vcglib/vcg/complex/algorithms/autoalign_4pcs.h

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* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2004                                                \/)\/    *
* Visual Computing Lab                                            /\/|      *
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* GNU General Public License (http://www.gnu.org/licenses/gpl.txt)          *
* for more details.                                                         *
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#ifndef _AUTOALIGN_4PCS_H_
#define _AUTOALIGN_4PCS_H_

/**
implementation of the 4PCS method from the paper:
"4-Points Congruent Sets for Robust Pairwise Surface Registration"
D.Aiger, N.Mitra D.Cohen-Or, SIGGRAPH 2008
ps: the name of the variables are out of vcg standard but like the one
used in the paper pseudocode.
*/

#include <vcg/space/point_matching.h>
#include <vcg/complex/algorithms/closest.h>
#include <vcg/complex/complex.h>
#include <wrap/io_trimesh/export_ply.h>

// note: temporary (callback.h should be moved inside vcg)
typedef bool AACb( const int pos,const char * str );

namespace vcg{
namespace tri{

template <class MeshType>
class FourPCS {
public:
    /* mesh only for using spatial indexing functions (to remove) */
  class PVertex;    // dummy prototype never used
  class PFace;

  class PUsedTypes: public vcg::UsedTypes < vcg::Use<PVertex>::template AsVertexType,
                                            vcg::Use<PFace  >::template AsFaceType >{};

  class PVertex : public vcg::Vertex< PUsedTypes,vcg::vertex::BitFlags,vcg::vertex::Coord3f ,vcg::vertex::Mark>{};
  class PFace   : public vcg::Face<   PUsedTypes> {};
  class PMesh   : public vcg::tri::TriMesh< std::vector<PVertex>, std::vector<PFace> > {};

  typedef typename MeshType::ScalarType ScalarType;
  typedef typename MeshType::CoordType CoordType;
  typedef typename MeshType::VertexIterator VertexIterator;
  typedef typename MeshType::VertexType VertexType;
  typedef vcg::Point4< vcg::Point3<ScalarType> > FourPoints;
  typedef vcg::GridStaticPtr<typename PMesh::VertexType, ScalarType > GridType;

  /* class for Parameters */
  struct Parameters
  {
    ScalarType delta;
    int feetsize;			// how many points in the neighborhood of each of the 4 points
    ScalarType f;			// overlap estimation
    int scoreFeet,		// how many of the feetsize points must match (max feetsize*4) to try an early interrupt
    scoreAln;			// how good must be the alignement	to end the process successfully

    void Default(){
      delta = 0.5;
      feetsize = 25;
      f = 0.5;
      scoreFeet = 50;
      scoreAln = 200;
    }
  };

    Parameters prs;	/// parameters

	public:
	void Init(MeshType &_P,MeshType &_Q);
	bool Align( int   L, vcg::Matrix44f & result, AACb * cb = NULL );		// main function

private:
	struct Couple: public std::pair<int,int>
	{
		Couple(const int & i, const int & j, float d):std::pair<int,int>(i,j),dist(d){}
		Couple(float d):std::pair<int,int>(0,0),dist(d){}
		float dist;
		const bool operator < (const   Couple & o) const {return dist < o.dist;}
		int & operator[](const int &i){return (i==0)? first : second;}
	};



	/* returns the closest point between to segments x1-x2 and x3-x4.  */
	void IntersectionLineLine(const CoordType & x1,const CoordType & x2,const CoordType & x3,const CoordType & x4, CoordType&x)
	{
		CoordType a = x2-x1, b = x4-x3, c = x3-x1;
		x = x1 + a * ((c^b).dot(a^b)) / (a^b).SquaredNorm();
	}




	struct CandiType{
		CandiType(){}
		CandiType(FourPoints _p,vcg::Matrix44<ScalarType>_T):p(_p),T(_T){}
		FourPoints  p;
		vcg::Matrix44<ScalarType> T;
		ScalarType err;
		int score;
		int base; // debug: for which base
		inline bool operator <(const CandiType & o) const {return score > o.score;}
	};


	MeshType	*P,											// mesh from which the coplanar base is selected
						*Q;											// mesh where to find the correspondences
	std::vector<int> mapsub;					// subset of index to the vertices in Q


	PMesh     Invr;										// invariants

	std::vector< CandiType > U;
	CandiType winner;
	int iwinner;											// winner == U[iwinner]

	FourPoints B;											// coplanar base
	std::vector<FourPoints> bases;		// used bases
	ScalarType side;									// side
	std::vector<VertexType*> ExtB[4]; // selection of vertices "close" to the four point
	std::vector<VertexType*> subsetP; // random selection on P
	ScalarType radius;

	ScalarType Bangle;
	std::vector<Couple > R1/*,R2*/;
	ScalarType r1,r2;

	// class for the point  'ei'
	struct EPoint{
		EPoint(vcg::Point3<ScalarType> _p, int _i):pos(_p),pi(_i){}
		vcg::Point3<ScalarType> pos;
		int pi;		//index to R[1|2]
		void GetBBox(vcg::Box3<ScalarType> & b){b.Add(pos);}
	};

	GridType *ugrid; // griglia
	vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType > ugridQ;
	vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType > ugridP;

	bool SelectCoplanarBase();												// on P
	bool FindCongruent() ;														// of base B, on Q, with approximation delta
	void ComputeR1R2(ScalarType d1,ScalarType d2);

	bool IsTransfCongruent(FourPoints fp,vcg::Matrix44<ScalarType> & mat, float &  trerr);
	int EvaluateSample(CandiType & fp, CoordType & tp, CoordType & np, const float &  angle);
	void EvaluateAlignment(CandiType & fp);
	void TestAlignment(CandiType & fp);

	/* debug tools */
public:
	std::vector<vcg::Matrix44f> allTr;// tutte le trasformazioni provate
	FILE * db;
	char namemesh1[255],namemesh2[255];
	int n_base;
	void InitDebug(const char * name1, const char * name2){
		db = fopen("debugPCS.txt","w");
		sprintf(&namemesh1[0],"%s",name1);
		sprintf(&namemesh2[0],"%s",name2);
		n_base = 0;
	}

	void FinishDebug(){
		fclose(db);
	}
	//void SaveALN(char * name,vcg::Matrix44f mat ){
	//	FILE * o = fopen(name,"w");
	//	fprintf(o,"2\n%s\n#\n",namemesh1);
	//	for(int  i = 0 ; i < 4; ++i)
	//		fprintf(o,"%f %f %f %f\n",mat[i][0],mat[i][1],mat[i][2],mat[i][3]);
	//	fprintf(o,"%s\n#\n",namemesh2);
	//	fprintf(o,"1.0 0.0 0.0 0.0 \n");
	//	fprintf(o,"0.0 1.0 0.0 0.0 \n");
	//	fprintf(o,"0.0 0.0 1.0 0.0 \n");
	//	fprintf(o,"0.0 0.0 0.0 1.0 \n");

	//	fclose(o);
	//}

};

template <class MeshType>
void
FourPCS<MeshType>:: Init(MeshType &_P,MeshType &_Q){

		P = &_P;Q=&_Q;
		ugridQ.Set(Q->vert.begin(),Q->vert.end());
		ugridP.Set(P->vert.begin(),P->vert.end());

		float ratio = 800 / (float) Q->vert.size();
		for(int vi = 0; vi < Q->vert.size(); ++vi)
		if(rand()/(float) RAND_MAX < ratio)
			mapsub.push_back(vi);

		for(int vi = 0; vi < P->vert.size(); ++vi)
		if(rand()/(float) RAND_MAX < ratio)
			subsetP.push_back(&P->vert[vi]);

		// estimate neigh distance
		float avD = 0.0;
		for(int i = 0 ; i < 100; ++i){
			int ri = rand()/(float) RAND_MAX * Q->vert.size() -1;
			std::vector< CoordType > samples,d_samples;
			std::vector<ScalarType > dists;
			std::vector<VertexType* > ress;
			vcg::tri::GetKClosestVertex<
					MeshType,
					vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType>,
					std::vector<VertexType*>,
					std::vector<ScalarType>,
					std::vector< CoordType > >(*Q,ugridQ,2,Q->vert[ri].cP(),Q->bbox.Diag(), ress,dists, samples);
			assert(ress.size() == 2);
			avD+=dists[1];
		}
		avD	/=100;						// average vertex-vertex distance
		avD /= sqrt(ratio);		// take into account the ratio

		prs.delta = avD * prs.delta;
		side = P->bbox.Dim()[P->bbox.MaxDim()]*prs.f; //rough implementation

	}

template <class MeshType>
bool
FourPCS<MeshType>::SelectCoplanarBase(){

	vcg::tri::UpdateBounding<MeshType>::Box(*P);

	// choose the inter point distance
	ScalarType dtol = side*0.1; //rough implementation

	//choose the first two points
	int i = 0,ch;

	// first point random
	ch = (rand()/(float)RAND_MAX)*(P->vert.size()-2);
	B[0] = P->vert[ch].P();
//printf("B[0] %d\n",ch);
	// second a point at distance d+-dtol
	for(i = 0; i < P->vert.size(); ++i){
		ScalarType dd = (P->vert[i].P() - B[0]).Norm();
		if(  ( dd < side + dtol) && (dd > side - dtol)){
			B[1] = P->vert[i].P();
//printf("B[1] %d\n",i);
			break;
		}
	}
	if(i ==  P->vert.size())
		return false;

	// third point at distance d from B[1] and forming a right angle
	int best = -1; ScalarType bestv=std::numeric_limits<float>::max();
	for(i = 0; i < P->vert.size(); ++i){
		int id = rand()/(float)RAND_MAX *  (P->vert.size()-1);
		ScalarType dd = (P->vert[id].P() - B[1]).Norm();
		if(  ( dd < side + dtol) && (dd > side - dtol)){
			ScalarType angle =  fabs( ( P->vert[id].P()-B[1]).normalized().dot((B[1]-B[0]).normalized()));
			if( angle < bestv){
				bestv = angle;
				best = id;
			}
		}
	}
	if(best == -1)
		return false;
	B[2] = P->vert[best].P();
//printf("B[2] %d\n",best);

	CoordType n = ((B[0]-B[1]).normalized() ^ (B[2]-B[1]).normalized()).normalized();
	CoordType B4 = B[1] +  (B[0]-B[1]) + (B[2]-B[1]);
	VertexType * v =0;
	ScalarType radius = dtol*4.0;

		std::vector<typename MeshType::VertexType*> closests;
		std::vector<ScalarType> distances;
		std::vector<CoordType> points;

		 vcg::tri::GetInSphereVertex<
					MeshType,
					vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType >,
					std::vector<typename MeshType::VertexType*>,
					std::vector<ScalarType>,
					std::vector<CoordType>
				>(*P,ugridP,B4,radius,closests,distances,points);

		if(closests.empty())
			return false;
	 best = -1;  bestv=std::numeric_limits<float>::max();
		for(i = 0; i <closests.size(); ++i){
		 ScalarType angle = fabs((closests[i]->P() - B[1]).normalized().dot(n));
			if( angle < bestv){
				bestv = angle;
				best = i;
			}
		}
		B[3] =  closests[best]->P();

//printf("B[3] %d\n", (typename MeshType::VertexType*)closests[best] - &(*P->vert.begin()));

		// compute r1 and r2
		CoordType x;
		std::swap(B[1],B[2]);
		IntersectionLineLine(B[0],B[1],B[2],B[3],x);

		r1 = (x - B[0]).dot(B[1]-B[0]) / (B[1]-B[0]).SquaredNorm();
		r2 = (x - B[2]).dot(B[3]-B[2]) / (B[3]-B[2]).SquaredNorm();

		if( ((B[0]+(B[1]-B[0])*r1)-(B[2]+(B[3]-B[2])*r2)).Norm() > prs.delta )
			return false;

		radius  =side*0.5;
		std::vector< CoordType > samples,d_samples;
		std::vector<ScalarType > dists;

		for(int i  = 0 ; i< 4; ++i){
			vcg::tri::GetKClosestVertex<
				MeshType,
				vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType >,
				std::vector<VertexType*>,
				std::vector<ScalarType>,
				std::vector< CoordType > >(*P,ugridP, prs.feetsize ,B[i],radius, ExtB[i],dists, samples);
		}

    //for(int i  = 0 ; i< 4; ++i)
 //		printf("%d ",ExtB[i].size());
    //	printf("\n");
return true;

}


template <class MeshType>
bool FourPCS<MeshType>::IsTransfCongruent(FourPoints fp, vcg::Matrix44<ScalarType> & mat, float &  trerr){

  std::vector<vcg::Point3<ScalarType> > fix;
  std::vector<vcg::Point3<ScalarType> > mov;
  for(int i = 0 ; i < 4; ++i) mov.push_back(B[i]);
  for(int i = 0 ; i < 4; ++i) fix.push_back(fp[i]);

  vcg::Point3<ScalarType> n,p;
  n = (( B[1]-B[0]).normalized() ^ ( B[2]- B[0]).normalized())*( B[1]- B[0]).Norm();
  p =  B[0] + n;
  mov.push_back(p);
  n = (( fp[1]-fp[0]).normalized() ^ (fp[2]- fp[0]).normalized())*( fp[1]- fp[0]).Norm();
  p =  fp[0] + n;
  fix.push_back(p);

  vcg::ComputeRigidMatchMatrix(fix,mov,mat);

  ScalarType err = 0.0;
  for(int i = 0; i < 4; ++i) err+= (mat * mov[i] - fix[i]).SquaredNorm();

  trerr = vcg::math::Sqrt(err);
  return  err  < prs.delta* prs.delta*4.0;
}

template <class MeshType>
void
FourPCS<MeshType>::ComputeR1R2(ScalarType d1,ScalarType d2){
	int vi,vj;
	R1.clear();
	//R2.clear();
	int start = clock();
	for(vi = 0; vi  < mapsub.size(); ++vi) for(vj = vi; vj < mapsub.size(); ++vj){
			ScalarType d = ((Q->vert[mapsub[vi]]).P()-(Q->vert[mapsub[vj]]).P()).Norm();
			if( (d < d1+ side*0.5) && (d > d1-side*0.5))
			{
				R1.push_back(Couple(mapsub[vi],mapsub[vj],d ));
				R1.push_back(Couple(mapsub[vj],mapsub[vi],d));
			}
	}
	//for( vi  = 0;  vi   < mapsub.size(); ++ vi ) for( vj  =  vi ;  vj  < mapsub.size(); ++ vj ){
	//		ScalarType d = ((Q->vert[mapsub[vi]]).P()-(Q->vert[mapsub[vj]]).P()).Norm();
	//	 	if( (d < d2+side*0.5) && (d > d2-side*0.5))
	//		{
	//			R2.push_back(Couple(mapsub[vi],mapsub[vj],d));
	//			R2.push_back(Couple(mapsub[vj],mapsub[vi],d));
	//		}
	//}

	std::sort(R1.begin(),R1.end());
//	std::sort(R2.begin(),R2.end());
}

template <class MeshType>
bool
FourPCS<MeshType>::FindCongruent() { // of base B, on Q, with approximation delta
	bool done = false;
	std::vector<EPoint> R2inv;
	int n_closests = 0, n_congr = 0;
	int ac =0 ,acf = 0,tr = 0,trf =0;
	ScalarType d1,d2;
	d1 = (B[1]-B[0]).Norm();
	d2 = (B[3]-B[2]).Norm();

	int start = clock();
	//int vi,vj;

	typename PMesh::VertexIterator vii;
	typename std::vector<Couple>::iterator bR1,eR1,bR2,eR2,ite,cite;
	bR1 = std::lower_bound<typename std::vector<Couple>::iterator,Couple>(R1.begin(),R1.end(),Couple(d1-prs.delta*2.0));
	eR1 = std::lower_bound<typename std::vector<Couple>::iterator,Couple>(R1.begin(),R1.end(),Couple(d1+prs.delta*2.0));
	bR2 = std::lower_bound<typename std::vector<Couple>::iterator,Couple>(R1.begin(),R1.end(),Couple(d2-prs.delta*2.0));
	eR2 = std::lower_bound<typename std::vector<Couple>::iterator,Couple>(R1.begin(),R1.end(),Couple(d2+prs.delta*2.0));

	// in  [bR1,eR1) there are all the pairs ad a distance d1 +- prs.delta
	// in  [bR1,eR1) there are all the pairs ad a distance d2 +- prs.delta

	if(bR1 == R1.end()) return false;// if there are no such pairs return
	if(bR2 == R1.end()) return false; // if there are no such pairs return

	// put [bR1,eR1) in a mesh to have the search operator for free (lazy me)
	Invr.Clear();
	int i = &(*bR1)-&(*R1.begin());
	for(ite = bR1; ite != eR1;++ite){
		vii = vcg::tri::Allocator<PMesh>::AddVertices(Invr,1);
		(*vii).P() = Q->vert[R1[i][0]].P() + (Q->vert[R1[i][1]].P()-Q->vert[R1[i][0]].P()) * r1;
		++i;
	}
	if(Invr.vert.empty() ) return false;

    // index remaps a vertex of Invr to its corresponding point in R1
    typename PMesh::template PerVertexAttributeHandle<int> id = vcg::tri::Allocator<PMesh>::template AddPerVertexAttribute<int>(Invr,std::string("index"));
    i = &(*bR1)-&(*R1.begin());
    for(vii = Invr.vert.begin(); vii != Invr.vert.end();++vii,++i)  id[vii] = i;

	vcg::tri::UpdateBounding<PMesh>::Box(Invr);
	//	printf("Invr size %d\n",Invr.vn);

	ugrid = new GridType();
	ugrid->Set(Invr.vert.begin(),Invr.vert.end());

	i = &(*bR2)-&(*R1.begin());
	// R2inv contains all the points generated by the couples in R2 (with the reference to remap into R2)
	for(ite = bR2; ite != eR2;++ite){
		R2inv.push_back( EPoint( Q->vert[R1[i][0]].P() + (Q->vert[R1[i][1]].P()-Q->vert[R1[i][0]].P()) * r2,i));
		++i;
	}

	n_closests = 0; n_congr = 0; ac =0 ; acf = 0; tr = 0; trf = 0;
	//	fprintf(db,"R2Inv.size  = %d \n",R2inv.size());
   for(uint i = 0 ; i < R2inv.size() ; ++i){

		std::vector<typename PMesh::VertexType*> closests;
		std::vector<ScalarType> distances;
		std::vector<CoordType> points;

		// for each point in R2inv get all the points in R1 closer than prs.delta
		vcg::Matrix44<ScalarType> mat;
		vcg::Box3f bb;
		bb.Add(R2inv[i].pos+vcg::Point3f(prs.delta * 0.1,prs.delta * 0.1 , prs.delta * 0.1 ));
		bb.Add(R2inv[i].pos-vcg::Point3f(prs.delta * 0.1,prs.delta* 0.1  , prs.delta* 0.1));

		vcg::tri::GetInBoxVertex<PMesh,GridType,std::vector<typename PMesh::VertexType*> >
			 (Invr,*ugrid,bb,closests);

		 n_closests+=closests.size();
		 for(uint ip = 0; ip < closests.size(); ++ip){
				FourPoints p;
				p[0] = Q->vert[R1[id[closests[ip]]][0]].P();
				p[1] = Q->vert[R1[id[closests[ip]]][1]].P();
				p[2] = Q->vert[R1[ R2inv[i].pi][0]].P();
				p[3] = Q->vert[R1[ R2inv[i].pi][1]].P();

				float trerr;
			  n_base++;
					if(!IsTransfCongruent(p,mat,trerr)) {
						trf++;
						//char name[255];
						//sprintf(name,"faileTR_%d_%f.aln",n_base,trerr);
						//fprintf(db,"TransCongruent %s\n", name);
						//SaveALN(name, mat);
					}
					else{
						tr++;
					n_congr++;
						U.push_back(CandiType(p,mat));
						EvaluateAlignment(U.back());
						U.back().base = bases.size()-1;

						if( U.back().score > prs.scoreFeet){
							TestAlignment(U.back());
							if(U.back().score > prs.scoreAln)
								{
									done = true; break;
								}
							}
						//char name[255];
						//sprintf(name,"passed_score_%5d_%d.aln",U.back().score,n_base);
						//fprintf(db,"OK TransCongruent %s, score: %d \n", name,U.back().score);
						//SaveALN(name, mat);
					}
				}
	 }

	 delete ugrid;
	 vcg::tri::Allocator<PMesh>::DeletePerVertexAttribute(Invr,id);
	 printf("n_closests %5d = (An %5d ) + ( Tr %5d ) + (OK) %5d\n",n_closests,acf,trf,n_congr);

	 return done;
//	 printf("done n_closests %d congr %d in %f s\n ",n_closests,n_congr,(clock()-start)/(float)CLOCKS_PER_SEC);
//	 printf("angle:%d %d, trasf %d %d\n",ac,acf,tr,trf);
}



template <class MeshType>
int FourPCS<MeshType>::EvaluateSample(CandiType & fp, CoordType & tp, CoordType & np, const float &  angle){
			VertexType*   v;
		ScalarType   dist ;
		radius = prs.delta;
		tp = fp.T * tp;

				vcg::Point4<ScalarType> np4;
				np4 = fp.T * vcg::Point4<ScalarType>(np[0],np[1],np[2],0.0);
				np[0] = np4[0]; np[1] = np4[1]; 	np[2] = np4[2];

	  v = 0;
			//v = vcg::tri::GetClosestVertex<
			//	MeshType,
			//	vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType >
			//  >(*Q,ugridQ,tp,radius,  dist  );
			typename MeshType::VertexType vq;
			vq.P() = tp;
			vq.N() = np;
			v = vcg::tri::GetClosestVertexNormal<
				MeshType,
				vcg::GridStaticPtr<typename MeshType::VertexType, ScalarType >
			  >(*Q,ugridQ,vq,radius,  dist  );

			if(v!=0)
				if( v->N().dot(np) -angle >0)  return 1; else return -1;

}


template <class MeshType>
void
FourPCS<MeshType>::EvaluateAlignment(CandiType  & fp){
		int n_delta_close = 0;
		for(int i  = 0 ; i< 4; ++i) {
			for(uint j = 0; j < ExtB[i].size();++j){
				CoordType np = ExtB[i][j]->cN();;
				CoordType tp  = ExtB[i][j]->P();
				n_delta_close+=EvaluateSample(fp,tp,np,0.9);
			}
		}
		fp.score = n_delta_close;
}

template <class MeshType>
void
FourPCS<MeshType>::TestAlignment(CandiType  & fp){
		radius = prs.delta;
		int n_delta_close = 0;
		for(uint j = 0; j < subsetP.size();++j){
				CoordType np = subsetP[j]->N();
				CoordType tp  = subsetP[j]->P();
				n_delta_close+=EvaluateSample(fp,tp,np,0.6);
			 }
		fp.score =  n_delta_close;
}


template <class MeshType>
bool
FourPCS<MeshType>::	 Align(  int L, vcg::Matrix44f & result, AACb * cb ){ // main loop

	int bestv = 0;
	bool found;
	int n_tries = 0;
	U.clear();

	if(L==0)
	{
		L = (log(1.0-0.9999) / log(1.0-pow((float)prs.f,3.f)))+1;
		printf("using %d bases\n",L);
	}

	ComputeR1R2(side*1.4,side*1.4);

	for(int t  = 0; t  < L; ++t ){
		do{
			n_tries = 0;
			do{
				n_tries++;
				found = SelectCoplanarBase();
				}
				while(!found && (n_tries <50));
				if(!found) {
					prs.f*=0.98;
					side = P->bbox.Dim()[P->bbox.MaxDim()]*prs.f; //rough implementation
					ComputeR1R2(side*1.4,side*1.4);
				}
		} while (!found && (prs.f >0.1));

		if(prs.f <0.1) {
			printf("FAILED");
			return false;
			}
		bases.push_back(B);
		if(cb) cb(t*100/L,"trying bases");
		if(FindCongruent())
			break;
	}

	if(U.empty()) return false;

	std::sort(U.begin(),U.end());

	bestv  = -std::numeric_limits<float>::max();
	iwinner = 0;

    for(int i = 0 ; i <  U.size() ;++i)
     {
        TestAlignment(U[i]);
        if(U[i].score > bestv){
            bestv = U[i].score;
            iwinner = i;
            }
    }

    printf("Best score: %d \n", bestv);

	winner =  U[iwinner];
	result = winner.T;

	// deallocations
	Invr.Clear();

	return true;
}

	} // namespace tri
} // namespace vcg
#endif