vcglib/vcg/complex/algorithms/smooth.h

/****************************************************************************
* VCGLib                                                            o o     *
* Visual and Computer Graphics Library                            o     o   *
*                                                                _   O  _   *
* Copyright(C) 2004-2016                                           \/)\/    *
* Visual Computing Lab                                            /\/|      *
* ISTI - Italian National Research Council                           |      *
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* All rights reserved.                                                      *
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* This program is free software; you can redistribute it and/or modify      *
* it under the terms of the GNU General Public License as published by      *
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* (at your option) any later version.                                       *
*                                                                           *
* This program is distributed in the hope that it will be useful,           *
* but WITHOUT ANY WARRANTY; without even the implied warranty of            *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the             *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt)          *
* for more details.                                                         *
*                                                                           *
****************************************************************************/

#ifndef __VCGLIB__SMOOTH
#define __VCGLIB__SMOOTH

#include <vcg/space/ray3.h>
#include <vcg/complex/algorithms/update/normal.h>
#include <vcg/complex/algorithms/update/halfedge_topology.h>
#include <vcg/complex/algorithms/closest.h>
#include <vcg/space/index/kdtree/kdtree.h>

namespace vcg
{
namespace tri
{
///
/** \addtogroup trimesh */
/*@{*/
/// Class of static functions to smooth and fair meshes and their attributes.

template <class SmoothMeshType>
class Smooth
{

  public:
    typedef SmoothMeshType MeshType;
    typedef typename MeshType::VertexType VertexType;
    typedef typename MeshType::VertexType::CoordType CoordType;
    typedef typename MeshType::VertexPointer VertexPointer;
    typedef typename MeshType::VertexIterator VertexIterator;
    typedef typename MeshType::ScalarType ScalarType;
    typedef typename MeshType::FaceType FaceType;
    typedef typename MeshType::FacePointer FacePointer;
    typedef typename MeshType::FaceIterator FaceIterator;
    typedef typename MeshType::FaceContainer FaceContainer;
    typedef typename vcg::Box3<ScalarType> Box3Type;
    typedef typename vcg::face::VFIterator<FaceType> VFLocalIterator;

    class ScaleLaplacianInfo
    {
      public:
        CoordType PntSum;
        ScalarType LenSum;
    };

    // This is precisely what curvature flow does.
    // Curvature flow smoothes the surface by moving along the surface
    // normal n with a speed equal to the mean curvature
    void VertexCoordLaplacianCurvatureFlow(MeshType & /*m*/, int /*step*/, ScalarType /*delta*/)
    {
    }

    // Another Laplacian smoothing variant,
    // here we sum the baricenter of the faces incidents on each vertex weighting them with the angle

    static void VertexCoordLaplacianAngleWeighted(MeshType &m, int step, ScalarType delta)
    {
        ScaleLaplacianInfo lpz;
        lpz.PntSum = CoordType(0, 0, 0);
        lpz.LenSum = 0;
        SimpleTempData<typename MeshType::VertContainer, ScaleLaplacianInfo> TD(m.vert, lpz);
        FaceIterator fi;
        for (int i = 0; i < step; ++i)
        {
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = lpz;
            ScalarType a[3];
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                {
                    CoordType mp = ((*fi).V(0)->P() + (*fi).V(1)->P() + (*fi).V(2)->P()) / 3.0;
                    CoordType e0 = ((*fi).V(0)->P() - (*fi).V(1)->P()).Normalize();
                    CoordType e1 = ((*fi).V(1)->P() - (*fi).V(2)->P()).Normalize();
                    CoordType e2 = ((*fi).V(2)->P() - (*fi).V(0)->P()).Normalize();

                    a[0] = AngleN(-e0, e2);
                    a[1] = AngleN(-e1, e0);
                    a[2] = AngleN(-e2, e1);
                    //assert(fabs(M_PI -a[0] -a[1] -a[2])<0.0000001);

                    for (int j = 0; j < 3; ++j)
                    {
                        CoordType dir = (mp - (*fi).V(j)->P()).Normalize();
                        TD[(*fi).V(j)].PntSum += dir * a[j];
                        TD[(*fi).V(j)].LenSum += a[j]; // well, it should be named angleSum
                    }
                }
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].LenSum > 0)
                    (*vi).P() = (*vi).P() + (TD[*vi].PntSum / TD[*vi].LenSum) * delta;
        }
    };

    // Scale dependent laplacian smoothing [Fujiwara 95]
    // as described in
    // Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow
    // Mathieu Desbrun, Mark Meyer, Peter Schroeder, Alan H. Barr
    // SIGGRAPH 99
    // REQUIREMENTS: Border Flags.
    //
    // Note the delta parameter is in a absolute unit
    // to get stability it should be a small percentage of the shortest edge.

    static void VertexCoordScaleDependentLaplacian_Fujiwara(MeshType &m, int step, ScalarType delta, bool SmoothSelected = false)
    {
        SimpleTempData<typename MeshType::VertContainer, ScaleLaplacianInfo> TD(m.vert);
        ScaleLaplacianInfo lpz;
        lpz.PntSum = CoordType(0, 0, 0);
        lpz.LenSum = 0;
        FaceIterator fi;
        for (int i = 0; i < step; ++i)
        {
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = lpz;

            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if (!(*fi).IsB(j))
                        {
                            CoordType edge = (*fi).V1(j)->P() - (*fi).V(j)->P();
                            ScalarType len = Norm(edge);
                            edge /= len;
                            TD[(*fi).V(j)].PntSum += edge;
                            TD[(*fi).V1(j)].PntSum -= edge;
                            TD[(*fi).V(j)].LenSum += len;
                            TD[(*fi).V1(j)].LenSum += len;
                        }

            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        // se l'edge j e' di bordo si riazzera tutto e si riparte
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].PntSum = CoordType(0, 0, 0);
                            TD[(*fi).V1(j)].PntSum = CoordType(0, 0, 0);
                            TD[(*fi).V(j)].LenSum = 0;
                            TD[(*fi).V1(j)].LenSum = 0;
                        }

            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            CoordType edge = (*fi).V1(j)->P() - (*fi).V(j)->P();
                            ScalarType len = Norm(edge);
                            edge /= len;
                            TD[(*fi).V(j)].PntSum += edge;
                            TD[(*fi).V1(j)].PntSum -= edge;
                            TD[(*fi).V(j)].LenSum += len;
                            TD[(*fi).V1(j)].LenSum += len;
                        }
            // The fundamental part:
            // We move the new point of a quantity
            //
            //  L(M) = 1/Sum(edgelen) * Sum(Normalized edges)
            //

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].LenSum > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                        (*vi).P() = (*vi).P() + (TD[*vi].PntSum / TD[*vi].LenSum) * delta;
                }
        }
    };

    class LaplacianInfo
    {
      public:
        LaplacianInfo(const CoordType &_p, const int _n) : sum(_p), cnt(_n) {}
        LaplacianInfo() {}
        CoordType sum;
        ScalarType cnt;
    };

    // Classical Laplacian Smoothing. Each vertex can be moved onto the average of the adjacent vertices.
    // Can smooth only the selected vertices and weight the smoothing according to the quality
    // In the latter case 0 means that the vertex is not moved and 1 means that the vertex is moved onto the computed position.
    //
    // From the Taubin definition "A signal proc approach to fair surface design"
    // We define the discrete Laplacian of a discrete surface signal by weighted averages over the neighborhoods
    //          \delta xi  = \Sum wij (xj - xi) ;
    // where xj are the adjacent vertices of xi and wij is usually 1/n_adj
    //
    // This function simply accumulate over a TempData all the positions of the ajacent vertices
    //
    static void AccumulateLaplacianInfo(MeshType &m, SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> &TD, bool cotangentFlag = false)
    {
        float weight = 1.0f;

        //if we are applying to a tetrahedral mesh:
        ForEachTetra(m, [&](MeshType::TetraType &t) {
            for (int i = 0; i < 4; ++i)
                if (!t.IsB(i))
                {
                    VertexPointer v0, v1, v2;
                    v0 = t.V(Tetra::VofF(i, 0));
                    v1 = t.V(Tetra::VofF(i, 1));
                    v2 = t.V(Tetra::VofF(i, 2));

                    TD[v0].sum += v1->P() * weight;
                    TD[v0].sum += v2->P() * weight;
                    TD[v0].cnt += 2 * weight;

                    TD[v1].sum += v0->P() * weight;
                    TD[v1].sum += v2->P() * weight;
                    TD[v1].cnt += 2 * weight;

                    TD[v2].sum += v0->P() * weight;
                    TD[v2].sum += v1->P() * weight;
                    TD[v2].cnt += 2 * weight;
                }
        });

        ForEachTetra(m, [&](MeshType::TetraType &t) {
            for (int i = 0; i < 4; ++i)
                if (t.IsB(i))
                {
                    VertexPointer v0, v1, v2;
                    v0 = t.V(Tetra::VofF(i, 0));
                    v1 = t.V(Tetra::VofF(i, 1));
                    v2 = t.V(Tetra::VofF(i, 2));

                    TD[v0].sum = v0->P();
                    TD[v1].sum = v1->P();
                    TD[v2].sum = v2->P();

                    TD[v0].cnt = 1;
                    TD[v1].cnt = 1;
                    TD[v2].cnt = 1;
                }
        });

        ForEachTetra(m, [&](MeshType::TetraType &t) {
            for (int i = 0; i < 4; ++i)
                if (t.IsB(i))
                {
                    VertexPointer v0, v1, v2;
                    v0 = t.V(Tetra::VofF(i, 0));
                    v1 = t.V(Tetra::VofF(i, 1));
                    v2 = t.V(Tetra::VofF(i, 2));

                    TD[v0].sum += v1->P() * weight;
                    TD[v0].sum += v2->P() * weight;
                    TD[v0].cnt += 2 * weight;

                    TD[v1].sum += v0->P() * weight;
                    TD[v1].sum += v2->P() * weight;
                    TD[v1].cnt += 2 * weight;

                    TD[v2].sum += v0->P() * weight;
                    TD[v2].sum += v1->P() * weight;
                    TD[v2].cnt += 2 * weight;
                }
        });

        FaceIterator fi;
        for (fi = m.face.begin(); fi != m.face.end(); ++fi)
        {
            if (!(*fi).IsD())
                for (int j = 0; j < 3; ++j)
                    if (!(*fi).IsB(j))
                    {
                        if (cotangentFlag)
                        {
                            float angle = Angle(fi->P1(j) - fi->P2(j), fi->P0(j) - fi->P2(j));
                            weight = tan((M_PI * 0.5) - angle);
                        }

                        TD[(*fi).V0(j)].sum += (*fi).P1(j) * weight;
                        TD[(*fi).V1(j)].sum += (*fi).P0(j) * weight;
                        TD[(*fi).V0(j)].cnt += weight;
                        TD[(*fi).V1(j)].cnt += weight;
                    }
        }
        // si azzaera i dati per i vertici di bordo
        for (fi = m.face.begin(); fi != m.face.end(); ++fi)
        {
            if (!(*fi).IsD())
                for (int j = 0; j < 3; ++j)
                    if ((*fi).IsB(j))
                    {
                        TD[(*fi).V0(j)].sum = (*fi).P0(j);
                        TD[(*fi).V1(j)].sum = (*fi).P1(j);
                        TD[(*fi).V0(j)].cnt = 1;
                        TD[(*fi).V1(j)].cnt = 1;
                    }
        }

        // se l'edge j e' di bordo si deve mediare solo con gli adiacenti
        for (fi = m.face.begin(); fi != m.face.end(); ++fi)
        {
            if (!(*fi).IsD())
                for (int j = 0; j < 3; ++j)
                    if ((*fi).IsB(j))
                    {
                        TD[(*fi).V(j)].sum += (*fi).V1(j)->P();
                        TD[(*fi).V1(j)].sum += (*fi).V(j)->P();
                        ++TD[(*fi).V(j)].cnt;
                        ++TD[(*fi).V1(j)].cnt;
                    }
        }
    }

    static void VertexCoordLaplacian(MeshType &m, int step, bool SmoothSelected = false, bool cotangentWeight = false, vcg::CallBackPos *cb = 0)
    {
        LaplacianInfo lpz(CoordType(0, 0, 0), 0);
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        for (int i = 0; i < step; ++i)
        {
            if (cb)
                cb(100 * i / step, "Classic Laplacian Smoothing");
            TD.Init(lpz);
            AccumulateLaplacianInfo(m, TD, cotangentWeight);
            for (auto vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                        (*vi).P() = ((*vi).P() + TD[*vi].sum) / (TD[*vi].cnt + 1);
                }
        }
    }

    // Same of above but moves only the vertices that do not change FaceOrientation more that the given threshold
    static void VertexCoordPlanarLaplacian(MeshType &m, int step, float AngleThrRad = math::ToRad(1.0), bool SmoothSelected = false, vcg::CallBackPos *cb = 0)
    {
        LaplacianInfo lpz(CoordType(0, 0, 0), 0);
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        for (int i = 0; i < step; ++i)
        {
            if (cb)
                cb(100 * i / step, "Planar Laplacian Smoothing");
            TD.Init(lpz);
            AccumulateLaplacianInfo(m, TD);
            // First normalize the AccumulateLaplacianInfo
            for (auto vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                        TD[*vi].sum = ((*vi).P() + TD[*vi].sum) / (TD[*vi].cnt + 1);
                }

            for (auto fi = m.face.begin(); fi != m.face.end(); ++fi)
            {
                if (!(*fi).IsD())
                {
                    for (int j = 0; j < 3; ++j)
                    {
                        if (Angle(Normal(TD[(*fi).V0(j)].sum, (*fi).P1(j), (*fi).P2(j)),
                                  Normal((*fi).P0(j), (*fi).P1(j), (*fi).P2(j))) > AngleThrRad)
                            TD[(*fi).V0(j)].sum = (*fi).P0(j);
                    }
                }
            }
            for (auto fi = m.face.begin(); fi != m.face.end(); ++fi)
            {
                if (!(*fi).IsD())
                {
                    for (int j = 0; j < 3; ++j)
                    {
                        if (Angle(Normal(TD[(*fi).V0(j)].sum, TD[(*fi).V1(j)].sum, (*fi).P2(j)),
                                  Normal((*fi).P0(j), (*fi).P1(j), (*fi).P2(j))) > AngleThrRad)
                        {
                            TD[(*fi).V0(j)].sum = (*fi).P0(j);
                            TD[(*fi).V1(j)].sum = (*fi).P1(j);
                        }
                    }
                }
            }

            for (auto vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                    if (!SmoothSelected || (*vi).IsS())
                        (*vi).P() = TD[*vi].sum;
        } // end step
    }

    static void VertexCoordLaplacianBlend(MeshType &m, int step, float alpha, bool SmoothSelected = false)
    {
        VertexIterator vi;
        LaplacianInfo lpz(CoordType(0, 0, 0), 0);
        assert(alpha <= 1.0);
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert);

        for (int i = 0; i < step; ++i)
        {
            TD.Init(lpz);
            AccumulateLaplacianInfo(m, TD);
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                    {
                        CoordType Delta = TD[*vi].sum / TD[*vi].cnt - (*vi).P();
                        (*vi).P() = (*vi).P() + Delta * alpha;
                    }
                }
        }
    }

    /* a couple of notes about the lambda mu values
We assume that 0 < lambda , and mu  is a negative scale factor such that mu < - lambda.
Holds mu+lambda < 0 (e.g in absolute value mu is greater)

let kpb be the pass-band frequency, taubin says that:
            kpb = 1/lambda + 1/mu >0

Values of kpb from 0.01 to 0.1 produce good results according to the original paper.

kpb * mu - mu/lambda = 1
mu = 1/(kpb-1/lambda )

So if
* lambda == 0.5, kpb==0.1  -> mu = 1/(0.1 - 2)  = -0.526
* lambda == 0.5, kpb==0.01 -> mu = 1/(0.01 - 2) = -0.502
*/

    static void VertexCoordTaubin(MeshType &m, int step, float lambda, float mu, bool SmoothSelected = false, vcg::CallBackPos *cb = 0)
    {
        LaplacianInfo lpz(CoordType(0, 0, 0), 0);
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        VertexIterator vi;
        for (int i = 0; i < step; ++i)
        {
            if (cb)
                cb(100 * i / step, "Taubin Smoothing");
            TD.Init(lpz);
            AccumulateLaplacianInfo(m, TD);
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                    {
                        CoordType Delta = TD[*vi].sum / TD[*vi].cnt - (*vi).P();
                        (*vi).P() = (*vi).P() + Delta * lambda;
                    }
                }
            TD.Init(lpz);
            AccumulateLaplacianInfo(m, TD);
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    if (!SmoothSelected || (*vi).IsS())
                    {
                        CoordType Delta = TD[*vi].sum / TD[*vi].cnt - (*vi).P();
                        (*vi).P() = (*vi).P() + Delta * mu;
                    }
                }
        } // end for step
    }

    static void VertexQualityTaubin(MeshType &m, int step, float lambda, float mu, bool SmoothSelected = false, vcg::CallBackPos *cb = 0)
    {
        SimpleTempData<typename MeshType::VertContainer, ScalarType> OldQ(m.vert, 0);

        VertexIterator vi;
        for (int i = 0; i < step; ++i)
        {
            for (size_t i = 0; i < m.vert.size(); i++)
                OldQ[i] = m.vert[i].Q();

            if (cb)
                cb(100 * i / step, "Taubin Smoothing");

            VertexQualityLaplacian(m, 1, SmoothSelected);

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            {
                if (!SmoothSelected || (*vi).IsS())
                {
                    ScalarType Delta = (*vi).Q() - OldQ[(*vi)];
                    (*vi).Q() = OldQ[(*vi)] + Delta * lambda;
                }
            }

            for (size_t i = 0; i < m.vert.size(); i++)
                OldQ[i] = m.vert[i].Q();
            VertexQualityLaplacian(m, 1, SmoothSelected);

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            {
                if (!SmoothSelected || (*vi).IsS())
                {
                    ScalarType Delta = m.vert[i].Q() - OldQ[(*vi)];
                    (*vi).Q() = OldQ[(*vi)] + Delta * mu;
                }
            }
        } // end for step
    }

    static void VertexCoordLaplacianQuality(MeshType &m, int step, bool SmoothSelected = false)
    {
        LaplacianInfo lpz;
        lpz.sum = CoordType(0, 0, 0);
        lpz.cnt = 1;
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        for (int i = 0; i < step; ++i)
        {
            for (VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                    if (!SmoothSelected || (*vi).IsS())
                    {
                        float q = (*vi).Q();
                        (*vi).P() = (*vi).P() * q + (TD[*vi].sum / TD[*vi].cnt) * (1.0 - q);
                    }
        } // end for
    };

    /*
  Improved Laplacian Smoothing of Noisy Surface Meshes
  J. Vollmer, R. Mencl, and H. M<EFBFBD>ller
  EUROGRAPHICS Volume 18 (1999), Number 3
*/

    class HCSmoothInfo
    {
      public:
        CoordType dif;
        CoordType sum;
        int cnt;
    };

    static void VertexCoordLaplacianHC(MeshType &m, int step, bool SmoothSelected = false)
    {
        ScalarType beta = 0.5;
        HCSmoothInfo lpz;
        lpz.sum = CoordType(0, 0, 0);
        lpz.dif = CoordType(0, 0, 0);
        lpz.cnt = 0;
        for (int i = 0; i < step; ++i)
        {
            SimpleTempData<typename MeshType::VertContainer, HCSmoothInfo> TD(m.vert, lpz);
            // First Loop compute the laplacian
            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                {
                    for (int j = 0; j < 3; ++j)
                    {
                        TD[(*fi).V(j)].sum += (*fi).V1(j)->P();
                        TD[(*fi).V1(j)].sum += (*fi).V(j)->P();
                        ++TD[(*fi).V(j)].cnt;
                        ++TD[(*fi).V1(j)].cnt;
                        // se l'edge j e' di bordo si deve sommare due volte
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->P();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->P();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }
                    }
                }
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD())
                    TD[*vi].sum /= (float)TD[*vi].cnt;

            // Second Loop compute average difference
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                {
                    for (int j = 0; j < 3; ++j)
                    {
                        TD[(*fi).V(j)].dif += TD[(*fi).V1(j)].sum - (*fi).V1(j)->P();
                        TD[(*fi).V1(j)].dif += TD[(*fi).V(j)].sum - (*fi).V(j)->P();
                        // se l'edge j e' di bordo si deve sommare due volte
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].dif += TD[(*fi).V1(j)].sum - (*fi).V1(j)->P();
                            TD[(*fi).V1(j)].dif += TD[(*fi).V(j)].sum - (*fi).V(j)->P();
                        }
                    }
                }

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            {
                TD[*vi].dif /= (float)TD[*vi].cnt;
                if (!SmoothSelected || (*vi).IsS())
                    (*vi).P() = TD[*vi].sum - (TD[*vi].sum - (*vi).P()) * beta + (TD[*vi].dif) * (1.f - beta);
            }
        } // end for step
    };

    // Laplacian smooth of the quality.

    class ColorSmoothInfo
    {
      public:
        unsigned int r;
        unsigned int g;
        unsigned int b;
        unsigned int a;
        int cnt;
    };

    static void VertexColorLaplacian(MeshType &m, int step, bool SmoothSelected = false, vcg::CallBackPos *cb = 0)
    {
        ColorSmoothInfo csi;
        csi.r = 0;
        csi.g = 0;
        csi.b = 0;
        csi.cnt = 0;
        SimpleTempData<typename MeshType::VertContainer, ColorSmoothInfo> TD(m.vert, csi);

        for (int i = 0; i < step; ++i)
        {
            if (cb)
                cb(100 * i / step, "Vertex Color Laplacian Smoothing");
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = csi;

            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if (!(*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].r += (*fi).V1(j)->C()[0];
                            TD[(*fi).V(j)].g += (*fi).V1(j)->C()[1];
                            TD[(*fi).V(j)].b += (*fi).V1(j)->C()[2];
                            TD[(*fi).V(j)].a += (*fi).V1(j)->C()[3];

                            TD[(*fi).V1(j)].r += (*fi).V(j)->C()[0];
                            TD[(*fi).V1(j)].g += (*fi).V(j)->C()[1];
                            TD[(*fi).V1(j)].b += (*fi).V(j)->C()[2];
                            TD[(*fi).V1(j)].a += (*fi).V(j)->C()[3];

                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            // si azzaera i dati per i vertici di bordo
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)] = csi;
                            TD[(*fi).V1(j)] = csi;
                        }

            // se l'edge j e' di bordo si deve mediare solo con gli adiacenti
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].r += (*fi).V1(j)->C()[0];
                            TD[(*fi).V(j)].g += (*fi).V1(j)->C()[1];
                            TD[(*fi).V(j)].b += (*fi).V1(j)->C()[2];
                            TD[(*fi).V(j)].a += (*fi).V1(j)->C()[3];

                            TD[(*fi).V1(j)].r += (*fi).V(j)->C()[0];
                            TD[(*fi).V1(j)].g += (*fi).V(j)->C()[1];
                            TD[(*fi).V1(j)].b += (*fi).V(j)->C()[2];
                            TD[(*fi).V1(j)].a += (*fi).V(j)->C()[3];

                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                    if (!SmoothSelected || (*vi).IsS())
                    {
                        (*vi).C()[0] = (unsigned int)ceil((double)(TD[*vi].r / TD[*vi].cnt));
                        (*vi).C()[1] = (unsigned int)ceil((double)(TD[*vi].g / TD[*vi].cnt));
                        (*vi).C()[2] = (unsigned int)ceil((double)(TD[*vi].b / TD[*vi].cnt));
                        (*vi).C()[3] = (unsigned int)ceil((double)(TD[*vi].a / TD[*vi].cnt));
                    }
        } // end for step
    };

    static void FaceColorLaplacian(MeshType &m, int step, bool SmoothSelected = false, vcg::CallBackPos *cb = 0)
    {
        ColorSmoothInfo csi;
        csi.r = 0;
        csi.g = 0;
        csi.b = 0;
        csi.cnt = 0;
        SimpleTempData<typename MeshType::FaceContainer, ColorSmoothInfo> TD(m.face, csi);

        for (int i = 0; i < step; ++i)
        {
            if (cb)
                cb(100 * i / step, "Face Color Laplacian Smoothing");
            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                TD[*fi] = csi;

            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
            {
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if (!(*fi).IsB(j))
                        {
                            TD[*fi].r += (*fi).FFp(j)->C()[0];
                            TD[*fi].g += (*fi).FFp(j)->C()[1];
                            TD[*fi].b += (*fi).FFp(j)->C()[2];
                            TD[*fi].a += (*fi).FFp(j)->C()[3];
                            ++TD[*fi].cnt;
                        }
            }
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD() && TD[*fi].cnt > 0)
                    if (!SmoothSelected || (*fi).IsS())
                    {
                        (*fi).C()[0] = (unsigned int)ceil((float)(TD[*fi].r / TD[*fi].cnt));
                        (*fi).C()[1] = (unsigned int)ceil((float)(TD[*fi].g / TD[*fi].cnt));
                        (*fi).C()[2] = (unsigned int)ceil((float)(TD[*fi].b / TD[*fi].cnt));
                        (*fi).C()[3] = (unsigned int)ceil((float)(TD[*fi].a / TD[*fi].cnt));
                    }
        } // end for step
    };

    // Laplacian smooth of the quality.

    class QualitySmoothInfo
    {
      public:
        ScalarType sum;
        int cnt;
    };

    static void PointCloudQualityAverage(MeshType &m, int neighbourSize = 8, int iter = 1)
    {
        tri::RequireCompactness(m);
        VertexConstDataWrapper<MeshType> ww(m);
        KdTree<ScalarType> kt(ww);
        typename KdTree<ScalarType>::PriorityQueue pq;
        for (int k = 0; k < iter; ++k)
        {
            std::vector<ScalarType> newQVec(m.vn);
            for (int i = 0; i < m.vn; ++i)
            {
                kt.doQueryK(m.vert[i].P(), neighbourSize, pq);
                float qAvg = 0;
                for (int j = 0; j < pq.getNofElements(); ++j)
                    qAvg += m.vert[pq.getIndex(j)].Q();
                newQVec[i] = qAvg / float(pq.getNofElements());
            }

            for (int i = 0; i < m.vn; ++i)
                m.vert[i].Q() = newQVec[i];
        }
    }

    static void PointCloudQualityMedian(MeshType &m, int medianSize = 8)
    {
        tri::RequireCompactness(m);
        VertexConstDataWrapper<MeshType> ww(m);
        KdTree<ScalarType> kt(ww);
        typename KdTree<ScalarType>::PriorityQueue pq;
        std::vector<ScalarType> newQVec(m.vn);
        for (int i = 0; i < m.vn; ++i)
        {
            kt.doQueryK(m.vert[i].P(), medianSize, pq);
            std::vector<ScalarType> qVec(pq.getNofElements());
            for (int j = 0; j < pq.getNofElements(); ++j)
                qVec[j] = m.vert[pq.getIndex(j)].Q();
            std::sort(qVec.begin(), qVec.end());
            newQVec[i] = qVec[qVec.size() / 2];
        }

        for (int i = 0; i < m.vn; ++i)
            m.vert[i].Q() = newQVec[i];
    }

    static void VertexQualityLaplacian(MeshType &m, int step = 1, bool SmoothSelected = false)
    {
        QualitySmoothInfo lpz;
        lpz.sum = 0;
        lpz.cnt = 0;
        SimpleTempData<typename MeshType::VertContainer, QualitySmoothInfo> TD(m.vert, lpz);
        //TD.Start(lpz);
        for (int i = 0; i < step; ++i)
        {
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = lpz;

            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < (*fi).VN(); ++j)
                        if (!(*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->Q();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->Q();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            // si azzaera i dati per i vertici di bordo
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < (*fi).VN(); ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)] = lpz;
                            TD[(*fi).V1(j)] = lpz;
                        }

            // se l'edge j e' di bordo si deve mediare solo con gli adiacenti
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < (*fi).VN(); ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->Q();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->Q();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            //VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                    if (!SmoothSelected || (*vi).IsS())
                        (*vi).Q() = TD[*vi].sum / TD[*vi].cnt;
        }

        //TD.Stop();
    };

    static void VertexNormalLaplacian(MeshType &m, int step, bool SmoothSelected = false)
    {
        LaplacianInfo lpz;
        lpz.sum = CoordType(0, 0, 0);
        lpz.cnt = 0;
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        for (int i = 0; i < step; ++i)
        {
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = lpz;

            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if (!(*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->N();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->N();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            // si azzaera i dati per i vertici di bordo
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)] = lpz;
                            TD[(*fi).V1(j)] = lpz;
                        }

            // se l'edge j e' di bordo si deve mediare solo con gli adiacenti
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->N();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->N();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            //VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                    if (!SmoothSelected || (*vi).IsS())
                        (*vi).N() = TD[*vi].sum / TD[*vi].cnt;
        }
    };

    // Smooth solo lungo la direzione di vista
    // alpha e' compreso fra 0(no smoot) e 1 (tutto smoot)
    // Nota che se smootare il bordo puo far fare bandierine.
    static void VertexCoordViewDepth(MeshType &m,
                                     const CoordType &viewpoint,
                                     const ScalarType alpha,
                                     int step, bool SmoothBorder = false)
    {
        LaplacianInfo lpz;
        lpz.sum = CoordType(0, 0, 0);
        lpz.cnt = 0;
        SimpleTempData<typename MeshType::VertContainer, LaplacianInfo> TD(m.vert, lpz);
        for (int i = 0; i < step; ++i)
        {
            VertexIterator vi;
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                TD[*vi] = lpz;

            FaceIterator fi;
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if (!(*fi).IsB(j))
                        {
                            TD[(*fi).V(j)].sum += (*fi).V1(j)->cP();
                            TD[(*fi).V1(j)].sum += (*fi).V(j)->cP();
                            ++TD[(*fi).V(j)].cnt;
                            ++TD[(*fi).V1(j)].cnt;
                        }

            // si azzaera i dati per i vertici di bordo
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                    for (int j = 0; j < 3; ++j)
                        if ((*fi).IsB(j))
                        {
                            TD[(*fi).V(j)] = lpz;
                            TD[(*fi).V1(j)] = lpz;
                        }

            // se l'edge j e' di bordo si deve mediare solo con gli adiacenti
            if (SmoothBorder)
                for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                    if (!(*fi).IsD())
                        for (int j = 0; j < 3; ++j)
                            if ((*fi).IsB(j))
                            {
                                TD[(*fi).V(j)].sum += (*fi).V1(j)->cP();
                                TD[(*fi).V1(j)].sum += (*fi).V(j)->cP();
                                ++TD[(*fi).V(j)].cnt;
                                ++TD[(*fi).V1(j)].cnt;
                            }

            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if (!(*vi).IsD() && TD[*vi].cnt > 0)
                {
                    CoordType np = TD[*vi].sum / TD[*vi].cnt;
                    CoordType d = (*vi).cP() - viewpoint;
                    d.Normalize();
                    ScalarType s = d.dot(np - (*vi).cP());
                    (*vi).P() += d * (s * alpha);
                }
        }
    }

    /****************************************************************************************************************/
    /****************************************************************************************************************/
    // Paso Double Smoothing
    //  The proposed
    // approach is a two step method where  in the first step the face normals
    // are adjusted and then, in a second phase, the vertex positions are updated.
    // Ref:
    // A. Belyaev and Y. Ohtake, A Comparison of Mesh Smoothing Methods, Proc. Israel-Korea Bi-Nat"l Conf. Geometric Modeling and Computer Graphics, pp. 83-87, 2003.

    /****************************************************************************************************************/
    /****************************************************************************************************************/
    // Classi di info

    class PDVertInfo
    {
      public:
        CoordType np;
    };

    class PDFaceInfo
    {
      public:
        PDFaceInfo() {}
        PDFaceInfo(const CoordType &_m) : m(_m) {}
        CoordType m;
    };
    /***************************************************************************/
    // Paso Doble Step 1 compute the smoothed normals
    /***************************************************************************/
    // Requirements:
    // VF Topology
    // Normalized Face Normals
    //
    // This is the Normal Smoothing approach of Shen and Berner
    // Fuzzy Vector Median-Based Surface Smoothing TVCG 2004

    void FaceNormalFuzzyVectorSB(MeshType &m,
                                 SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> &TD,
                                 ScalarType sigma)
    {
        int i;

        FaceIterator fi;

        for (fi = m.face.begin(); fi != m.face.end(); ++fi)
        {
            CoordType bc = (*fi).Barycenter();
            // 1) Clear all the visited flag of faces that are vertex-adjacent to fi
            for (i = 0; i < 3; ++i)
            {
                vcg::face::VFIterator<FaceType> ep(&*fi, i);
                while (!ep.End())
                {
                    ep.f->ClearV();
                    ++ep;
                }
            }

            // 1) Effectively average the normals weighting them with
            (*fi).SetV();
            CoordType mm = CoordType(0, 0, 0);
            for (i = 0; i < 3; ++i)
            {
                vcg::face::VFIterator<FaceType> ep(&*fi, i);
                while (!ep.End())
                {
                    if (!(*ep.f).IsV())
                    {
                        if (sigma > 0)
                        {
                            ScalarType dd = SquaredDistance(ep.f->Barycenter(), bc);
                            ScalarType ang = AngleN(ep.f->N(), (*fi).N());
                            mm += ep.f->N() * exp((-sigma) * ang * ang / dd);
                        }
                        else
                            mm += ep.f->N();
                        (*ep.f).SetV();
                    }
                    ++ep;
                }
            }
            mm.Normalize();
            TD[*fi].m = mm;
        }
    }

    // Replace the normal of the face with the average of normals of the vertex adijacent faces.
    // Normals are weighted with face area.
    // It assumes that:
    // VF adjacency is present.

    static void FaceNormalLaplacianVF(MeshType &m)
    {
        tri::RequireVFAdjacency(m);
        PDFaceInfo lpzf(CoordType(0, 0, 0));
        SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> TDF(m.face, lpzf);

        tri::UpdateNormal<MeshType>::NormalizePerFaceByArea(m);

        for (FaceIterator fi = m.face.begin(); fi != m.face.end(); ++fi)
            if (!(*fi).IsD())
            {
                // 1) Clear all the visited flag of faces that are vertex-adjacent to fi
                for (int i = 0; i < 3; ++i)
                {
                    VFLocalIterator ep(&*fi, i);
                    for (; !ep.End(); ++ep)
                        ep.f->ClearV();
                }

                // 2) Effectively average the normals
                CoordType normalSum = (*fi).N();

                for (int i = 0; i < 3; ++i)
                {
                    VFLocalIterator ep(&*fi, i);
                    for (; !ep.End(); ++ep)
                    {
                        if (!(*ep.f).IsV())
                        {
                            normalSum += ep.f->N();
                            (*ep.f).SetV();
                        }
                    }
                }
                normalSum.Normalize();
                TDF[*fi].m = normalSum;
            }
        for (FaceIterator fi = m.face.begin(); fi != m.face.end(); ++fi)
            (*fi).N() = TDF[*fi].m;

        tri::UpdateNormal<MeshType>::NormalizePerFace(m);
    }

    // Replace the normal of the face with the average of normals of the face adijacent faces.
    // Normals are weighted with face area.
    // It assumes that:
    // Normals are normalized:
    // FF adjacency is present.

    static void FaceNormalLaplacianFF(MeshType &m, int step = 1, bool SmoothSelected = false)
    {
        PDFaceInfo lpzf(CoordType(0, 0, 0));
        SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> TDF(m.face, lpzf);
        tri::RequireFFAdjacency(m);

        FaceIterator fi;
        tri::UpdateNormal<MeshType>::NormalizePerFaceByArea(m);
        for (int iStep = 0; iStep < step; ++iStep)
        {
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!(*fi).IsD())
                {
                    CoordType normalSum = (*fi).N();

                    for (int i = 0; i < 3; ++i)
                        normalSum += (*fi).FFp(i)->N();

                    TDF[*fi].m = normalSum;
                }
            for (fi = m.face.begin(); fi != m.face.end(); ++fi)
                if (!SmoothSelected || (*fi).IsS())
                    (*fi).N() = TDF[*fi].m;

            tri::UpdateNormal<MeshType>::NormalizePerFace(m);
        }
    }

    /***************************************************************************/
    // Paso Doble Step 1 compute the smoothed normals
    /***************************************************************************/
    // Requirements:
    // VF Topology
    // Normalized Face Normals
    //
    // This is the Normal Smoothing approach based on a angle thresholded weighting
    // sigma is in the 0 .. 1 range, it represent the cosine of a threshold angle.
    // sigma == 0 All the normals are averaged
    // sigma == 1 Nothing is averaged.
    // Only within the specified range are averaged toghether. The averagin is weighted with the

    static void FaceNormalAngleThreshold(MeshType &m,
                                         SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> &TD,
                                         ScalarType sigma)
    {
        for (FaceIterator fi = m.face.begin(); fi != m.face.end(); ++fi)
        {
            // 1) Clear all the visited flag of faces that are vertex-adjacent to fi
            for (int i = 0; i < 3; ++i)
            {
                VFLocalIterator ep(&*fi, i);
                for (; !ep.End(); ++ep)
                    ep.f->ClearV();
            }

            // 1) Effectively average the normals weighting them with the squared difference of the angle similarity
            // sigma is the cosine of a threshold angle. sigma \in 0..1
            // sigma == 0 All the normals are averaged
            // sigma == 1 Nothing is averaged.
            // The averaging is weighted with the difference between normals. more similar the normal more important they are.

            CoordType normalSum = CoordType(0, 0, 0);
            for (int i = 0; i < 3; ++i)
            {
                VFLocalIterator ep(&*fi, i);
                for (; !ep.End(); ++ep)
                {
                    if (!(*ep.f).IsV())
                    {
                        ScalarType cosang = ep.f->N().dot((*fi).N());
                        // Note that if two faces form an angle larger than 90 deg, their contribution should be very very small.
                        // Without this clamping
                        cosang = math::Clamp(cosang, ScalarType(0.0001), ScalarType(1.f));
                        if (cosang >= sigma)
                        {
                            ScalarType w = cosang - sigma;
                            normalSum += ep.f->N() * (w * w); // similar normals have a cosang very close to 1 so cosang - sigma is maximized
                        }
                        (*ep.f).SetV();
                    }
                }
            }
            normalSum.Normalize();
            TD[*fi].m = normalSum;
        }

        for (FaceIterator fi = m.face.begin(); fi != m.face.end(); ++fi)
            (*fi).N() = TD[*fi].m;
    }

    /****************************************************************************************************************/
    // Restituisce il gradiente dell'area del triangolo nel punto p.
    // Nota che dovrebbe essere sempre un vettore che giace nel piano del triangolo e perpendicolare al lato opposto al vertice p.
    // Ottimizzato con Maple e poi pesantemente a mano.

    static CoordType TriAreaGradient(CoordType &p, CoordType &p0, CoordType &p1)
    {
        CoordType dd = p1 - p0;
        CoordType d0 = p - p0;
        CoordType d1 = p - p1;
        CoordType grad;

        ScalarType t16 = d0[1] * d1[2] - d0[2] * d1[1];
        ScalarType t5 = -d0[2] * d1[0] + d0[0] * d1[2];
        ScalarType t4 = -d0[0] * d1[1] + d0[1] * d1[0];

        ScalarType delta = sqrtf(t4 * t4 + t5 * t5 + t16 * t16);

        grad[0] = (t5 * (-dd[2]) + t4 * (dd[1])) / delta;
        grad[1] = (t16 * (-dd[2]) + t4 * (-dd[0])) / delta;
        grad[2] = (t16 * (dd[1]) + t5 * (dd[0])) / delta;

        return grad;
    }

    template <class ScalarType>
    static CoordType CrossProdGradient(CoordType &p, CoordType &p0, CoordType &p1, CoordType &m)
    {
        CoordType grad;
        CoordType p00 = p0 - p;
        CoordType p01 = p1 - p;
        grad[0] = (-p00[2] + p01[2]) * m[1] + (-p01[1] + p00[1]) * m[2];
        grad[1] = (-p01[2] + p00[2]) * m[0] + (-p00[0] + p01[0]) * m[2];
        grad[2] = (-p00[1] + p01[1]) * m[0] + (-p01[0] + p00[0]) * m[1];

        return grad;
    }

    /*
Deve Calcolare il gradiente di
E(p) = A(p,p0,p1) (n - m)^2 =
A(...) (2-2nm)   =
(p0-p)^(p1-p)
2A - 2A * ------------- m  =
2A

2A  -  2 (p0-p)^(p1-p) * m
*/

    static CoordType FaceErrorGrad(CoordType &p, CoordType &p0, CoordType &p1, CoordType &m)
    {
        return TriAreaGradient(p, p0, p1) * 2.0f - CrossProdGradient(p, p0, p1, m) * 2.0f;
    }
    /***************************************************************************/
    // Paso Doble Step 2 Fitta la mesh a un dato insieme di normali
    /***************************************************************************/

    static void FitMesh(MeshType &m,
                        SimpleTempData<typename MeshType::VertContainer, PDVertInfo> &TDV,
                        SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> &TDF,
                        float lambda)
    {
        vcg::face::VFIterator<FaceType> ep;
        VertexIterator vi;
        for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
        {
            CoordType ErrGrad = CoordType(0, 0, 0);

            ep.f = (*vi).VFp();
            ep.z = (*vi).VFi();
            while (!ep.End())
            {
                ErrGrad += FaceErrorGrad(ep.f->V(ep.z)->P(), ep.f->V1(ep.z)->P(), ep.f->V2(ep.z)->P(), TDF[ep.f].m);
                ++ep;
            }
            TDV[*vi].np = (*vi).P() - ErrGrad * (ScalarType)lambda;
        }

        for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            (*vi).P() = TDV[*vi].np;
    }
    /****************************************************************************************************************/

    static void FastFitMesh(MeshType &m,
                            SimpleTempData<typename MeshType::VertContainer, PDVertInfo> &TDV,
                            //SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> &TDF,
                            bool OnlySelected = false)
    {
        //vcg::face::Pos<FaceType> ep;
        vcg::face::VFIterator<FaceType> ep;
        VertexIterator vi;

        for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
        {
            CoordType Sum(0, 0, 0);
            ScalarType cnt = 0;
            VFLocalIterator ep(&*vi);
            for (; !ep.End(); ++ep)
            {
                CoordType bc = Barycenter<FaceType>(*ep.F());
                Sum += ep.F()->N() * (ep.F()->N().dot(bc - (*vi).P()));
                ++cnt;
            }
            TDV[*vi].np = (*vi).P() + Sum * (1.0 / cnt);
        }

        if (OnlySelected)
        {
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                if ((*vi).IsS())
                    (*vi).P() = TDV[*vi].np;
        }
        else
        {
            for (vi = m.vert.begin(); vi != m.vert.end(); ++vi)
                (*vi).P() = TDV[*vi].np;
        }
    }

    // The sigma parameter affect the normal smoothing step

    static void VertexCoordPasoDoble(MeshType &m, int NormalSmoothStep, typename MeshType::ScalarType Sigma = 0, int FitStep = 50, bool SmoothSelected = false)
    {
        tri::RequireCompactness(m);
        tri::RequireVFAdjacency(m);
        PDVertInfo lpzv;
        lpzv.np = CoordType(0, 0, 0);
        PDFaceInfo lpzf(CoordType(0, 0, 0));

        assert(HasPerVertexVFAdjacency(m) && HasPerFaceVFAdjacency(m));
        SimpleTempData<typename MeshType::VertContainer, PDVertInfo> TDV(m.vert, lpzv);
        SimpleTempData<typename MeshType::FaceContainer, PDFaceInfo> TDF(m.face, lpzf);

        for (int j = 0; j < NormalSmoothStep; ++j)
            FaceNormalAngleThreshold(m, TDF, Sigma);

        for (int j = 0; j < FitStep; ++j)
            FastFitMesh(m, TDV, SmoothSelected);
    }

    static void VertexNormalPointCloud(MeshType &m, int neighborNum, int iterNum, KdTree<ScalarType> *tp = 0)
    {
        SimpleTempData<typename MeshType::VertContainer, CoordType> TD(m.vert, CoordType(0, 0, 0));
        VertexConstDataWrapper<MeshType> ww(m);
        KdTree<ScalarType> *tree = 0;
        if (tp == 0)
            tree = new KdTree<ScalarType>(ww);
        else
            tree = tp;
        typename KdTree<ScalarType>::PriorityQueue nq;

        //  tree->setMaxNofNeighbors(neighborNum);
        for (int ii = 0; ii < iterNum; ++ii)
        {
            for (VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            {
                tree->doQueryK(vi->cP(), neighborNum, nq);
                int neighbours = nq.getNofElements();
                for (int i = 0; i < neighbours; i++)
                {
                    int neightId = nq.getIndex(i);
                    if (m.vert[neightId].cN() * vi->cN() > 0)
                        TD[vi] += m.vert[neightId].cN();
                    else
                        TD[vi] -= m.vert[neightId].cN();
                }
            }
            for (VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
            {
                vi->N() = TD[vi];
                TD[vi] = CoordType(0, 0, 0);
            }
            tri::UpdateNormal<MeshType>::NormalizePerVertex(m);
        }

        if (tp == 0)
            delete tree;
    }

    //! Laplacian smoothing with a reprojection on a target surface.
    // grid must be a spatial index that contains all triangular faces of the target mesh gridmesh
    template <class GRID, class MeshTypeTri>
    static void VertexCoordLaplacianReproject(MeshType &m, GRID &grid, MeshTypeTri &gridmesh)
    {
        for (VertexIterator vi = m.vert.begin(); vi != m.vert.end(); ++vi)
        {
            if (!(*vi).IsD())
                VertexCoordLaplacianReproject(m, grid, gridmesh, &*vi);
        }
    }

    template <class GRID, class MeshTypeTri>
    static void VertexCoordLaplacianReproject(MeshType &m, GRID &grid, MeshTypeTri &gridmesh, typename MeshType::VertexType *vp)
    {
        assert(MeshType::HEdgeType::HasHVAdjacency());

        // compute barycenter
        typedef std::vector<VertexPointer> VertexSet;
        VertexSet verts;
        verts = HalfEdgeTopology<MeshType>::getVertices(vp);

        typename MeshType::CoordType ct(0, 0, 0);
        for (typename VertexSet::iterator it = verts.begin(); it != verts.end(); ++it)
        {
            ct += (*it)->P();
        }
        ct /= verts.size();

        // move vertex
        vp->P() = ct;

        vector<FacePointer> faces2 = HalfEdgeTopology<MeshType>::get_incident_faces(vp);

        // estimate normal
        typename MeshType::CoordType avgn(0, 0, 0);

        for (unsigned int i = 0; i < faces2.size(); i++)
            if (faces2[i])
            {
                vector<VertexPointer> vertices = HalfEdgeTopology<MeshType>::getVertices(faces2[i]);

                assert(vertices.size() == 4);

                avgn += vcg::Normal<typename MeshType::CoordType>(vertices[0]->cP(), vertices[1]->cP(), vertices[2]->cP());
                avgn += vcg::Normal<typename MeshType::CoordType>(vertices[2]->cP(), vertices[3]->cP(), vertices[0]->cP());
            }
        avgn.Normalize();

        // reproject
        ScalarType diag = m.bbox.Diag();
        typename MeshType::CoordType raydir = avgn;
        Ray3<typename MeshType::ScalarType> ray;

        ray.SetOrigin(vp->P());
        ScalarType t;
        typename MeshTypeTri::FaceType *f = 0;
        typename MeshTypeTri::FaceType *fr = 0;

        vector<typename MeshTypeTri::CoordType> closests;
        vector<typename MeshTypeTri::ScalarType> minDists;
        vector<typename MeshTypeTri::FaceType *> faces;
        ray.SetDirection(-raydir);
        f = vcg::tri::DoRay<MeshTypeTri, GRID>(gridmesh, grid, ray, diag / 4.0, t);

        if (f)
        {
            closests.push_back(ray.Origin() + ray.Direction() * t);
            minDists.push_back(fabs(t));
            faces.push_back(f);
        }
        ray.SetDirection(raydir);
        fr = vcg::tri::DoRay<MeshTypeTri, GRID>(gridmesh, grid, ray, diag / 4.0, t);
        if (fr)
        {
            closests.push_back(ray.Origin() + ray.Direction() * t);
            minDists.push_back(fabs(t));
            faces.push_back(fr);
        }

        if (fr)
            if (fr->N() * raydir < 0)
                fr = 0; // discard: inverse normal;
        typename MeshType::CoordType newPos;
        if (minDists.size() == 0)
        {
            newPos = vp->P();
            f = 0;
        }
        else
        {
            int minI = std::min_element(minDists.begin(), minDists.end()) - minDists.begin();
            newPos = closests[minI];
            f = faces[minI];
        }

        if (f)
            vp->P() = newPos;
    }

}; //end Smooth class

} // End namespace tri
} // End namespace vcg

#endif //  VCG_SMOOTH