vcglib/vcg/complex/algorithms/point_sampling.h

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/****************************************************************************

The sampling Class has a set of static functions, that you can call to sample the surface of a mesh.
Each function is templated on the mesh and on a Sampler object s.
Each function calls many time the sample object with the sampling point as parameter.

Sampler Classes and Sampling algorithms are independent.
Sampler classes exploits the sample that are generated with various algorithms.
For example, you can compute Hausdorff distance (that is a sampler) using various
sampling strategies (montecarlo, stratified etc).

****************************************************************************/
#ifndef __VCGLIB_POINT_SAMPLING
#define __VCGLIB_POINT_SAMPLING


#include <vcg/math/random_generator.h>
#include <vcg/complex/algorithms/closest.h>
#include <vcg/space/index/spatial_hashing.h>
#include <vcg/complex/algorithms/stat.h>
#include <vcg/complex/algorithms/create/platonic.h>
#include <vcg/complex/algorithms/update/topology.h>
#include <vcg/complex/algorithms/update/normal.h>
#include <vcg/complex/algorithms/update/bounding.h>
#include <vcg/complex/algorithms/update/flag.h>
#include <vcg/space/segment2.h>

namespace vcg
{
namespace tri
{
/// \ingroup trimesh
/// \headerfile point_sampling.h vcg/complex/algorithms/point_sampling.h

/**
 \brief An basic sampler class that show the required interface used by the SurfaceSampling class.

 Most of the methods of sampling classes call the AddFace method of this class with the face containing the sample and its barycentric coord.
 Beside being an example of how to write a sampler it provides a simple way to use the various sampling classes.
 For example if you just want to get a vector with positions over the surface You have just to write

     vector<Point3f> myVec;
     TrivialSampler<MyMesh> ts(myVec);
     SurfaceSampling<MyMesh, TrivialSampler<MyMesh> >::Montecarlo(M, ts, SampleNum);

**/

template <class MeshType>
class TrivialSampler
{
public:
  typedef typename MeshType::CoordType  CoordType;
  typedef typename MeshType::VertexType VertexType;
  typedef typename MeshType::FaceType   FaceType;

  TrivialSampler()
  {
    sampleVec = new std::vector<CoordType>();
    vectorOwner=true;
  }

  TrivialSampler(std::vector<CoordType> &Vec)
  {
    sampleVec = &Vec;
    sampleVec->clear();
    vectorOwner=false;
  }

  ~TrivialSampler()
  {
    if(vectorOwner) delete sampleVec;
  }

private:
  std::vector<CoordType> *sampleVec;
  bool vectorOwner;
public:

  void AddVert(const VertexType &p)
  {
    sampleVec->push_back(p.cP());
  }
  void AddFace(const FaceType &f, const CoordType &p)
  {
    sampleVec->push_back(f.cP(0)*p[0] + f.cP(1)*p[1] +f.cP(2)*p[2] );
  }

  void AddTextureSample(const FaceType &, const CoordType &, const Point2i &, float )
  {
    // Retrieve the color of the sample from the face f using the barycentric coord p
    // and write that color in a texture image at position <tp[0], texHeight-tp[1]>
    // if edgeDist is > 0 then the corrisponding point is affecting face color even if outside the face area (in texture space)
  }
}; // end class TrivialSampler





template <class MeshType>
class MeshSampler
{
public:
  typedef typename MeshType::VertexType  VertexType;
  typedef typename MeshType::FaceType    FaceType;
  typedef typename MeshType::CoordType   CoordType;

  MeshSampler(MeshType &_m):m(_m){}
  MeshType &m;

  void AddVert(const VertexType &p)
  {
    tri::Allocator<MeshType>::AddVertices(m,1);
    m.vert.back().ImportData(p);
  }

  void AddFace(const FaceType &f, CoordType p)
  {
    tri::Allocator<MeshType>::AddVertices(m,1);
    m.vert.back().P() = f.cP(0)*p[0] + f.cP(1)*p[1] +f.cP(2)*p[2];
    m.vert.back().N() = f.cV(0)->N()*p[0] + f.cV(1)->N()*p[1] + f.cV(2)->N()*p[2];
    if(tri::HasPerVertexQuality(m) )
       m.vert.back().Q() = f.cV(0)->Q()*p[0] + f.cV(1)->Q()*p[1] + f.cV(2)->Q()*p[2];
  }
}; // end class BaseSampler





/**
 \brief Main Class of the Sampling framework.

This class allows you to perform various kind of random/procedural point sampling over a triangulated surface.
The class is templated over the PointSampler object that allows to customize the use of the generated samples.


**/


template <class MetroMesh, class VertexSampler = TrivialSampler< MetroMesh> >
class SurfaceSampling
{
  typedef typename MetroMesh::CoordType       CoordType;
  typedef typename MetroMesh::ScalarType      ScalarType;
  typedef typename MetroMesh::VertexType      VertexType;
  typedef typename MetroMesh::VertexPointer   VertexPointer;
  typedef typename MetroMesh::VertexIterator  VertexIterator;
  typedef typename MetroMesh::FacePointer     FacePointer;
  typedef typename MetroMesh::FaceIterator    FaceIterator;
  typedef typename MetroMesh::FaceType        FaceType;
  typedef typename MetroMesh::FaceContainer   FaceContainer;
  typedef typename vcg::Box3<ScalarType>      BoxType;

  typedef typename vcg::SpatialHashTable<FaceType, ScalarType> MeshSHT;
  typedef typename vcg::SpatialHashTable<FaceType, ScalarType>::CellIterator MeshSHTIterator;
  typedef typename vcg::SpatialHashTable<VertexType, ScalarType> MontecarloSHT;
  typedef typename vcg::SpatialHashTable<VertexType, ScalarType>::CellIterator MontecarloSHTIterator;
  typedef typename vcg::SpatialHashTable<VertexType, ScalarType> SampleSHT;
  typedef typename vcg::SpatialHashTable<VertexType, ScalarType>::CellIterator SampleSHTIterator;

public:

static math::MarsenneTwisterRNG &SamplingRandomGenerator()
{
	static math::MarsenneTwisterRNG rnd;
	return rnd;
}

// Returns an integer random number in the [0,i-1] interval using the improve Marsenne-Twister method.
static unsigned int RandomInt(unsigned int i)
{
	return (SamplingRandomGenerator().generate(0) % i);
}

// Returns a random number in the [0,1) real interval using the improved Marsenne-Twister method.
static double RandomDouble01()
{
    return SamplingRandomGenerator().generate01();
}

#define FAK_LEN 1024
static double LnFac(int n) {
   // Tabled log factorial function. gives natural logarithm of n!

   // define constants
   static const double                 // coefficients in Stirling approximation
      C0 =  0.918938533204672722,      // ln(sqrt(2*pi))
      C1 =  1./12.,
      C3 = -1./360.;
   // C5 =  1./1260.,                  // use r^5 term if FAK_LEN < 50
   // C7 = -1./1680.;                  // use r^7 term if FAK_LEN < 20
   // static variables
   static double fac_table[FAK_LEN];   // table of ln(n!):
   static bool initialized = false;         // remember if fac_table has been initialized


   if (n < FAK_LEN) {
      if (n <= 1) {
         if (n < 0) assert(0);//("Parameter negative in LnFac function");
         return 0;
      }
      if (!initialized) {              // first time. Must initialize table
         // make table of ln(n!)
         double sum = fac_table[0] = 0.;
         for (int i=1; i<FAK_LEN; i++) {
            sum += log(double(i));
            fac_table[i] = sum;
         }
         initialized = true;
      }
      return fac_table[n];
   }
   // not found in table. use Stirling approximation
   double  n1, r;
   n1 = n;  r  = 1. / n1;
   return (n1 + 0.5)*log(n1) - n1 + C0 + r*(C1 + r*r*C3);
}

static int  PoissonRatioUniforms(double L) {
   /*

   This subfunction generates a integer with the poisson
   distribution using the ratio-of-uniforms rejection method (PRUAt).
   This approach is STABLE even for large L (e.g. it does not suffer from the overflow limit of the classical Knuth implementation)
   Execution time does not depend on L, except that it matters whether
   is within the range where ln(n!) is tabulated.

   Reference:

   E. Stadlober
   "The ratio of uniforms approach for generating discrete random variates".
   Journal of Computational and Applied Mathematics,
   vol. 31, no. 1, 1990, pp. 181-189.

   Partially adapted/inspired from some subfunctions of the Agner Fog stocc library ( www.agner.org/random )
   Same licensing scheme.

   */
  // constants

  const double SHAT1 = 2.943035529371538573;    // 8/e
  const double SHAT2 = 0.8989161620588987408;   // 3-sqrt(12/e)
  double u;                                          // uniform random
  double lf;                                         // ln(f(x))
  double x;                                          // real sample
  int k;                                         // integer sample

  double   pois_a = L + 0.5;                               // hat center
  int mode = (int)L;                      // mode
  double   pois_g  = log(L);
  double    pois_f0 = mode * pois_g - LnFac(mode);          // value at mode
  double   pois_h = sqrt(SHAT1 * (L+0.5)) + SHAT2;         // hat width
  double   pois_bound = (int)(pois_a + 6.0 * pois_h);  // safety-bound

  while(1) {
      u = RandomDouble01();
      if (u == 0) continue;                           // avoid division by 0
      x = pois_a + pois_h * (RandomDouble01() - 0.5) / u;
      if (x < 0 || x >= pois_bound) continue;         // reject if outside valid range
      k = (int)(x);
      lf = k * pois_g - LnFac(k) - pois_f0;
      if (lf >= u * (4.0 - u) - 3.0) break;           // quick acceptance
      if (u * (u - lf) > 1.0) continue;               // quick rejection
      if (2.0 * log(u) <= lf) break;                  // final acceptance
   }
   return k;
}


/**
  algorithm poisson random number (Knuth):
    init:
         Let L ← e^−λ, k ← 0 and p ← 1.
    do:
         k ← k + 1.
         Generate uniform random number u in [0,1] and let p ← p × u.
    while p > L.
    return k − 1.

  */
static int Poisson(double lambda)
{
  if(lambda>50) return PoissonRatioUniforms(lambda);
  double L = exp(-lambda);
  int k =0;
  double p = 1.0;
  do
  {
    k = k+1;
    p = p*RandomDouble01();
  } while (p>L);

  return k -1;
}


static void AllVertex(MetroMesh & m, VertexSampler &ps)
{
	VertexIterator vi;
	for(vi=m.vert.begin();vi!=m.vert.end();++vi)
	{
		if(!(*vi).IsD())
		{
			ps.AddVert(*vi);
		}
	}
}

/// Sample the vertices in a weighted way. Each vertex has a probability of being chosen
/// that is proportional to its quality.
/// It assumes that you are asking a number of vertices smaller than nv;
/// Algorithm:
/// 1) normalize quality so that sum q == 1;
/// 2) shuffle vertices.
/// 3) for each vertices choose it if rand > thr;

static void VertexWeighted(MetroMesh & m, VertexSampler &ps, int sampleNum)
{
	ScalarType qSum = 0;
	VertexIterator vi;
	for(vi = m.vert.begin(); vi != m.vert.end(); ++vi)
			if(!(*vi).IsD())
						qSum += (*vi).Q();

	ScalarType samplePerUnit = sampleNum/qSum;
	ScalarType floatSampleNum =0;
	std::vector<VertexPointer> vertVec;
	FillAndShuffleVertexPointerVector(m,vertVec);

	std::vector<bool> vertUsed(m.vn,false);

	int i=0; int cnt=0;
	while(cnt < sampleNum)
		{
			if(vertUsed[i])
				{
						floatSampleNum += vertVec[i]->Q() * samplePerUnit;
						int vertSampleNum   = (int) floatSampleNum;
						floatSampleNum -= (float) vertSampleNum;

						// for every sample p_i in T...
						if(vertSampleNum > 1)
							{
								ps.AddVert(*vertVec[i]);
								cnt++;
								vertUsed[i]=true;
							}
				}
			i = (i+1)%m.vn;
		}
}

/// Sample the vertices in a uniform way. Each vertex has a probability of being chosen
/// that is proportional to the area it represent.
static void VertexAreaUniform(MetroMesh & m, VertexSampler &ps, int sampleNum)
{
	VertexIterator vi;
	for(vi = m.vert.begin(); vi != m.vert.end(); ++vi)
		if(!(*vi).IsD())
						(*vi).Q() = 0;

	FaceIterator fi;
	for(fi = m.face.begin(); fi != m.face.end(); ++fi)
		if(!(*fi).IsD())
		{
			ScalarType areaThird = DoubleArea(*fi)/6.0;
			(*fi).V(0)->Q()+=areaThird;
			(*fi).V(1)->Q()+=areaThird;
			(*fi).V(2)->Q()+=areaThird;
		}

	VertexWeighted(m,ps,sampleNum);
}

static void	FillAndShuffleFacePointerVector(MetroMesh & m, std::vector<FacePointer> &faceVec)
{
	FaceIterator fi;
	for(fi=m.face.begin();fi!=m.face.end();++fi)
		if(!(*fi).IsD())	faceVec.push_back(&*fi);

	assert((int)faceVec.size()==m.fn);

	unsigned int (*p_myrandom)(unsigned int) = RandomInt;
	std::random_shuffle(faceVec.begin(),faceVec.end(), p_myrandom);
}
static void	FillAndShuffleVertexPointerVector(MetroMesh & m, std::vector<VertexPointer> &vertVec)
{
	VertexIterator vi;
	for(vi=m.vert.begin();vi!=m.vert.end();++vi)
				if(!(*vi).IsD())	vertVec.push_back(&*vi);

	assert((int)vertVec.size()==m.vn);

	unsigned int (*p_myrandom)(unsigned int) = RandomInt;
	std::random_shuffle(vertVec.begin(),vertVec.end(), p_myrandom);
}

/// Sample the vertices in a uniform way. Each vertex has the same probabiltiy of being chosen.
static void VertexUniform(MetroMesh & m, VertexSampler &ps, int sampleNum)
{
	if(sampleNum>=m.vn) {
	  AllVertex(m,ps);
		return;
	}

	std::vector<VertexPointer> vertVec;
	FillAndShuffleVertexPointerVector(m,vertVec);

	for(int i =0; i< sampleNum; ++i)
		ps.AddVert(*vertVec[i]);
}

/// \brief Sample all the border corner vertices
///
/// It assumes that the border flag have been set over the mesh.
/// All the vertices on the border where the surface forms an angle smaller than the given threshold are sampled.
///
static void VertexBorderCorner(MetroMesh & m, VertexSampler &ps, float angleRad)
{
  typename MetroMesh::template PerVertexAttributeHandle  <float> angleSumH = tri::Allocator<MetroMesh>:: template GetPerVertexAttribute<float> (m);

  for(VertexIterator vi=m.vert.begin();vi!=m.vert.end();++vi)
    angleSumH[vi]=0;

  for(FaceIterator fi=m.face.begin();fi!=m.face.end();++fi)
  {
    for(int i=0;i<3;++i)
    {
      angleSumH[fi->V(i)] += vcg::Angle(fi->P2(i) - fi->P0(i),fi->P1(i) - fi->P0(i));
    }
  }

  for(VertexIterator vi=m.vert.begin();vi!=m.vert.end();++vi)
  {
    if(angleSumH[vi]<angleRad)
        ps.AddVert(*vi);
  }

  tri::Allocator<MetroMesh>:: template DeletePerVertexAttribute<float> (m,angleSumH);
}

/// Sample all the crease vertices.
/// It assumes that the crease edges had been marked as non-faux edges
/// for example by using
/// tri::UpdateFlags<MeshType>::FaceFauxCrease(mesh,creaseAngleRad);
/// Then it chooses all the vertices where there are at least three non faux edges.
///
static void VertexCrease(MetroMesh & m, VertexSampler &ps)
{
  typedef typename UpdateTopology<MetroMesh>::PEdge SimpleEdge;
  std::vector< SimpleEdge > Edges;
  typename std::vector< SimpleEdge >::iterator ei;
  UpdateTopology<MetroMesh>::FillUniqueEdgeVector(m,Edges,false);

  typename MetroMesh::template PerVertexAttributeHandle  <int> hv = tri::Allocator<MetroMesh>:: template GetPerVertexAttribute<int> (m);

  for(ei=Edges.begin(); ei!=Edges.end(); ++ei)
  {
    hv[ei->v[0]]++;
    hv[ei->v[1]]++;
  }

  for(VertexIterator vi=m.vert.begin();vi!=m.vert.end();++vi)
  {
    if(hv[vi]>2)
        ps.AddVert(*vi);
  }
}


static void FaceUniform(MetroMesh & m, VertexSampler &ps, int sampleNum)
{
	if(sampleNum>=m.fn) {
	  AllFace(m,ps);
		return;
	}

	std::vector<FacePointer> faceVec;
	FillAndShuffleFacePointerVector(m,faceVec);

	for(int i =0; i< sampleNum; ++i)
		ps.AddFace(*faceVec[i],Barycenter(*faceVec[i]));
}

static void AllFace(MetroMesh & m, VertexSampler &ps)
{
	FaceIterator fi;
	for(fi=m.face.begin();fi!=m.face.end();++fi)
				if(!(*fi).IsD())
				{
					ps.AddFace(*fi,Barycenter(*fi));
				}
}


static void AllEdge(MetroMesh & m, VertexSampler &ps)
{
  // Edge sampling.
  typedef typename UpdateTopology<MetroMesh>::PEdge SimpleEdge;
  std::vector< SimpleEdge > Edges;
  typename std::vector< SimpleEdge >::iterator ei;
  UpdateTopology<MetroMesh>::FillUniqueEdgeVector(m,Edges);

  for(ei=Edges.begin(); ei!=Edges.end(); ++ei)
    ps.AddFace(*(*ei).f,ei->EdgeBarycentricToFaceBarycentric(0.5));
}

// Regular Uniform Edge sampling
// Each edge is subdivided in a number of pieces proprtional to its length
// Sample are choosen without touching the vertices.

static void EdgeUniform(MetroMesh & m, VertexSampler &ps,int sampleNum, bool sampleFauxEdge=true)
{
		typedef typename UpdateTopology<MetroMesh>::PEdge SimpleEdge;
		std::vector< SimpleEdge > Edges;
	UpdateTopology<MetroMesh>::FillUniqueEdgeVector(m,Edges,sampleFauxEdge);
		// First loop compute total edge length;
		float edgeSum=0;
		typename std::vector< SimpleEdge >::iterator ei;
		for(ei=Edges.begin(); ei!=Edges.end(); ++ei)
					edgeSum+=Distance((*ei).v[0]->P(),(*ei).v[1]->P());

		float sampleLen = edgeSum/sampleNum;
		float rest=0;
		for(ei=Edges.begin(); ei!=Edges.end(); ++ei)
				{
					float len = Distance((*ei).v[0]->P(),(*ei).v[1]->P());
					float samplePerEdge = floor((len+rest)/sampleLen);
					rest = (len+rest) - samplePerEdge * sampleLen;
					float step = 1.0/(samplePerEdge+1);
					for(int i=0;i<samplePerEdge;++i)
					{
						Point3f interp(0,0,0);
						interp[ (*ei).z     ]=step*(i+1);
						interp[((*ei).z+1)%3]=1.0-step*(i+1);
						ps.AddFace(*(*ei).f,interp);
					}
				}
}

// Generate the barycentric coords of a random point over a single face,
// with a uniform distribution over the triangle.
// It uses the parallelogram folding trick.
static CoordType RandomBarycentric()
{
  return math::GenerateBarycentricUniform<ScalarType>(SamplingRandomGenerator());
}

// Given a triangle return a random point over it
static CoordType RandomPointInTriangle(const FaceType &f)
{
	CoordType u = RandomBarycentric();
	return f.cP(0)*u[0] + f.cP(1)*u[1] + f.cP(2)*u[2];
}

static void StratifiedMontecarlo(MetroMesh & m, VertexSampler &ps,int sampleNum)
{
	ScalarType area = Stat<MetroMesh>::ComputeMeshArea(m);
	ScalarType samplePerAreaUnit = sampleNum/area;
	// Montecarlo sampling.
	double  floatSampleNum = 0.0;

	FaceIterator fi;
	for(fi=m.face.begin(); fi != m.face.end(); fi++)
		if(!(*fi).IsD())
		{
			// compute # samples in the current face (taking into account of the remainders)
			floatSampleNum += 0.5*DoubleArea(*fi) * samplePerAreaUnit;
			int faceSampleNum   = (int) floatSampleNum;

			// for every sample p_i in T...
			for(int i=0; i < faceSampleNum; i++)
					ps.AddFace(*fi,RandomBarycentric());
			floatSampleNum -= (double) faceSampleNum;
		}
}

/**
  This function compute montecarlo distribution with an approximate number of
  samples exploiting the poisson distribution approximation of the binomial distribution.

  For a given triangle t of area a_t, in a Mesh of area A,
  if we take n_s sample over the mesh, the number of samples that falls in t
  follows the poisson distribution of P(lambda ) with lambda = n_s * (a_t/A).

  To approximate the Binomial we use a Poisson distribution with parameter
  \lambda = np can be used as an approximation to B(n,p)
  (it works if n is sufficiently large and p is sufficiently small).

  */

static void MontecarloPoisson(MetroMesh & m, VertexSampler &ps,int sampleNum)
{
  ScalarType area = Stat<MetroMesh>::ComputeMeshArea(m);
  ScalarType samplePerAreaUnit = sampleNum/area;

  FaceIterator fi;
  for(fi=m.face.begin(); fi != m.face.end(); fi++)
    if(!(*fi).IsD())
    {
      float areaT=DoubleArea(*fi) * 0.5f;
      int faceSampleNum = Poisson(areaT*samplePerAreaUnit);

      // for every sample p_i in T...
      for(int i=0; i < faceSampleNum; i++)
          ps.AddFace(*fi,RandomBarycentric());
//      SampleNum -= (double) faceSampleNum;
    }
}


/**
  This function computes a montecarlo distribution with an EXACT number of samples.
  it works by generating a sequence of consecutive segments proportional to the triangle areas
  and actually shooting sample over this line
  */

static void EdgeMontecarlo(MetroMesh & m, VertexSampler &ps, int sampleNum, bool sampleAllEdges)
{
  typedef typename UpdateTopology<MetroMesh>::PEdge SimpleEdge;
  std::vector< SimpleEdge > Edges;
  UpdateTopology<MetroMesh>::FillUniqueEdgeVector(m,Edges,sampleAllEdges);

  assert(!Edges.empty());

  typedef  std::pair<ScalarType, SimpleEdge*> IntervalType;
  std::vector< IntervalType > intervals (Edges.size()+1);
  int i=0;
  intervals[i]=std::make_pair(0,(SimpleEdge*)(0));
  // First loop: build a sequence of consecutive segments proportional to the edge lenghts.
  typename std::vector< SimpleEdge >::iterator ei;
  for(ei=Edges.begin(); ei != Edges.end(); ei++)
  {
    intervals[i+1]=std::make_pair(intervals[i].first+Distance((*ei).v[0]->P(),(*ei).v[1]->P()), &*ei);
    ++i;
  }

  // Second Loop get a point on the line 0...Sum(edgeLen) to pick a point;
  ScalarType edgeSum = intervals.back().first;
  for(i=0;i<sampleNum;++i)
  {
    ScalarType val = edgeSum * RandomDouble01();
    // lower_bound returns the furthermost iterator i in [first, last) such that, for every iterator j in [first, i), *j < value.
    // E.g. An iterator pointing to the first element "not less than" val, or end() if every element is less than val.
    typename std::vector<IntervalType>::iterator it = lower_bound(intervals.begin(),intervals.end(),std::make_pair(val,(SimpleEdge*)(0)) );
    assert(it != intervals.end() && it != intervals.begin());
    assert( ( (*(it-1)).first < val ) && ((*(it)).first >= val) );
    SimpleEdge * ep=(*it).second;
    ps.AddFace( *(ep->f), ep->EdgeBarycentricToFaceBarycentric(RandomDouble01()) );
  }
}

/**
  This function computes a montecarlo distribution with an EXACT number of samples.
  it works by generating a sequence of consecutive segments proportional to the triangle areas
  and actually shooting sample over this line
  */

static void Montecarlo(MetroMesh & m, VertexSampler &ps,int sampleNum)
{
	typedef  std::pair<ScalarType, FacePointer> IntervalType;
	std::vector< IntervalType > intervals (m.fn+1);
	FaceIterator fi;
	int i=0;
	intervals[i]=std::make_pair(0,FacePointer(0));
	// First loop: build a sequence of consecutive segments proportional to the triangle areas.
	for(fi=m.face.begin(); fi != m.face.end(); fi++)
		if(!(*fi).IsD())
		{
			intervals[i+1]=std::make_pair(intervals[i].first+0.5*DoubleArea(*fi), &*fi);
			++i;
		}
	ScalarType meshArea = intervals.back().first;
	for(i=0;i<sampleNum;++i)
		{
			ScalarType val = meshArea * RandomDouble01();
			// lower_bound returns the furthermost iterator i in [first, last) such that, for every iterator j in [first, i), *j < value.
			// E.g. An iterator pointing to the first element "not less than" val, or end() if every element is less than val.
			typename std::vector<IntervalType>::iterator it = lower_bound(intervals.begin(),intervals.end(),std::make_pair(val,FacePointer(0)) );
			assert(it != intervals.end());
			assert(it != intervals.begin());
			assert( (*(it-1)).first <val );
			assert( (*(it)).first >= val);
			ps.AddFace( *(*it).second, RandomBarycentric() );
		}
}

static ScalarType WeightedArea(FaceType f)
{
	ScalarType averageQ = ( f.V(0)->Q() + f.V(1)->Q() + f.V(2)->Q() ) /3.0;
	return DoubleArea(f)*averageQ/2.0;
}

/// Compute a sampling of the surface that is weighted by the quality
/// the area of each face is multiplied by the average of the quality of the vertices.
/// So the a face with a zero quality on all its vertices is never sampled and a face with average quality 2 get twice the samples of a face with the same area but with an average quality of 1;
static void WeightedMontecarlo(MetroMesh & m, VertexSampler &ps, int sampleNum)
{
	assert(tri::HasPerVertexQuality(m));

	ScalarType weightedArea = 0;
	FaceIterator fi;
	for(fi = m.face.begin(); fi != m.face.end(); ++fi)
			if(!(*fi).IsD())
						weightedArea += WeightedArea(*fi);

	ScalarType samplePerAreaUnit = sampleNum/weightedArea;
	// Montecarlo sampling.
	double  floatSampleNum = 0.0;
	for(fi=m.face.begin(); fi != m.face.end(); fi++)
		if(!(*fi).IsD())
	{
			// compute # samples in the current face (taking into account of the remainders)
			floatSampleNum += WeightedArea(*fi) * samplePerAreaUnit;
			int faceSampleNum   = (int) floatSampleNum;

			// for every sample p_i in T...
			for(int i=0; i < faceSampleNum; i++)
					ps.AddFace(*fi,RandomBarycentric());

            floatSampleNum -= (double) faceSampleNum;
    }
}


// Subdivision sampling of a single face.
// return number of added samples

static int SingleFaceSubdivision(int sampleNum, const CoordType & v0, const CoordType & v1, const CoordType & v2, VertexSampler &ps, FacePointer fp, bool randSample)
{
    // recursive face subdivision.
    if(sampleNum == 1)
    {
        // ground case.
        CoordType SamplePoint;
        if(randSample)
        {
            CoordType rb=RandomBarycentric();
            SamplePoint=v0*rb[0]+v1*rb[1]+v2*rb[2];
        }
        else SamplePoint=((v0+v1+v2)*(1.0f/3.0f));

        ps.AddFace(*fp,SamplePoint);
        return 1;
    }

    int s0 = sampleNum /2;
    int s1 = sampleNum-s0;
    assert(s0>0);
    assert(s1>0);

    ScalarType w0 = ScalarType(s1)/ScalarType(sampleNum);
    ScalarType w1 = 1.0-w0;
    // compute the longest edge.
    ScalarType  maxd01 = SquaredDistance(v0,v1);
    ScalarType  maxd12 = SquaredDistance(v1,v2);
    ScalarType  maxd20 = SquaredDistance(v2,v0);
    int     res;
    if(maxd01 > maxd12)
        if(maxd01 > maxd20)     res = 0;
    else                    res = 2;
    else
        if(maxd12 > maxd20)     res = 1;
    else                    res = 2;

    int faceSampleNum=0;
    // break the input triangle along the midpoint of the longest edge.
    CoordType  pp;
    switch(res)
    {
    case 0 :    pp = v0*w0 + v1*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,pp,v2,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,pp,v1,v2,ps,fp,randSample);
        break;
    case 1 :    pp =  v1*w0 + v2*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,v1,pp,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,v0,pp,v2,ps,fp,randSample);
        break;
    case 2 :    pp = v0*w0 + v2*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,v1,pp,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,pp,v1,v2,ps,fp,randSample);
        break;
    }
    return faceSampleNum;
}


/// Compute a sampling of the surface where the points are regularly scattered over the face surface using a recursive longest-edge subdivision rule.
static void FaceSubdivision(MetroMesh & m, VertexSampler &ps,int sampleNum, bool randSample)
{

	ScalarType area = Stat<MetroMesh>::ComputeMeshArea(m);
	ScalarType samplePerAreaUnit = sampleNum/area;
	std::vector<FacePointer> faceVec;
	FillAndShuffleFacePointerVector(m,faceVec);
	vcg::tri::UpdateNormal<MetroMesh>::PerFaceNormalized(m);
	double  floatSampleNum = 0.0;
	int faceSampleNum;
	// Subdivision sampling.
	typename std::vector<FacePointer>::iterator fi;
	for(fi=faceVec.begin(); fi!=faceVec.end(); fi++)
	{
		const CoordType b0(1.0, 0.0, 0.0);
		const CoordType b1(0.0, 1.0, 0.0);
		const CoordType b2(0.0, 0.0, 1.0);
		// compute # samples in the current face.
		floatSampleNum += 0.5*DoubleArea(**fi) * samplePerAreaUnit;
		faceSampleNum          = (int) floatSampleNum;
		if(faceSampleNum>0)
			faceSampleNum = SingleFaceSubdivision(faceSampleNum,b0,b1,b2,ps,*fi,randSample);
		floatSampleNum -= (double) faceSampleNum;
	}
}
//---------
// Subdivision sampling of a single face.
// return number of added samples

static int SingleFaceSubdivisionOld(int sampleNum, const CoordType & v0, const CoordType & v1, const CoordType & v2, VertexSampler &ps, FacePointer fp, bool randSample)
{
    // recursive face subdivision.
    if(sampleNum == 1)
    {
        // ground case.
        CoordType SamplePoint;
        if(randSample)
        {
            CoordType rb=RandomBarycentric();
            SamplePoint=v0*rb[0]+v1*rb[1]+v2*rb[2];
        }
        else SamplePoint=((v0+v1+v2)*(1.0f/3.0f));

        CoordType SampleBary;
        InterpolationParameters(*fp,SamplePoint,SampleBary);
        ps.AddFace(*fp,SampleBary);
        return 1;
    }

    int s0 = sampleNum /2;
    int s1 = sampleNum-s0;
    assert(s0>0);
    assert(s1>0);

    ScalarType w0 = ScalarType(s1)/ScalarType(sampleNum);
    ScalarType w1 = 1.0-w0;
    // compute the longest edge.
    ScalarType  maxd01 = SquaredDistance(v0,v1);
    ScalarType  maxd12 = SquaredDistance(v1,v2);
    ScalarType  maxd20 = SquaredDistance(v2,v0);
    int     res;
    if(maxd01 > maxd12)
        if(maxd01 > maxd20)     res = 0;
    else                    res = 2;
    else
        if(maxd12 > maxd20)     res = 1;
    else                    res = 2;

    int faceSampleNum=0;
    // break the input triangle along the midpoint of the longest edge.
    CoordType  pp;
    switch(res)
    {
    case 0 :    pp = v0*w0 + v1*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,pp,v2,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,pp,v1,v2,ps,fp,randSample);
        break;
    case 1 :    pp =  v1*w0 + v2*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,v1,pp,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,v0,pp,v2,ps,fp,randSample);
        break;
    case 2 :    pp = v0*w0 + v2*w1;
        faceSampleNum+=SingleFaceSubdivision(s0,v0,v1,pp,ps,fp,randSample);
        faceSampleNum+=SingleFaceSubdivision(s1,pp,v1,v2,ps,fp,randSample);
        break;
    }
    return faceSampleNum;
}


/// Compute a sampling of the surface where the points are regularly scattered over the face surface using a recursive longest-edge subdivision rule.
static void FaceSubdivisionOld(MetroMesh & m, VertexSampler &ps,int sampleNum, bool randSample)
{

    ScalarType area = Stat<MetroMesh>::ComputeMeshArea(m);
    ScalarType samplePerAreaUnit = sampleNum/area;
    std::vector<FacePointer> faceVec;
    FillAndShuffleFacePointerVector(m,faceVec);
    tri::UpdateNormal<MetroMesh>::PerFaceNormalized(m);
    double  floatSampleNum = 0.0;
    int faceSampleNum;
    // Subdivision sampling.
    typename std::vector<FacePointer>::iterator fi;
    for(fi=faceVec.begin(); fi!=faceVec.end(); fi++)
    {
        // compute # samples in the current face.
        floatSampleNum += 0.5*DoubleArea(**fi) * samplePerAreaUnit;
        faceSampleNum          = (int) floatSampleNum;
        if(faceSampleNum>0)
            faceSampleNum = SingleFaceSubdivision(faceSampleNum,(**fi).V(0)->cP(), (**fi).V(1)->cP(), (**fi).V(2)->cP(),ps,*fi,randSample);
        floatSampleNum -= (double) faceSampleNum;
    }
}


//---------

// Similar Triangles sampling.
// Skip vertex and edges
// Sample per edges includes vertexes, so here we should expect  n_samples_per_edge >=4

static int SingleFaceSimilar(FacePointer fp, VertexSampler &ps, int n_samples_per_edge)
{
        int n_samples=0;
    int         i, j;
    float segmentNum=n_samples_per_edge -1 ;
        float segmentLen = 1.0/segmentNum;
        // face sampling.
    for(i=1; i < n_samples_per_edge-1; i++)
        for(j=1; j < n_samples_per_edge-1-i; j++)
        {
            //AddSample( v0 + (V1*(double)i + V2*(double)j) );
                        CoordType sampleBary(i*segmentLen,j*segmentLen, 1.0 - (i*segmentLen+j*segmentLen) ) ;
            n_samples++;
                        ps.AddFace(*fp,sampleBary);
        }
    return n_samples;
}
static int SingleFaceSimilarDual(FacePointer fp, VertexSampler &ps, int n_samples_per_edge, bool randomFlag)
{
        int n_samples=0;
    float         i, j;
    float segmentNum=n_samples_per_edge -1 ;
        float segmentLen = 1.0/segmentNum;
        // face sampling.
    for(i=0; i < n_samples_per_edge-1; i++)
        for(j=0; j < n_samples_per_edge-1-i; j++)
        {
            //AddSample( v0 + (V1*(double)i + V2*(double)j) );
                        CoordType V0((i+0)*segmentLen,(j+0)*segmentLen, 1.0 - ((i+0)*segmentLen+(j+0)*segmentLen) ) ;
                        CoordType V1((i+1)*segmentLen,(j+0)*segmentLen, 1.0 - ((i+1)*segmentLen+(j+0)*segmentLen) ) ;
                        CoordType V2((i+0)*segmentLen,(j+1)*segmentLen, 1.0 - ((i+0)*segmentLen+(j+1)*segmentLen) ) ;
                        n_samples++;
                        if(randomFlag) 	{
                                            CoordType rb=RandomBarycentric();
                                            ps.AddFace(*fp, V0*rb[0]+V1*rb[1]+V2*rb[2]);
                            } else  ps.AddFace(*fp,(V0+V1+V2)/3.0);

				if( j < n_samples_per_edge-i-2 )
							{
									CoordType V3((i+1)*segmentLen,(j+1)*segmentLen, 1.0 - ((i+1)*segmentLen+(j+1)*segmentLen) ) ;
									n_samples++;
									if(randomFlag) 	{
														CoordType rb=RandomBarycentric();
														ps.AddFace(*fp, V3*rb[0]+V1*rb[1]+V2*rb[2]);
										} else  ps.AddFace(*fp,(V3+V1+V2)/3.0);
							}
		}
	return n_samples;
}

// Similar sampling
// Each triangle is subdivided into similar triangles following a generalization of the classical 1-to-4 splitting rule of triangles.
// According to the level of subdivision <k> you get 1, 4 , 9, 16 , <k^2> triangles.
// Depending on the kind of the sampling strategies we can have two different approach to choosing the sample points.
// 1) you have already sampled both edges and vertices
// 2) you are not going to take samples on edges and vertices.
//
// In the first case you have to consider only internal vertices of the subdivided triangles (to avoid multiple sampling of edges and vertices).
// Therefore the number of internal points is ((k-3)*(k-2))/2. where k is the number of points on an edge (vertex included)
// E.g. for k=4 you get 3 segments on each edges and the original triangle is subdivided
// into 9 smaller triangles and you get (1*2)/2 == 1 only a single internal point.
// So if you want N samples in a triangle you have to solve  k^2 -5k +6 - 2N = 0
// from which you get:
//
//      5 + sqrt( 1 + 8N )
// k = -------------------
//             2
//
// In the second case if you are not interested to skip the sampling on edges and vertices you have to consider as sample number the number of triangles.
// So if you want N samples in a triangle, the number <k> of points on  an edge (vertex included) should be simply:
//      k = 1 + sqrt(N)
// examples:
// N = 4 -> k = 3
// N = 9 -> k = 4



//template <class MetroMesh>
//void Sampling<MetroMesh>::SimilarFaceSampling()
static void FaceSimilar(MetroMesh & m, VertexSampler &ps,int sampleNum, bool dualFlag, bool randomFlag)
{
		ScalarType area = Stat<MetroMesh>::ComputeMeshArea(m);
		ScalarType samplePerAreaUnit = sampleNum/area;

        // Similar Triangles sampling.
    int n_samples_per_edge;
    double  n_samples_decimal = 0.0;
    FaceIterator fi;

    for(fi=m.face.begin(); fi != m.face.end(); fi++)
    {
        // compute # samples in the current face.
        n_samples_decimal += 0.5*DoubleArea(*fi) * samplePerAreaUnit;
        int n_samples          = (int) n_samples_decimal;
        if(n_samples>0)
        {
            // face sampling.
            if(dualFlag)
                            {
                                    n_samples_per_edge = (int)((sqrt(1.0+8.0*(double)n_samples) +5.0)/2.0); // original for non dual case
                                    n_samples = SingleFaceSimilar(&*fi,ps, n_samples_per_edge);
                            } else {
                                    n_samples_per_edge = (int)(sqrt((double)n_samples) +1.0);
                                    n_samples = SingleFaceSimilarDual(&*fi,ps, n_samples_per_edge,randomFlag);
                        }
        }
        n_samples_decimal -= (double) n_samples;
    }
}


	// Rasterization fuction
	// Take a triangle
	// T deve essere una classe funzionale che ha l'operatore ()
	// con due parametri x,y di tipo S esempio:
	// class Foo { public void operator()(int x, int y ) { ??? } };

// This function does rasterization with a safety buffer area, thus accounting some points actually outside triangle area
// The safety area samples are generated according to face flag BORDER which should be true for texture space border edges
// Use correctSafePointsBaryCoords = true to map safety texels to closest point barycentric coords (on edge).
    static void SingleFaceRaster(typename MetroMesh::FaceType &f,  VertexSampler &ps,
                            const Point2<typename MetroMesh::ScalarType> & v0,
                            const Point2<typename MetroMesh::ScalarType> & v1,
                            const Point2<typename MetroMesh::ScalarType> & v2,
                            bool correctSafePointsBaryCoords=true)
    {
    typedef typename MetroMesh::ScalarType S;
    // Calcolo bounding box
    Box2i bbox;
    Box2<S> bboxf;
    bboxf.Add(v0);
    bboxf.Add(v1);
    bboxf.Add(v2);

	bbox.min[0] = floor(bboxf.min[0]);
	bbox.min[1] = floor(bboxf.min[1]);
	bbox.max[0] = ceil(bboxf.max[0]);
	bbox.max[1] = ceil(bboxf.max[1]);

    // Calcolo versori degli spigoli
    Point2<S> d10 = v1 - v0;
    Point2<S> d21 = v2 - v1;
    Point2<S> d02 = v0 - v2;

    // Preparazione prodotti scalari
    S b0  = (bbox.min[0]-v0[0])*d10[1] - (bbox.min[1]-v0[1])*d10[0];
    S b1  = (bbox.min[0]-v1[0])*d21[1] - (bbox.min[1]-v1[1])*d21[0];
    S b2  = (bbox.min[0]-v2[0])*d02[1] - (bbox.min[1]-v2[1])*d02[0];
    // Preparazione degli steps
    S db0 = d10[1];
    S db1 = d21[1];
    S db2 = d02[1];
    // Preparazione segni
    S dn0 = -d10[0];
    S dn1 = -d21[0];
    S dn2 = -d02[0];

    //Calculating orientation
    bool flipped = !(d02 * vcg::Point2<S>(-d10[1], d10[0]) >= 0);

    // Calculating border edges
    Segment2<S> borderEdges[3];
    S edgeLength[3];
    unsigned char edgeMask = 0;

    if (f.IsB(0)) {
        borderEdges[0] = Segment2<S>(v0, v1);
        edgeLength[0] = borderEdges[0].Length();
        edgeMask |= 1;
    }
    if (f.IsB(1)) {
        borderEdges[1] = Segment2<S>(v1, v2);
        edgeLength[1] = borderEdges[1].Length();
        edgeMask |= 2;
    }
    if (f.IsB(2)) {
        borderEdges[2] = Segment2<S>(v2, v0);
        edgeLength[2] = borderEdges[2].Length();
        edgeMask |= 4;
    }

    // Rasterizzazione
    double de = v0[0]*v1[1]-v0[0]*v2[1]-v1[0]*v0[1]+v1[0]*v2[1]-v2[0]*v1[1]+v2[0]*v0[1];

    for(int x=bbox.min[0]-1;x<=bbox.max[0]+1;++x)
    {
        bool in = false;
        S n[3]  = { b0-db0-dn0, b1-db1-dn1, b2-db2-dn2};
        for(int y=bbox.min[1]-1;y<=bbox.max[1]+1;++y)
        {
            if( ((n[0]>=0 && n[1]>=0 && n[2]>=0) || (n[0]<=0 && n[1]<=0 && n[2]<=0))  && (de != 0))
            {
                typename MetroMesh::CoordType baryCoord;
                baryCoord[0] =  double(-y*v1[0]+v2[0]*y+v1[1]*x-v2[0]*v1[1]+v1[0]*v2[1]-x*v2[1])/de;
                baryCoord[1] = -double( x*v0[1]-x*v2[1]-v0[0]*y+v0[0]*v2[1]-v2[0]*v0[1]+v2[0]*y)/de;
                baryCoord[2] = 1-baryCoord[0]-baryCoord[1];

                ps.AddTextureSample(f, baryCoord, Point2i(x,y), 0);
                in = true;
            } else {
                // Check whether a pixel outside (on a border edge side) triangle affects color inside it
                Point2<S> px(x, y);
                Point2<S> closePoint;
                int closeEdge = -1;
                S minDst = FLT_MAX;

                // find the closest point (on some edge) that lies on the 2x2 squared neighborhood of the considered point
                for (int i=0; i<3; ++i)
                {
                    if (edgeMask & (1 << i))
                    {
                        Point2<S> close;
                        S dst;
                        if ( ((!flipped) && (n[i]<0)) ||
                             (  flipped  && (n[i]>0))   )
                        {
                            dst = ((close = ClosestPoint(borderEdges[i], px)) - px).Norm();
                            if(dst < minDst &&
                               close.X() > px.X()-1 && close.X() < px.X()+1 &&
                               close.Y() > px.Y()-1 && close.Y() < px.Y()+1)
                            {
                                minDst = dst;
                                closePoint = close;
                                closeEdge = i;
                            }
                        }
                    }
                }

                if (closeEdge >= 0)
                {
                    typename MetroMesh::CoordType baryCoord;
                    if (correctSafePointsBaryCoords)
                    {
                        // Add x,y sample with closePoint barycentric coords (on edge)
                        baryCoord[closeEdge] = (closePoint - borderEdges[closeEdge].P(1)).Norm()/edgeLength[closeEdge];
                        baryCoord[(closeEdge+1)%3] = 1 - baryCoord[closeEdge];
                        baryCoord[(closeEdge+2)%3] = 0;
                    } else {
                        // Add x,y sample with his own barycentric coords (off edge)
                        baryCoord[0] =  double(-y*v1[0]+v2[0]*y+v1[1]*x-v2[0]*v1[1]+v1[0]*v2[1]-x*v2[1])/de;
                        baryCoord[1] = -double( x*v0[1]-x*v2[1]-v0[0]*y+v0[0]*v2[1]-v2[0]*v0[1]+v2[0]*y)/de;
                        baryCoord[2] = 1-baryCoord[0]-baryCoord[1];
                    }
                    ps.AddTextureSample(f, baryCoord, Point2i(x,y), minDst);
                    in = true;
                }
            }
            n[0] += dn0;
            n[1] += dn1;
            n[2] += dn2;
        }
        b0 += db0;
        b1 += db1;
        b2 += db2;
    }
}

// check the radius constrain
static bool checkPoissonDisk(SampleSHT & sht, const Point3<ScalarType> & p, ScalarType radius)
{
	// get the samples closest to the given one
	std::vector<VertexType*> closests;
  typedef VertTmark<MetroMesh> MarkerVert;
  static MarkerVert mv;

	Box3f bb(p-Point3f(radius,radius,radius),p+Point3f(radius,radius,radius));
	GridGetInBox(sht, mv, bb, closests);

  ScalarType r2 = radius*radius;
    for(int i=0; i<closests.size(); ++i)
        if(SquaredDistance(p,closests[i]->cP()) < r2)
            return false;

    return true;
}

struct PoissonDiskParam
{
  PoissonDiskParam()
  {
    adaptiveRadiusFlag = false;
    bestSampleChoiceFlag = true;
    bestSamplePoolSize = 10;
    radiusVariance =1;
    MAXLEVELS = 5;
    invertQuality = false;
    preGenFlag = false;
    preGenMesh = NULL;
    geodesicDistanceFlag = false;
    pds=NULL;
  }

  struct Stat
  {
    int montecarloTime;
    int gridTime;
    int pruneTime;
    int totalTime;
    Point3i gridSize;
    int gridCellNum;
    int sampleNum;
    int montecarloSampleNum;
  };

  bool geodesicDistanceFlag;
  bool bestSampleChoiceFlag; // In poisson disk pruning when we choose a sample in a cell, we choose the sample that remove the minimal number of other samples. This previlege the "on boundary" samples.
  int bestSamplePoolSize;
  bool adaptiveRadiusFlag;
  float radiusVariance;
  bool invertQuality;
  bool preGenFlag;        // when generating a poisson distribution, you can initialize the set of computed points with ALL the vertices of another mesh. Useful for building progressive refinements.
  MetroMesh *preGenMesh;
  int MAXLEVELS;

  Stat *pds;
};


// generate Poisson-disk sample using a set of pre-generated samples (with the Montecarlo algorithm)
// It always return a point.
static VertexPointer getSampleFromCell(Point3i &cell, MontecarloSHT & samplepool)
{
	MontecarloSHTIterator cellBegin, cellEnd;
	samplepool.Grid(cell, cellBegin, cellEnd);
	return *cellBegin;
}

// Given a cell of the grid it search the point that remove the minimum number of other samples
// it linearly scan all the points of a cell.

static VertexPointer getBestPrecomputedMontecarloSample(Point3i &cell, MontecarloSHT & samplepool, ScalarType diskRadius, const PoissonDiskParam &pp)
{
  MontecarloSHTIterator cellBegin,cellEnd;
  samplepool.Grid(cell, cellBegin, cellEnd);
  VertexPointer bestSample=0;
  int minRemoveCnt = std::numeric_limits<int>::max();
  std::vector<typename MontecarloSHT::HashIterator> inSphVec;
  int i=0;
  for(MontecarloSHTIterator ci=cellBegin; ci!=cellEnd && i<pp.bestSamplePoolSize; ++ci,i++)
  {
    VertexPointer sp = *ci;
    if(pp.adaptiveRadiusFlag)  diskRadius = sp->Q();
    int curRemoveCnt = samplepool.CountInSphere(sp->cP(),diskRadius,inSphVec);
    if(curRemoveCnt < minRemoveCnt)
    {
      bestSample = sp;
      minRemoveCnt = curRemoveCnt;
    }
  }
  return bestSample;
}


static ScalarType ComputePoissonDiskRadius(MetroMesh &origMesh, int sampleNum)
{
	ScalarType meshArea = Stat<MetroMesh>::ComputeMeshArea(origMesh);
	// Manage approximately the PointCloud Case, use the half a area of the bbox.
	// TODO: If you had the radius a much better approximation could be done.
	if(meshArea ==0)
		{
					meshArea = (origMesh.bbox.DimX()*origMesh.bbox.DimY() +
											origMesh.bbox.DimX()*origMesh.bbox.DimZ() +
											origMesh.bbox.DimY()*origMesh.bbox.DimZ());
		}
	ScalarType diskRadius = sqrt(meshArea / (0.7 * M_PI * sampleNum)); // 0.7 is a density factor
	return diskRadius;
}

static int ComputePoissonSampleNum(MetroMesh &origMesh, ScalarType diskRadius)
{
	ScalarType meshArea = Stat<MetroMesh>::ComputeMeshArea(origMesh);
	int sampleNum = meshArea /  (diskRadius*diskRadius *M_PI *0.7)  ; // 0.7 is a density factor
	return sampleNum;
}

// Note that this function actually CHANGE the quality of the montecarlo samples so that it represents the expected radius of each sample
// the expected radius of the sample is computed so that it linearly maps the quality between diskradius / variance and diskradius*variance

static void ComputePoissonSampleRadii(MetroMesh &sampleMesh, ScalarType diskRadius, ScalarType radiusVariance, bool invert)
{
	VertexIterator vi;
	std::pair<float,float> minmax = tri::Stat<MetroMesh>::ComputePerVertexQualityMinMax( sampleMesh);
	float minRad = diskRadius / radiusVariance;
	float maxRad = diskRadius * radiusVariance;
	float deltaQ = minmax.second-minmax.first;
	float deltaRad = maxRad-minRad;
	for (vi = sampleMesh.vert.begin(); vi != sampleMesh.vert.end(); vi++)
	{
	 (*vi).Q() = minRad + deltaRad*((invert ? minmax.second - (*vi).Q() : (*vi).Q() - minmax.first )/deltaQ);
	}
}

// initialize spatial hash table for searching
// radius is the radius of empty disk centered over the samples (e.g. twice of the empty space disk)
// This radius implies that when we pick a sample in a cell all that cell probably will not be touched again.
// Howvever we must ensure that we do not put too many vertices inside each hash cell

static void InitSpatialHashTable(MetroMesh &montecarloMesh, MontecarloSHT &montecarloSHT, ScalarType diskRadius,
                                 const struct PoissonDiskParam pp=PoissonDiskParam())
{
  ScalarType cellsize = 2.0f* diskRadius / sqrt(3.0);
  float occupancyRatio=0;
  do
  {
    // inflating
    BoxType bb=montecarloMesh.bbox;
    assert(!bb.IsNull());
    bb.Offset(cellsize);

    int sizeX = std::max(1.0f,bb.DimX() / cellsize);
    int sizeY = std::max(1.0f,bb.DimY() / cellsize);
    int sizeZ = std::max(1.0f,bb.DimZ() / cellsize);
    Point3i gridsize(sizeX, sizeY, sizeZ);

    montecarloSHT.InitEmpty(bb, gridsize);

    for (VertexIterator vi = montecarloMesh.vert.begin(); vi != montecarloMesh.vert.end(); vi++)
      montecarloSHT.Add(&(*vi));
    montecarloSHT.UpdateAllocatedCells();
    if(pp.pds)
    {
      pp.pds->gridSize = gridsize;
      pp.pds->gridCellNum = (int)montecarloSHT.AllocatedCells.size();
    }
    cellsize/=2.0f;
    occupancyRatio = float(montecarloMesh.vn) / float(montecarloSHT.AllocatedCells.size());
//    qDebug(" %i / %i = %6.3f", montecarloMesh.vn , montecarloSHT.AllocatedCells.size(),occupancyRatio);
  }
  while( occupancyRatio> 100);
}

// Trivial approach that puts all the samples in a UG and removes all the ones that surely do not fit the
static void PoissonDiskPruning(VertexSampler &ps, MetroMesh &montecarloMesh,
                               ScalarType diskRadius, const struct PoissonDiskParam pp=PoissonDiskParam())
{
    int t0 = clock();
    // spatial index of montecarlo samples - used to choose a new sample to insert
    MontecarloSHT montecarloSHT;
    InitSpatialHashTable(montecarloMesh,montecarloSHT,diskRadius,pp);

    // if we are doing variable density sampling we have to prepare the random samples quality with the correct expected radii.
    // At this point we just assume that there is the quality values as sampled from the base mesh
    if(pp.adaptiveRadiusFlag)
        ComputePoissonSampleRadii(montecarloMesh, diskRadius, pp.radiusVariance, pp.invertQuality);

    unsigned int (*p_myrandom)(unsigned int) = RandomInt;
    std::random_shuffle(montecarloSHT.AllocatedCells.begin(),montecarloSHT.AllocatedCells.end(), p_myrandom);
    int t1 = clock();
    if(pp.pds) {
      pp.pds->montecarloSampleNum = montecarloMesh.vn;
    }
    int removedCnt=0;
    if(pp.preGenFlag && pp.preGenMesh !=0)
    {
      // Initial pass for pruning the Hashed grid with the an eventual pre initialized set of samples
      for(VertexIterator vi =pp.preGenMesh->vert.begin(); vi!=pp.preGenMesh->vert.end();++vi)
      {
        ps.AddVert(*vi);
        removedCnt += montecarloSHT.RemoveInSphere(vi->cP(),diskRadius);
      }
      montecarloSHT.UpdateAllocatedCells();
    }
    vertex::ApproximateGeodesicDistanceFunctor<VertexType> GDF;
    while(!montecarloSHT.AllocatedCells.empty())
    {
        removedCnt=0;
        for (size_t i = 0; i < montecarloSHT.AllocatedCells.size(); i++)
        {
            if( montecarloSHT.EmptyCell(montecarloSHT.AllocatedCells[i])  ) continue;
            ScalarType currentRadius =diskRadius;
            VertexPointer sp;
            if(pp.bestSampleChoiceFlag)
              sp = getBestPrecomputedMontecarloSample(montecarloSHT.AllocatedCells[i], montecarloSHT, diskRadius, pp);
            else
              sp = getSampleFromCell(montecarloSHT.AllocatedCells[i], montecarloSHT);

            if(pp.adaptiveRadiusFlag)
              currentRadius = sp->Q();

            ps.AddVert(*sp);
            if(pp.geodesicDistanceFlag) removedCnt += montecarloSHT.RemoveInSphereNormal(sp->cP(),sp->cN(),GDF,currentRadius);
                            else        removedCnt += montecarloSHT.RemoveInSphere(sp->cP(),currentRadius);
        }
        montecarloSHT.UpdateAllocatedCells();
    }
    int t2 = clock();
    if(pp.pds)
    {
      pp.pds->gridTime = t1-t0;
      pp.pds->pruneTime = t2-t1;
    }
}

/** Compute a Poisson-disk sampling of the surface.
 *  The radius of the disk is computed according to the estimated sampling density.
 *
 * This algorithm is an adaptation of the algorithm of White et al. :
 *
 * "Poisson Disk Point Set by Hierarchical Dart Throwing"
 * K. B. White, D. Cline, P. K. Egbert,
 * IEEE Symposium on Interactive Ray Tracing, 2007,
 * 10-12 Sept. 2007, pp. 129-132.
 */
static void HierarchicalPoissonDisk(MetroMesh &origMesh, VertexSampler &ps, MetroMesh &montecarloMesh, ScalarType diskRadius, const struct PoissonDiskParam pp=PoissonDiskParam())
{
//  int t0=clock();
    // spatial index of montecarlo samples - used to choose a new sample to insert
    MontecarloSHT montecarloSHTVec[5];



    // initialize spatial hash table for searching
    // radius is the radius of empty disk centered over the samples (e.g. twice of the empty space disk)
    // This radius implies that when we pick a sample in a cell all that cell will not be touched again.
    ScalarType cellsize = 2.0f* diskRadius / sqrt(3.0);

	// inflating
	BoxType bb=origMesh.bbox;
	bb.Offset(cellsize);

  int sizeX = std::max(1.0f,bb.DimX() / cellsize);
  int sizeY = std::max(1.0f,bb.DimY() / cellsize);
  int sizeZ = std::max(1.0f,bb.DimZ() / cellsize);
    Point3i gridsize(sizeX, sizeY, sizeZ);

    // spatial hash table of the generated samples - used to check the radius constrain
    SampleSHT checkSHT;
    checkSHT.InitEmpty(bb, gridsize);


	// sampling algorithm
	// ------------------
	//
	// - generate millions of samples using montecarlo algorithm
	// - extract a cell (C) from the active cell list (with probability proportional to cell's volume)
	// - generate a sample inside C by choosing one of the contained pre-generated samples
	//   - if the sample violates the radius constrain discard it, and add the cell to the cells-to-subdivide list
	// - iterate until the active cell list is empty or a pre-defined number of subdivisions is reached
	//

	int level = 0;

    // initialize spatial hash to index pre-generated samples
    montecarloSHTVec[0].InitEmpty(bb, gridsize);
    // create active cell list
    for (VertexIterator vi = montecarloMesh.vert.begin(); vi != montecarloMesh.vert.end(); vi++)
        montecarloSHTVec[0].Add(&(*vi));
    montecarloSHTVec[0].UpdateAllocatedCells();

  // if we are doing variable density sampling we have to prepare the random samples quality with the correct expected radii.
    if(pp.adaptiveRadiusFlag)
            ComputePoissonSampleRadii(montecarloMesh, diskRadius, pp.radiusVariance, pp.invertQuality);

	do
	{
		MontecarloSHT &montecarloSHT = montecarloSHTVec[level];

        if(level>0)
        {// initialize spatial hash with the remaining points
            montecarloSHT.InitEmpty(bb, gridsize);
            // create active cell list
            for (typename MontecarloSHT::HashIterator hi = montecarloSHTVec[level-1].hash_table.begin(); hi != montecarloSHTVec[level-1].hash_table.end(); hi++)
                montecarloSHT.Add((*hi).second);
            montecarloSHT.UpdateAllocatedCells();
        }
        // shuffle active cells
        unsigned int (*p_myrandom)(unsigned int) = RandomInt;
        std::random_shuffle(montecarloSHT.AllocatedCells.begin(),montecarloSHT.AllocatedCells.end(), p_myrandom);

		// generate a sample inside C by choosing one of the contained pre-generated samples
		//////////////////////////////////////////////////////////////////////////////////////////
	int removedCnt=montecarloSHT.hash_table.size();
	int addedCnt=checkSHT.hash_table.size();
		for (int i = 0; i < montecarloSHT.AllocatedCells.size(); i++)
		{
			for(int j=0;j<4;j++)
			{
				if( montecarloSHT.EmptyCell(montecarloSHT.AllocatedCells[i])  ) continue;

            // generate a sample chosen from the pre-generated one
            typename MontecarloSHT::HashIterator hi = montecarloSHT.hash_table.find(montecarloSHT.AllocatedCells[i]);

            if(hi==montecarloSHT.hash_table.end()) {break;}
            VertexPointer sp = (*hi).second;
            // vr spans between 3.0*r and r / 4.0 according to vertex quality
            ScalarType sampleRadius = diskRadius;
            if(pp.adaptiveRadiusFlag)  sampleRadius = sp->Q();
            if (checkPoissonDisk(checkSHT, sp->cP(), sampleRadius))
            {
               ps.AddVert(*sp);
               montecarloSHT.RemoveCell(sp);
               checkSHT.Add(sp);
               break;
            }
            else
                montecarloSHT.RemovePunctual(sp);
        }
        }
        addedCnt = checkSHT.hash_table.size()-addedCnt;
        removedCnt = removedCnt-montecarloSHT.hash_table.size();

        // proceed to the next level of subdivision
        // increase grid resolution
        gridsize *= 2;

		//
		level++;
	} while(level < 5);
}

//template <class MetroMesh>
//void Sampling<MetroMesh>::SimilarFaceSampling()

// This function also generates samples outside faces if those affects faces in texture space.
// Use correctSafePointsBaryCoords = true to map safety texels to closest point barycentric coords (on edge)
// otherwise obtained samples will map to barycentric coord actually outside face
//
// If you don't need to get those extra points clear faces Border Flags
// vcg::tri::UpdateFlags<Mesh>::FaceClearB(m);
//
// Else make sure to update border flags from texture space FFadj
// vcg::tri::UpdateTopology<Mesh>::FaceFaceFromTexCoord(m);
// vcg::tri::UpdateFlags<Mesh>::FaceBorderFromFF(m);
static void Texture(MetroMesh & m, VertexSampler &ps, int textureWidth, int textureHeight, bool correctSafePointsBaryCoords=true)
{
		FaceIterator fi;

		printf("Similar Triangles face sampling\n");
		for(fi=m.face.begin(); fi != m.face.end(); fi++)
			if (!fi->IsD())
			{
				Point2f ti[3];
				for(int i=0;i<3;++i)
					ti[i]=Point2f((*fi).WT(i).U() * textureWidth - 0.5, (*fi).WT(i).V() * textureHeight - 0.5);
					// - 0.5 constants are used to obtain correct texture mapping

                SingleFaceRaster(*fi,  ps, ti[0],ti[1],ti[2], correctSafePointsBaryCoords);
            }
}

typedef GridStaticPtr<FaceType, ScalarType > TriMeshGrid;

class RRParam
{
public:
float offset;
float minDiag;
tri::FaceTmark<MetroMesh> markerFunctor;
TriMeshGrid gM;
};

static void RegularRecursiveOffset(MetroMesh & m, std::vector<Point3f> &pvec, ScalarType offset, float minDiag)
{
	Box3<ScalarType> bb=m.bbox;
	bb.Offset(offset*2.0);

	RRParam rrp;

	rrp.markerFunctor.SetMesh(&m);

	rrp.gM.Set(m.face.begin(),m.face.end(),bb);


	rrp.offset=offset;
	rrp.minDiag=minDiag;
	SubdivideAndSample(m, pvec, bb, rrp, bb.Diag());
}

static void SubdivideAndSample(MetroMesh & m, std::vector<Point3f> &pvec, const Box3<ScalarType> bb, RRParam &rrp, float curDiag)
{
	Point3f startPt = bb.Center();

	ScalarType dist;
	// Compute mesh point nearest to bb center
	FaceType   *nearestF=0;
	float dist_upper_bound = curDiag+rrp.offset;
	Point3f closestPt;
	vcg::face::PointDistanceBaseFunctor<ScalarType> PDistFunct;
	dist=dist_upper_bound;
	nearestF =  rrp.gM.GetClosest(PDistFunct,rrp.markerFunctor,startPt,dist_upper_bound,dist,closestPt);
  curDiag /=2;
    if(dist < dist_upper_bound)
        {
            if(curDiag/3 < rrp.minDiag) //store points only for the last level of recursion (?)
                {
                    if(rrp.offset==0)
                            pvec.push_back(closestPt);
                    else
                        {
                            if(dist>rrp.offset) // points below the offset threshold cannot be displaced at the right offset distance, we can only make points nearer.
                            {
                                Point3f delta = startPt-closestPt;
                                pvec.push_back(closestPt+delta*(rrp.offset/dist));
                            }
                        }
                }
            if(curDiag < rrp.minDiag) return;
            Point3f hs = (bb.max-bb.min)/2;
            for(int i=0;i<2;i++)
                for(int j=0;j<2;j++)
                    for(int k=0;k<2;k++)
                        SubdivideAndSample(m,pvec,
                                                                            Box3f(Point3f( bb.min[0]+i*hs[0], bb.min[1]+j*hs[1], bb.min[2]+k*hs[2]),
                                                                                        Point3f(startPt[0]+i*hs[0],startPt[1]+j*hs[1],startPt[2]+k*hs[2])),rrp,curDiag);

        }
}
}; // end class


template <class MeshType>
void MontecarloSampling(MeshType &m, // the mesh that has to be sampled
                     MeshType &mm, // the mesh that will contain the samples
                     int sampleNum) // the desired number sample, if zero you must set the radius to the wanted value
{
  typedef tri::MeshSampler<MeshType> BaseSampler;
  MeshSampler<MeshType> mcSampler(&mm);
  tri::SurfaceSampling<MeshType,BaseSampler>::Montecarlo(m, mcSampler, sampleNum);
}


template <class MeshType>
void MontecarloSampling(MeshType &m, // the mesh that has to be sampled
                     std::vector<Point3f> &montercarloSamples, // the vector that will contain the set of points
                     int sampleNum) // the desired number sample, if zero you must set the radius to the wanted value
{
  typedef tri::TrivialSampler<MeshType> BaseSampler;
  BaseSampler mcSampler(montercarloSamples);
  tri::SurfaceSampling<MeshType,BaseSampler>::Montecarlo(m, mcSampler, sampleNum);
}

// Yet another simpler wrapper for the generation of a poisson disk distribution over a mesh.
//
template <class MeshType>
void PoissonSampling(MeshType &m, // the mesh that has to be sampled
                     std::vector<Point3f> &poissonSamples, // the vector that will contain the set of points
                     int sampleNum, // the desired number sample, if zero you must set the radius to the wanted value
                     float &radius,  // the Poisson Disk Radius (used if sampleNum==0, setted if sampleNum!=0)
                     float radiusVariance=1)

{
  typedef tri::TrivialSampler<MeshType> BaseSampler;
  typedef tri::MeshSampler<MeshType> MontecarloSampler;

  typename tri::SurfaceSampling<MeshType, BaseSampler>::PoissonDiskParam pp;
  typename tri::SurfaceSampling<MeshType, BaseSampler>::PoissonDiskParam::Stat stat;
  pp.pds = &stat;
  int t0=clock();

  if(sampleNum>0) radius = tri::SurfaceSampling<MeshType,BaseSampler>::ComputePoissonDiskRadius(m,sampleNum);
  if(radius>0 && sampleNum==0) sampleNum = tri::SurfaceSampling<MeshType,BaseSampler>::ComputePoissonSampleNum(m,radius);

  pp.pds->sampleNum = sampleNum;
  poissonSamples.clear();
//  std::vector<Point3f> MontecarloSamples;
  MeshType MontecarloMesh;

  // First step build the sampling
  MontecarloSampler mcSampler(MontecarloMesh);
  BaseSampler pdSampler(poissonSamples);

  tri::SurfaceSampling<MeshType,MontecarloSampler>::Montecarlo(m, mcSampler, std::max(10000,sampleNum*40));
  tri::UpdateBounding<MeshType>::Box(MontecarloMesh);
//  tri::Build(MontecarloMesh, MontecarloSamples);
  int t1=clock();
  pp.pds->montecarloTime = t1-t0;

  if(radiusVariance !=1)
  {
    pp.adaptiveRadiusFlag=true;
    pp.radiusVariance=radiusVariance;
  }
  tri::SurfaceSampling<MeshType,BaseSampler>::PoissonDiskPruning(pdSampler, MontecarloMesh, radius,pp);
  int t2=clock();
  pp.pds->totalTime = t2-t0;
}
// simpler wrapper for the pruning
//
template <class MeshType>
void PoissonPruning(MeshType &m, // the mesh that has to be pruned
                    std::vector<Point3f> &poissonSamples, // the vector that will contain the chosen set of points
                    float & radius)
{
    typedef tri::TrivialSampler<MeshType> BaseSampler;
    typename tri::SurfaceSampling<MeshType, BaseSampler>::PoissonDiskParam pp;
    typename tri::SurfaceSampling<MeshType, BaseSampler>::PoissonDiskParam::Stat stat;
    pp.pds = &stat;
    tri::UpdateBounding<MeshType>::Box(m);
    BaseSampler pdSampler(poissonSamples);
    tri::SurfaceSampling<MeshType,BaseSampler>::PoissonDiskPruning(pdSampler, m, radius,pp);
}
} // end namespace tri
} // end namespace vcg

#endif