
              allow_rotation=1;
              allow_translation=1;
              sample_size=2;
              break;
          case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
              orientation_preserving=0;
              allow_scaling=1;
              allow_rotation=1;
              allow_translation=1;
              sample_size=3;
              break;
          case DB_HOMOGRAPHY_TYPE_TRANSLATION:
              orientation_preserving=1;
              allow_scaling=0;
              allow_rotation=0;
              allow_translation=1;
              sample_size=1;
              break;
          case DB_HOMOGRAPHY_TYPE_ROTATION:
              orientation_preserving=1;
              allow_scaling=0;
              allow_rotation=1;
              allow_translation=0;
              sample_size=1;
              break;
          case DB_HOMOGRAPHY_TYPE_ROTATION_U:
              orientation_preserving=0;
              allow_scaling=0;
              allow_rotation=1;
              allow_translation=0;
              sample_size=2;
              break;
          case DB_HOMOGRAPHY_TYPE_SCALING:
              orientation_preserving=1;
              allow_scaling=1;
              allow_rotation=0;
              allow_translation=0;
              sample_size=1;
              break;
          case DB_HOMOGRAPHY_TYPE_S_T:
              orientation_preserving=1;
              allow_scaling=1;
              allow_rotation=0;
              allow_translation=1;
              sample_size=2;
              break;
          case DB_HOMOGRAPHY_TYPE_R_T:
              orientation_preserving=1;
              allow_scaling=0;
              allow_rotation=1;
              allow_translation=1;
              sample_size=2;
              break;
          case DB_HOMOGRAPHY_TYPE_R_S:
              orientation_preserving=1;
              allow_scaling=1;
              allow_rotation=0;
              allow_translation=0;
              sample_size=1;
              break;
          }
  
          if(point_count>=sample_size) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,3,point_count,r_seed);
              X[0]= &x_i[s[0]<<1];
              X[1]= &x_i[s[1]<<1];
              X[2]= &x_i[s[2]<<1];
              Xp[0]= &xp_i[s[0]<<1];
              Xp[1]= &xp_i[s[1]<<1];
              Xp[2]= &xp_i[s[2]<<1];
              db_StitchSimilarity2D(&hyp_H_array[9*hyp_count],Xp,X,sample_size,orientation_preserving,
                                    allow_scaling,allow_rotation,allow_translation);
              hyp_count++;
          }
          break;
  
      case DB_HOMOGRAPHY_TYPE_CAMROTATION:
          if(point_count>=2) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,2,point_count,r_seed);
              db_StitchCameraRotation_2Points(&hyp_H_array[9*hyp_count],
                                        &x_h[3*s[0]],&x_h[3*s[1]],
                                        &xp_h[3*s[0]],&xp_h[3*s[1]]);
              hyp_count++;
          }
          break;
  
      case DB_HOMOGRAPHY_TYPE_CAMROTATION_F:
          if(point_count>=3) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,3,point_count,r_seed);
              hyp_count+=db_StitchRotationCommonFocalLength_3Points(&hyp_H_array[9*hyp_count],
                                        &x_h[3*s[0]],&x_h[3*s[1]],&x_h[3*s[2]],
                                        &xp_h[3*s[0]],&xp_h[3*s[1]],&xp_h[3*s[2]]);
          }
          break;
  
      case DB_HOMOGRAPHY_TYPE_CAMROTATION_F_UD:
          if(point_count>=3) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,3,point_count,r_seed);
              hyp_count+=db_StitchRotationCommonFocalLength_3Points(&hyp_H_array[9*hyp_count],
                                        &x_h[3*s[0]],&x_h[3*s[1]],&x_h[3*s[2]],
                                        &xp_h[3*s[0]],&xp_h[3*s[1]],&xp_h[3*s[2]],NULL,0);
          }
          break;
  
      case DB_HOMOGRAPHY_TYPE_AFFINE:
          if(point_count>=3) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,3,point_count,r_seed);
              db_StitchAffine2D_3Points(&hyp_H_array[9*hyp_count],
                                        &x_h[3*s[0]],&x_h[3*s[1]],&x_h[3*s[2]],
                                        &xp_h[3*s[0]],&xp_h[3*s[1]],&xp_h[3*s[2]]);
              hyp_count++;
          }
          break;
  
      case DB_HOMOGRAPHY_TYPE_PROJECTIVE:
      default:
          if(point_count>=4) for(i=0;i<nr_samples;i++)
          {
              db_RandomSample(s,4,point_count,r_seed);
              db_StitchProjective2D_4Points(&hyp_H_array[9*hyp_count],
                                        &x_h[3*s[0]],&x_h[3*s[1]],&x_h[3*s[2]],&x_h[3*s[3]],
                                        &xp_h[3*s[0]],&xp_h[3*s[1]],&xp_h[3*s[2]],&xp_h[3*s[3]]);
              hyp_count++;
          }
      }
  
      if(hyp_count)
      {
          /*Count cost in chunks and decimate hypotheses
          until only one remains or the correspondences are
          exhausted*/
          for(i=0;i<hyp_count;i++)
          {
              hyp_perm[i]=i;
              hyp_cost_array[i]=0.0;
          }
          for(i=0,last_hyp=hyp_count-1;(last_hyp>0) && (i<point_count);i+=chunk_size)
          {
              /*Update cost with the next chunk*/
              last_corr=db_mini(i+chunk_size-1,point_count-1);
              for(j=0;j<=last_hyp;j++)
              {
                  hyp_point=hyp_H_array+9*hyp_perm[j];
                  for(c=i;c<=last_corr;)
                  {
                      /*Take log of product of ten reprojection
                      errors to reduce nr of expensive log operations*/
                      if(c+9<=last_corr)
                      {
                          x_i_temp=x_i+(c<<1);
                          xp_i_temp=xp_i+(c<<1);
  
                          acc=db_ExpCauchyInhomogenousHomographyError(xp_i_temp,hyp_point,x_i_temp,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+2,hyp_point,x_i_temp+2,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+4,hyp_point,x_i_temp+4,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+6,hyp_point,x_i_temp+6,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+8,hyp_point,x_i_temp+8,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+10,hyp_point,x_i_temp+10,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+12,hyp_point,x_i_temp+12,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+14,hyp_point,x_i_temp+14,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+16,hyp_point,x_i_temp+16,one_over_scale2);
                          acc*=db_ExpCauchyInhomogenousHomographyError(xp_i_temp+18,hyp_point,x_i_temp+18,one_over_scale2);
                          c+=10;
                      }
                      else
                      {
                          for(acc=1.0;c<=last_corr;c++)
                          {
                              acc*=db_ExpCauchyInhomogenousHomographyError(xp_i+(c<<1),hyp_point,x_i+(c<<1),one_over_scale2);
                          }
                      }
                      hyp_cost_array[j]+=log(acc);
                  }
              }
              if (chunk_size<point_count){
                  /*Prune out half of the hypotheses*/
                  new_last_hyp=(last_hyp+1)/2-1;
                  pivot=db_LeanQuickSelect(hyp_cost_array,last_hyp+1,new_last_hyp,temp_select);
                  for(j=0,c=0;(j<=last_hyp) && (c<=new_last_hyp);j++)
                  {
                      if(hyp_cost_array[j]<=pivot)
                      {
                          hyp_cost_array[c]=hyp_cost_array[j];
                          hyp_perm[c]=hyp_perm[j];
                          c++;
                      }
                  }
                  last_hyp=new_last_hyp;
              }
          }
          /*Find the best hypothesis*/
          lowest_cost=hyp_cost_array[0];
          best_pos=0;
          for(j=1;j<=last_hyp;j++)
          {
              if(hyp_cost_array[j]<lowest_cost)
              {
                  lowest_cost=hyp_cost_array[j];
                  best_pos=j;
              }
          }
  
          /*Move the best hypothesis*/
          db_Copy9(H_temp,hyp_H_array+9*hyp_perm[best_pos]);
  
          // outlier removal
          if (outlierremoveflagE) // no polishment needed
          {
              point_count_new = db_RemoveOutliers_Homography(H_temp,x_i,xp_i,wp,im,im_p,im_r,im_raw,im_raw_p,point_count,one_over_scale2);
          }
          else
          {
              /*Polish*/
              switch(homography_type)
              {
              case DB_HOMOGRAPHY_TYPE_SIMILARITY:
              case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
              case DB_HOMOGRAPHY_TYPE_TRANSLATION:
              case DB_HOMOGRAPHY_TYPE_ROTATION:
              case DB_HOMOGRAPHY_TYPE_ROTATION_U:
              case DB_HOMOGRAPHY_TYPE_SCALING:
              case DB_HOMOGRAPHY_TYPE_S_T:
              case DB_HOMOGRAPHY_TYPE_R_T:
              case DB_HOMOGRAPHY_TYPE_R_S:
              case DB_HOMOGRAPHY_TYPE_AFFINE:
              case DB_HOMOGRAPHY_TYPE_PROJECTIVE:
              case DB_HOMOGRAPHY_TYPE_CAMROTATION_F:
              case DB_HOMOGRAPHY_TYPE_CAMROTATION_F_UD:
                  db_RobCamRotation_Polish_Generic(H_temp,db_mini(point_count,max_points),homography_type,x_i,xp_i,one_over_scale2,max_iterations);
                  break;
              case DB_HOMOGRAPHY_TYPE_CAMROTATION:
                  db_RobCamRotation_Polish(H_temp,db_mini(point_count,max_points),x_i,xp_i,one_over_scale2,max_iterations);
                  break;
              }
  
          }
  
      }
      else db_Identity3x3(H_temp);
  
      switch(homography_type)
      {
      case DB_HOMOGRAPHY_TYPE_PROJECTIVE:
          if(stat) stat->nr_parameters=8;
          break;
      case DB_HOMOGRAPHY_TYPE_AFFINE:
          if(stat) stat->nr_parameters=6;
          break;
      case DB_HOMOGRAPHY_TYPE_SIMILARITY:
      case DB_HOMOGRAPHY_TYPE_SIMILARITY_U:
      case DB_HOMOGRAPHY_TYPE_CAMROTATION_F:
      case DB_HOMOGRAPHY_TYPE_CAMROTATION_F_UD:
          if(stat) stat->nr_parameters=4;
          break;
      case DB_HOMOGRAPHY_TYPE_CAMROTATION:
          if(stat) stat->nr_parameters=3;
          break;
      case DB_HOMOGRAPHY_TYPE_TRANSLATION:
      case DB_HOMOGRAPHY_TYPE_S_T:
      case DB_HOMOGRAPHY_TYPE_R_T:
      case DB_HOMOGRAPHY_TYPE_R_S:
          if(stat) stat->nr_parameters=2;
          break;
      case DB_HOMOGRAPHY_TYPE_ROTATION:
      case DB_HOMOGRAPHY_TYPE_ROTATION_U:
      case DB_HOMOGRAPHY_TYPE_SCALING:
          if(stat) stat->nr_parameters=1;
          break;
      }
  
      db_RobImageHomography_Statistics(H_temp,db_mini(point_count,max_points),x_i,xp_i,one_over_scale2,stat);
  
      /*Put on the calibration matrices*/
      db_Multiply3x3_3x3(H_temp2,H_temp,K_inv);
      db_Multiply3x3_3x3(H,Kp,H_temp2);
  
      if (finalNumE)
          *finalNumE = point_count_new;
  
  }
  